Computer Software

MATLAB

Julia Programming Language

Maxima

Mathematica

• Some elementary topics in mathematics
  • Lecture notes created by MATLAB Live Script (*.mlx)
    • The Nyquist-Shannon Sampling Theorem and Aliasing
    • X-ray transform on \(\mathbb{R}^2\)   material grayscale image
    • Differentiability of One-Variable Functions
    • Convergence and Divergence of Taylor Series
    • Riemann Integrability of One-Variable Functions
    • SVD and Best Low Rank Approximation   material RGB image
    • Ordinary Differential Equations
    • Numerical Solutions to ODEs
    • Convergence and Divergence of Fourier Series
      Slideshow   Decomposition of Piecewise Smooth Functions
  • Lecture notes created by Julia Programming Language with Pluto.jl
    • Numerical Solutions to ODEs (including the SIR model)
      Copy URL https://rencho.me/julia/pluto/pluto_numerical_ode.jl
    • Convergence and Divergence of Fourier Series
      Copy URL https://rencho.me/julia/pluto/pluto_fourier.jl
    • The Nyquist-Shannon Sampling Theorem and Aliasing
      Copy URL https://rencho.me/julia/pluto/pluto_aliasing.jl
    • X-ray transform on \(\mathbb{R}^2\)
      Copy URL https://rencho.me/julia/pluto/pluto_ct.jl
      package: xray.jl   material grayscale image
    • Differentiability
      Copy URL https://rencho.me/julia/pluto/pluto_differentiability.jl
    • Riemann integrability
      Copy URL https://rencho.me/julia/pluto/pluto_riemann.jl
    • The Haar wavelet and multiresolution decomposition
      Copy URL https://rencho.me/julia/pluto/pluto_wavelet.jl
      material RGB image
    • Low rank approximation of an RGB image
      Copy URL https://rencho.me/julia/pluto/pluto_svd.jl
      material RGB image
    • Taylor series
      Copy URL https://rencho.me/julia/pluto/pluto_taylor.jl
  • Lecture notes created by Julia Programming Language with IJulia.jl and Jupyter Notebook (*.ipynb)
    • Taylor series
    • Low rank approximation of an RGB image   material RGB image
    • What is data?
  • Lecture notes created by LaTeX (*.pdf)
    • Slides: Taylor's Theorem and Taylor Expansion, video 1, video 2
    • One-variable complex analysis and two-dimensional incompressible irrotational steady flow
    • A note on the Bargmann transform on the Euclidean space
    • X-ray transform on the plane and reconstruction formula: The principles of computed tomography
    • d'Alembert's formula for the one-dimensional wave equation
    • A first order system including the one-dimensional wave equation
  
  
• MATLAB is a commercial numerical analysis software widely used in the world except for Japan.
  • YouTube Playlist
  • Number Theory
    • Green-Tao Theorem
    • The abc Theorem
    • Prime Number Theorem
  • Calculus
    • Convergence and divergence of Taylor expansions
    • Gibbs phenomenon of Fourier series
  • CSV
    • COVID-19
  • X-ray transform and reconstruction
    • square, double squares, half plane regular triangle, stomach
  • 2D wave front set (in progress)
    • simple planar shapes
  • Wavelet filters
    • Wavelet filters and their Fourier transforms
  • Multiresolution decomposition of audio signals
    • the Haar wavelet, the Daubechies 2 wavelet
  • Discrete Fourier transform and spectrogram of audio signals
    • human voice, computer sound
  • Low rank approximation of grayscale images based on SVD
    • Marina Bay Area, Singapore, Eckhart, U of Chicago, flowers, shaved ice
  • Multiresolution decomposition of grayscale images
    • flowers, Marina Bay Area, Eckhart, U of Chicago
  • Multichannel images
    • Data of an RGB image is a triplet of matrices,
    • shaved ice, cactuses, fried hokkien mee
  • Solving ODEs and systems
    • fluid flow, SIR model, SIR model for COVID-19.
  • Solving PDEs with FFT
    • examples (heat, Schrödinger, KdV, Burgers).
  
