https://leonardschneider.github.io/Recent content on Leo and AI's blogHugo -- 0.151.2en-usSat, 08 Nov 2025 00:00:00 +0000https://leonardschneider.github.io/posts/measuring-the-cosmos/Sat, 08 Nov 2025 00:00:00 +0000https://leonardschneider.github.io/posts/measuring-the-cosmos/Starting with just a stick and basic geometry, rebuild humanity&rsquo;s knowledge of cosmic scales from scratch. Learn how Eratosthenes, Galileo, Newton, and others measured Earth, the Moon, the Sun, and distant stars using ingenious observations.https://leonardschneider.github.io/posts/wkb-classical-limit-localized-wavefunctions/Sat, 08 Nov 2025 00:00:00 +0000https://leonardschneider.github.io/posts/wkb-classical-limit-localized-wavefunctions/The Ehrenfest theorem fails to explain how quantum mechanics becomes classical. But there is another way: when wavefunctions become sufficiently localized, they follow classical trajectories not through averaging, but through the geometry of quantum phase. Discover how the WKB approximation reveals the true classical limit—and why entanglement marks its absolute boundary.https://leonardschneider.github.io/posts/mersenne-primes-hunt-for-giants/Sun, 02 Nov 2025 00:00:00 +0000https://leonardschneider.github.io/posts/mersenne-primes-hunt-for-giants/Every largest known prime since 1952 has been a Mersenne number. Discover the elegant Lucas-Lehmer test that makes these exponentially growing numbers the most efficient targets for finding record-breaking primes.https://leonardschneider.github.io/posts/ehrenfest-theorem-classical-limit/Mon, 27 Oct 2025 00:00:00 +0000https://leonardschneider.github.io/posts/ehrenfest-theorem-classical-limit/The Ehrenfest theorem suggests quantum mechanics reduces to classical physics for average values, but this beautiful result is deeply misleading. Discover why quantum uncertainty persists even when expectation values obey classical equations.https://leonardschneider.github.io/posts/probability-modeling-extinction/Sun, 26 Oct 2025 00:00:00 +0000https://leonardschneider.github.io/posts/probability-modeling-extinction/Is extinction inevitable if there is any chance of reproductive failure? Explore branching processes, born from Victorian concerns about aristocratic surnames, to understand when populations face certain doom and when survival remains possible.https://leonardschneider.github.io/posts/markov-chains-generating-functions/Mon, 20 Oct 2025 00:00:00 +0000https://leonardschneider.github.io/posts/markov-chains-generating-functions/Learn how Markov chains emerged from a 19th-century academic rivalry over Pushkin&rsquo;s poetry, and discover how generating functions solve complex probability problems, from predicting weather to powering Google&rsquo;s PageRank algorithm.https://leonardschneider.github.io/posts/fibonacci-generating-functions/Mon, 20 Oct 2025 00:00:00 +0000https://leonardschneider.github.io/posts/fibonacci-generating-functions/Discover how generating functions transform the recursive Fibonacci sequence into a closed-form formula involving the golden ratio - a powerful mathematical technique that converts a simple recursive pattern into an elegant analytical solution.https://leonardschneider.github.io/about/Sun, 19 Oct 2025 18:42:17 -0400https://leonardschneider.github.io/about/<p>Hi, I&rsquo;m Leo Schneider. This is my personal blog where I write about things that capture my curiosity.</p> <p><strong>A Note on Collaboration:</strong> The articles on this blog are coauthored with AI assistants, primarily ChatGPT and Claude. This collaborative approach allows me to explore ideas more deeply, refine explanations, and present complex topics with greater clarity.</p> <p>My main interests include mathematics, physics, and artificial intelligence—particularly the deep connections between them. I&rsquo;m fascinated by how mathematical structures reveal patterns in the physical world and how both inform our understanding of intelligence and computation.</p> <p>Beyond these core topics, I explore whatever else sparks my interest. You&rsquo;ll find articles ranging from rigorous technical explorations to more casual investigations of ideas I find compelling.</p> <p>This blog is a space for learning in public, sharing insights, and connecting with others who enjoy thinking deeply about interesting problems.</p>https://leonardschneider.github.io/posts/exponential-self-similarity/Sun, 19 Oct 2025 00:00:00 +0000https://leonardschneider.github.io/posts/exponential-self-similarity/Explore why exponential functions are unique: they look identical at every scale. This self-similarity property provides an intuitive foundation for understanding why the exponential is its own derivative.