Mathematics
Set Theory
Foundations of sets and mappings, axiomatic set theory, ZFC axioms, ordinals and cardinals.
Proof Techniques
How to write proofs: direct proof, proof by contradiction, mathematical induction, and other proof methods.
Mathematical Logic
What a proof is, formally: formal systems, proof theory, type theory, the Curry–Howard correspondence, incompleteness theorems, and the foundations of proof assistants.
Functions
Basic concepts of functions, composition, inverse functions, and elementary functions.
Sequences
Arithmetic and geometric sequences, recurrence relations, convergence of series.
Abstract Algebra
Groups, rings, and fields, Galois theory, homological algebra.
Linear Algebra
Vector spaces, determinants, eigenvalues, diagonalization, and function spaces.
Lie Algebras
Fundamentals of Lie groups and Lie algebras, representation theory.
Number Theory
Foundations of integer theory, primes, congruences, and applications to cryptography.
Combinatorics
Permutations and combinations, generating functions, graph coloring, Ramsey theory.
Graph Theory
Graph fundamentals, trees, network flows, matchings.
Geometry
From trigonometry and coordinate geometry to projective and computational geometry.
Differential Geometry
Differential geometry of curves and surfaces, Riemannian geometry, connections and curvature.
Algebraic Geometry
From polynomials and shapes to algebraic varieties and schemes.
Differentiation
Fundamentals and applications of differentiation: limits, derivatives, partial derivatives.
Matrix Calculus
Differentiation with respect to matrices and vectors — essential techniques for machine learning and optimization.
Integration
From indefinite and definite integrals to multiple integrals and line integrals.
Real Analysis
Lebesgue measure and integration, Lp spaces, foundations of Fourier analysis.
Complex Analysis
Foundations of complex function theory, conformal mappings, and the residue theorem.
Functional Analysis
Banach spaces, Hilbert spaces, operator theory.
Ordinary Differential Equations
First-order and higher-order ODEs, existence and uniqueness of solutions, dynamical systems.
Partial Differential Equations
Wave, heat, and Laplace equations, Sobolev spaces.
Optimization
Convex optimization, Lagrange multipliers, numerical optimization.
Fourier Analysis
Fourier series and the Fourier transform, from fundamentals to applications.
Laplace Transform
Definition of the Laplace transform, inverse transforms, and applications to differential equations.
Z-Transform
The Z-transform for discrete signals, inverse transforms, and applications to difference equations.
Radon Transform
Theory of the Radon transform and applications to CT image reconstruction.
Probability Theory
From the foundations of probability to stochastic processes.
Statistics
Descriptive statistics, inferential statistics, and Bayesian statistics.
Computer Algebra & Arbitrary-Precision Arithmetic
Symbolic computation (polynomial GCD, factorization, Gröbner bases, symbolic integration) and arbitrary-precision arithmetic (fast multiplication, floating-point, root extraction, constant computation) — 29 chapters in all.
Numerical Analysis
Interpolation and approximation, numerical integration, linear systems, eigenvalue problems, and numerical methods for ODEs/PDEs.
Verified Numerics & Interval Arithmetic
Interval arithmetic, verified numerical computation, random number generation.
Signal Processing
Filter Design
Design methods for analog and digital filters. Chebyshev, Butterworth, elliptic filters, and more.
Wiener Filter
Optimal linear filter in the MMSE sense. Frequency-domain and time-domain derivation with Wirtinger derivative.
Kalman Filter
Recursive optimal estimation based on state-space models. Comparison with Wiener filter, prerequisites and extensions.
Control Theory
State-Space Representation
Foundation of modern control. Link to transfer functions, controllable/observable canonical forms, ZOH discretization.
LQR (Linear Quadratic Regulator)
Optimal state feedback minimizing a quadratic cost. Riccati equation and Bryson's rule.
PID Control
The industrial workhorse. Ziegler-Nichols tuning, anti-windup, and implementation caveats.
→ See all 8 articles: controllability, pole placement, observer, Lyapunov, Riccati, and more
Machine Learning
Introduction
Overview of machine learning. From the three major categories — supervised, unsupervised, and reinforcement learning — to deep learning and generative AI.
Basics
Theory and implementation of classical methods: linear regression, logistic regression, decision trees, k-NN, and more.
Intermediate
Neural networks, CNNs, and RNNs. Gradient descent, regularization, and optimization techniques.
Advanced
Attention, Transformer, ViT, VAE, GAN, diffusion models, and other cutting-edge methods with mathematical rigor.
Electronic Instruments
A History of Electronic Instruments
A lineage not of devices but of how instrument sound is represented mathematically, from the Fourier series to differentiable synthesis, in a single chart.
Additive Synthesis
Approximating a tone as a sum of sinusoids. Sawtooth coefficients, the Gibbs overshoot, and implementations from the Telharmonium to the ANS.
FM Synthesis
Approximating a tone by frequency modulation of a sinusoid. The Jacobi–Anger expansion shows the sideband amplitudes are Bessel functions.
About This Site
Since the 1980s, we have carried out research and development in acoustic signal processing, image processing, machine learning, and numerical computation. This site shares the mathematical and engineering knowledge accumulated along the way. The learning and reading methods we refined through that research live on in our reading-management app, Reading Forest.