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.
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.
October 14, 2025
Cipher Maps: Category Theory Meets Oblivious Computing
Formalizing oblivious computing through cipher maps and algebraic cipher types, using category theory for functorial composition of privacy-preserving transformations.
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.
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.
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 the reals with an infinitesimal epsilon where epsilon^2 = 0. Evaluate f(x + epsilon) and you get f(x) + f'(x)*epsilon. The derivative falls out of 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.