A long-term collection of mathematical investigations spanning logic, proof, algebra, analysis, probability, optimization, numerical methods, and dynamical systems.
The project focuses particularly on assumptions that are often left implicit: why mathematical statements are true, where their proofs depend on specific hypotheses, and what happens when those hypotheses are weakened or removed.
Investigations range from foundational logic and proof techniques to graduate-level applied mathematics, with computational experiments used where they illuminate the underlying theory.
| # | Investigation | Area | Type |
|---|---|---|---|
| 001 | Implication and the Contrapositive | Logic | Proof |
| 002 | Quantifier Order | Logic | Foundations |
| 003 | Irrationality of √2 | Number Theory | Proof |
| 004 | Infinitely Many Primes | Number Theory | Proof |
| 005 | Why Induction Works | Foundations | Proof |
| 006 | Linear Independence | Linear Algebra | Proof |
| 007 | Matrix Inverses | Linear Algebra | Proof |
| 008 | Eigenvectors as Invariant Directions | Linear Algebra | Proof + Computation |
| 009 | Derivative from First Principles | Analysis | Derivation |
| 010 | Failure Modes of Newton's Method | Numerical Analysis | Experiment |
| 011 | Cauchy Sequences and Why Completeness Matters | Analysis | Foundations |
| 012 | Uniform versus Pointwise Convergence | Analysis | Comparison |
| 013 | Taylor Approximation and Remainder Bounds | Analysis | Derivation |
| 014 | Gram-Schmidt from Projection | Linear Algebra | Derivation |
| 015 | Least Squares from Orthogonal Projection | Linear Algebra | Derivation |
| 016 | Positive-Definite Matrices | Linear Algebra | Proof + Computation |
| 017 | Spectral Theorem | Linear Algebra | Proof |
| 018 | Singular Value Decomposition | Linear Algebra | Proof + Computation |
| 019 | Condition Numbers and Numerical Instability | Numerical Analysis | Experiment |
| 020 | Floating-Point Arithmetic versus Real Arithmetic | Numerical Analysis | Comparison |
| 021 | Gradient Descent on Quadratic Functions | Optimization | Experiment |
| 022 | Convexity and Why Local Minima Become Global Minima | Optimization | Proof |
| 023 | Lagrange Multipliers from Geometry | Optimization | Derivation |
| 024 | Bayes' Theorem from Conditional Probability | Probability | Derivation |
| 025 | Law of Large Numbers Simulation | Probability | Experiment |
| 026 | Central Limit Theorem Experiment | Probability | Experiment |
| 027 | Markov Chains and Stationary Distributions | Probability | Theory + Computation |
| 028 | Existence and Uniqueness for Differential Equations | Differential Equations | Foundations |
| 029 | Phase Portraits of Nonlinear Systems | Dynamical Systems | Visualization |
| 030 | Linearization around Equilibria | Dynamical Systems | Derivation |
| 031 | Stability and Lyapunov Functions | Dynamical Systems | Proof |
| 032 | Discretization of Continuous Dynamical Systems | Numerical Analysis | Derivation + Experiment |
| 033 | Numerical Integration Error | Numerical Analysis | Experiment |
| 034 | State-Space Systems | Control Theory | Foundations |
| 035 | Controllability and Observability | Control Theory | Theory + Computation |
| 036 | Kalman Filtering from Gaussian Estimation | Control Theory | Derivation + Computation |
Each investigation aims to do more than record a result. Where appropriate, entries examine:
- Definitions, axioms, and prerequisite results
- Hidden or easily overlooked assumptions
- Formal derivations and proofs
- Counterexamples and failure cases
- Computational experiments and visualizations
- Connections to broader mathematical ideas
The goal is to build a cumulative mathematical reference that moves from first principles toward increasingly advanced topics while preserving rigor at every level.