Notes on various math and software topics

## Table of contents

- Math diagrams
- Numerical computing
- Probability
- Differential equations
- Python
- Probability approximations
- Regular expressions
- C++
- Special functions
- Typesetting: TeX, HTML, Unicode
- Emacs
- R
- Miscellaneous math

## Math diagrams

- Diagram of probability distribution relationships
- Modes of convergence
- Topological properties diagram
- Category Relationships in Mathematical Physics
- Category theory definition dependencies
- Conjugate prior diagram
- Topological vector spaces
- How areas of math are connected
- Kinds of rings

## Numerical computing

- Numerical integration
- IEEE floating-point exceptions in C++
- Accurately computing running variance
- High dimensional integration
- Double exponential integration
- Math.h in POSIX, ISO, and Visual Studio
- Stand-alone code for numerical computing
- Applied linear algebra

## Probability

- How to test a random number generator
- Explaining probability to jurors
- Diagram of probability distribution relationships
- Central Limit Theorems
- Counting selections with replacement
- Distributions in Mathematica, R, Python (SciPy), and Excel
- Adult heights and mixture distributions
- Predictive probabilities for normal outcomes
- One-arm binary predictive probability
- Relating two definitions of expectation
- Illustrating the error in the delta method
- Relating the error function erf and Φ
- Inverse gamma distribution
- Negative binomial distribution
- Upper and lower bounds for the normal distribution function
- Canonical example of Bayes’ theorem in detail
- Functions of regular variation
- Student-
*t*as a mixture of normals - Linear regression is more than curve-fitting

## Differential equations

- Picard’s theorem (for Banach-valued functions)
- Undetermined coefficients
- Picking the step size for numerical ODEs
- Complex networks
- Outline of Laplace transforms
- Navier-Stokes equations
- Multi-index notation
- Abstract PDE operators
- Euler-Lagrange equations
- Applied functional analysis

## Python

- Moving to the Python scientific computing stack
- Bessel functions in SciPy
- Gamma and related functions in SciPy
- Distributions in SciPy
- Python counterparts for C math functions

## Probability approximations

- Camp-Paulson normal approximation to the binomial distribution
- Details for error bound on normal approximation to the Poisson distribution
- Error in the normal approximation to the beta distribution
- Error in the normal approximation to the binomial distribution
- Error in the normal approximation to the gamma distribution
- Error in the normal approximation to the Poisson distribution
- Error in the normal approximation to the t distribution
- Error in the Poisson approximation to the binomial distribution
- Relative error in normal approximations

## Regular expressions

- Regular expressions in Mathematica
- Regular expressions in PowerShell and Perl
- C++ TR1 regular expressions
- Regular expressions in R
- Comparing regular expressions in Perl, Python, and Emacs

## C++

- C++ TR1 regular expressions
- IEEE floating-point exceptions in C++
- Unraveling Strings in Visual C++
- Random number generation in C++

## Special functions

- Bessel functions in SciPy
- Gamma and related functions in SciPy
- Identities for gamma and related functions
- Relations between Bessel functions
- Diagram of relations between special functions

## Typesetting: TeX, HTML, Unicode

- Common Math Symbols in HTML, XML, TeX, and Unicode
- Accented letters in HTML, TeX, and Microsoft Word
- Greek letters in HTML, XML, TeX, and Unicode
- Unicode resources

## Emacs

- Emacs kill (cut) commands
- Emacs point (cursor) movement
- Getting started with Emacs on Windows
- Notes on Unicode in Emacs

## R

- Distributions in R and S-PLUS
- Moving data between R and Excel via the clipboard
- Sweave: First steps toward reproducible analyses
- Troubleshooting Sweave
- R language for programmers
- Regular expressions in R

## Misc math

- Notes on Spherical Trigonometry
- Big-O and related notation
- Differentiation in Banach spaces
- Applied complex analysis
- Solving quadratic congruences
- The difference between an unbiased estimator and a consistent estimator
- Binomial coefficients
- Cubic equation solver
- Online calculators
- How to calculate binomial coefficients
- The
*pqr*theorem for seminorms - Chebyshev polynomials
- Legendre polynomials
- Richard Stanley’s twelvefold way (combinatorics)
- Hypergeometric functions
- Orthogonal polynomials
- Theory of interest
- Applied number theory
- Fourier transforms under varying conventions