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- Monte Carlo (MC) simulation of spins--Ising model
- Numerical vs. MC integration: Simpson's rule, Gaussian quadrature (orthogonal functions--recursive function evaluation, generating function)
- Probability: Importance sampling, Markov chain, Metropolis algorithm
- Random number generation (RNG)
- Statistics: Variance, standard deviation, standard deviation of the MC mean
- Cluster analysis: Graphs, search, stack

- MC simulation of stock price--geometric Brownian motion
- Random walk: Einstein's law, central-limit theorem
- Random variable: Black-Scholes analysis
- Coordinate transformation: Jacobian, Box-Muller algorithm for RNG of normal distribution
- Interpolation: Least square fit of data
- Quantum MC and kinetic MC simulations

- Molecular dynamics (MD) simulation of particles--Newton's second law of motion
- Numerical differentiation
- Ordinary differential equation (ODE): Symplectic integrators
- Minimization of functions: Conjugate gradient method
- Hybrid MD/MC simulation

- Quantum dynamics simulation of an electron--time-dependent Schrodinger equation
- Partial differential equation (PDE)
- Fourier analysis: Spectral analysis, fast Fourier transform (FFT)

- Electronic structures of molecules--quantum mechanical eigenvalue problem
- Linear algebra: Matrix, orthogonal transformation, rank, singular value decomposition, Krylov subspace
- Matrix eigensystems: Housholder transformation, QL decomposition
- Root finding: Newton-Raphson method