  
• Julia Programming Language is a well-known free programing language. Here are some outputs.
  • YouTube Playlist
  • Number Theory
    • Green-Tao Theorem
    • The abc Theorem
    • Prime Number Theorem
  • CSV
    • COVID-19
  • X-ray transform and reconstruction
    • square, stomach, implants
  • Low rank approximation of grayscale images based on SVD
    • Marina Bay Area, Singapore, Eckhart, U of Chicago, flowers, shaved ice
  • Wavelet filters
    • Wavelet filters and their Fourier transforms
  • Fourier analysis and multiresolution analysis of audio signals
    • human voice, computer sound
  • Multiresolution decomposition of grayscale images
    • shaved ice, flowers.
  • Multichannel images
    • Data of an RGB image is a triplet of matrices,
    • flowers, mp4: low rank approximation, mp4: multiresolution decomposition, LRA at YouTube, MRD at YouTube,
    • fried hokkien mee.
  • Solving ODEs and systems
    • SIR model for COVID-19.
  • Solving PDEs with FFT
    • examples (heat, Schrödinger, KdV, Burgers).
  
  
• Maxima is a well-known free computer algebra system. Here are some maxima files and outputs used in my courses on calculus, linear algebra and etc.
  • YouTube Playlist
  • Number Theory
    • Green-Tao Theorem: wxmx, pdf
    • Prime Number Theorem: wxmx, pdf
    • RSA algorithm in cryptography: wxmx, pdf
    • approximation of the base of the natural logarithm e: wxmx, pdf
    • approximation of π: wxmx, pdf
    • area of isoperimetric regular polygons: wxmx, pdf
  • Linear Algebra
    • system of linear equations, determinant and inverse: wxmx, pdf
    • eigenvalues, eigenvectors and normalization of square matrices: wxmx, pdf
    • matrix exponentials of square matrices: wxmx, pdf
    • LU factorization, QR factorization, singular value decomposition (SVD): wxmx, pdf
    • l^p unit spheres on the plane: wxmx, pdf, mp4, YouTube
  • Calculus
    • differentiation, integration, Taylor's polynomial and quadrature: wxmx, pdf
    • convergence and divergence of the Taylor series: wxmx, mp4, YouTube
    • Newton's method of one variable: wxmx, pdf
    • the graphs of some functions of two variables: wxmx, pdf 1, pdf 2
    • successive integrations: wxmx, pdf
    • planar vector fields and integral curves: wxmx, pdf
    • complex numbers and residue: wxmx, pdf
  • Fourier Analysis
    • Fourier transform and Fourier series: wxmx, pdf, mp4, YouTube
    • Gibbs phenomenon of Fourier series: wxmx, png 1, png 2, mp4, YouTube
    • A continuous but nowhere differentiable function: wxmx, png, mp4, YouTube
    • Weyl's equidistribution theorem: wxmx, png, mp4, YouTube
    • The Nyquist-Shannon sampling theorem and reconstruction formula: wxmx, png 1, png 2,
  
  
• Mathematica is a well-known commercial computer algebra system. Here are some mathematica files and outputs used in my courses on calculus, linear algebra and etc.
  • YouTube Playlist
  • Number Theory
    • Green-Tao Theorem: nb, pdf
    • The abc Theorem: nb, pdf
    • Prime Number Theorem: nb, mp4, YouTube, mp4, YouTube
    • RSA algorithm in cryptography: nb, pdf
  • Linear Algebra
    • l^p unit spheres on the plane: nb, mp4, YouTube
  • Calculus
    • Gradient Descent: nb, pdf, mp4, YouTube
  • one-dimensional Fourier series:
    • The Dirichelet kernel: nb, mp4, YouTube
    • Gibbs phenomenon: nb, mp4, YouTube
  • equidustribuion of sequences on unit circle: nb
    • vertexes of regular polygons: mp4, YouTube
    • Weyl's equidistribution theorem 1: mp4, YouTube
    • Weyl's equidistribution theorem 2: mp4, YouTube
    • The progression of the golden ratio: mp4, YouTube