June 25, 2024
Reverse-Process Synthetic Data Generation for Math Reasoning
Training LLMs on mathematical reasoning by inverting easy-to-solve problems: generate derivatives, reverse them into integration exercises with full step-by-step solutions.
Mathematics
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February 19, 2024
Master's Project: Reliability Estimation in Series Systems
My master's project on maximum likelihood estimation for series systems with right-censored and masked failure data.
March 16, 2026
What Happens When You Let an AI Loose on 1,000 Erdős Problems
I pointed Claude Code at the Erdős problem database with vague instructions to 'find interesting things.' It built 92 Python modules, ran 131 subagents, and computed exact Ramsey numbers nobody had computed before. I mostly watched.
March 13, 2026
Free Algebras: Why Lists and Polynomials Are Universal
The free monoid on a set is the type of lists over that set. The universal property says fold is the unique homomorphism from lists to any monoid. This explains why lists, multisets, and polynomials appear everywhere.
March 13, 2026
Homomorphisms: The Maps Between Structures
A homomorphism preserves structure. fold is the universal homomorphism from the free monoid. This is the algebraic reason that fold, evaluation, and parallelism work.
March 13, 2026
Lattices: Fixed Points and Iteration
A lattice has two operations, meet and join, satisfying absorption laws. Tarski's theorem gives a generic fixed-point algorithm. Lattice structure determines the iteration, just as monoid structure determines power-by-squaring.
March 13, 2026
Semirings: One Algorithm, Six Graph Problems
A semiring has two monoidal operations linked by distributivity. Matrix multiplication over different semirings gives shortest paths, longest paths, widest paths, reachability, and path counting, all from the same code.
March 13, 2026
Streaming Statistics, One Monoid at a Time
Online accumulators are monoids. Default construction is the identity, combination via += is the binary operation, and parallel composition gives the product monoid, computing arbitrary statistics in a single pass.
January 19, 2026
Duality: The Hidden Structure of Opposites
Many structures come in pairs: forward/reverse AD, push/pull iteration, encode/decode. Recognizing duality lets you transfer theorems and insights between domains.
January 18, 2026
Seeing Structure First
A reflection on eleven explorations in generic programming, and how algorithms arise from algebraic structure.
December 2, 2025
Closed-Form Results for Masked Exponential Series Systems
Closed-form MLEs and Fisher information for exponential series systems with masked failure data. No numerical optimization required.
November 30, 2025
CBT: Computational Basis Transforms
A C++17 header-only library that formalizes a pattern behind FFT, logarithmic arithmetic, and Bayesian inference: transform to a domain where your target operation is cheap.
November 30, 2025
libdis: Disjoint Interval Sets as a Complete Boolean Algebra
A C++ header-only library that treats disjoint interval sets as proper mathematical objects with Boolean algebra operations.
November 30, 2025
XTK: A Symbolic Expression Toolkit for Term Rewriting
A Python library for rule-based term rewriting with pattern matching, multiple input formats, and an interactive REPL.
January 15, 2025
Differentiation: Three Ways
Three approaches to computing derivatives, forward-mode AD, reverse-mode AD, and finite differences, each with different trade-offs for numerical computing and machine learning.
February 1, 2024
Perfect Hashing: Space Bounds, Entropy, and Cryptographic Security
Space bounds, entropy requirements, and cryptographic security properties of perfect hash functions.
August 28, 2023
Numerical Integration with Generic Concepts
Numerical integration meets generic programming. By requiring only ordered field operations, the quadrature routines work with dual numbers, giving you differentiation under the integral for free.
June 17, 2023
The Bernoulli Model: A Probabilistic Framework for Data Structures and Types
The Bernoulli Model is a framework for reasoning about probabilistic data structures by treating noisy outputs as Bernoulli-distributed approximations of latent values, from Booleans to set-indicator functions.
June 17, 2023
The Bernoulli Model: A Probabilistic Framework for Data Structures and Types
The Bernoulli Model is a framework for reasoning about probabilistic data structures by treating noisy outputs as Bernoulli-distributed approximations of latent values, from Booleans to set-indicator functions.
January 17, 2023
Reverse-Mode Automatic Differentiation
Reverse-mode automatic differentiation is just the chain rule applied systematically. I built one in C++20 to understand what PyTorch and JAX are actually doing.
October 15, 2022
How to Teach a Computer Calculus with Pattern Matching
Define patterns, define replacements, repeat until done. Watch a 90-line rewrite engine learn to differentiate.
April 12, 2022
Numerical Differentiation
Choosing step size h for finite differences: small enough for a good approximation, not so small that floating-point errors eat your lunch.
September 20, 2021
Forward-Mode Automatic Differentiation
Dual numbers extend our number system with an infinitesimal epsilon where epsilon^2 = 0. Evaluating f(x + epsilon) yields f(x) + epsilon * f'(x)—the derivative emerges automatically from the algebra.
March 8, 2021
Teaching Linear Algebra with C++20 Concepts
elementa is a linear algebra library built to teach. Every design decision prioritizes clarity over cleverness. Code that reads like a textbook and compiles.
July 14, 2020
Polynomials as Euclidean Domains
The same GCD algorithm works for integers and polynomials because both are Euclidean domains. One structure, many types, same algorithms.
February 18, 2020
Exact Rational Arithmetic
Rational numbers give exact arithmetic where floating-point fails. The implementation connects GCD, the Stern-Brocot tree, and the algebraic structure of fields.
September 10, 2019
Is It Prime?
The Miller-Rabin primality test demonstrates how probabilistic algorithms achieve arbitrary certainty, trading absolute truth for practical efficiency.
June 22, 2019
Modular Arithmetic as Rings
Integers modulo N form a ring, an algebraic structure that determines which algorithms apply. Understanding this structure unlocks algorithms from cryptography to competitive programming.
March 15, 2019
One Algorithm, Infinite Powers
The Russian peasant algorithm computes products, powers, Fibonacci numbers, and more, once you see the underlying algebraic structure.