Mathematics

Introduction Chaos theory is a branch of mathematics and physics that studies how tiny variations in initial conditions can lead to dramatically different outcomes in deterministic systems. Although the underlying equations are perfectly predictable, the resulting behavior appears random and unpredictable to the naked eye. This paradox—order hidden within apparent disorder—has fascinated scientists, engineers, and artists for decades. In this article we will: Trace the historical development of chaos theory from Poincaré to modern computational methods. Explain the core concepts such as sensitive dependence on initial conditions, strange attractors, and fractal geometry. Dive into canonical models (the logistic map, the Lorenz system, the double pendulum) with concrete Python code examples. Illustrate real‑world applications in weather forecasting, finance, biology, engineering, and art. Discuss the philosophical and practical implications of living in a chaotic world. By the end of the post you should have a solid conceptual foundation, a toolbox of simple simulations, and a sense of where chaos theory can be applied in your own field. ...

Introduction A random walk is one of the most fundamental stochastic processes in probability theory. At its core, it describes a path that consists of a succession of random steps. Despite its deceptively simple definition, the random walk model underpins a surprisingly wide range of phenomena—from the diffusion of particles in physics to stock‑price dynamics in finance, from the spread of diseases in epidemiology to algorithmic techniques in computer science. ...

SorryDB: Can AI Really Prove Real-World Math Theorems? Imagine you’re a mathematician knee-deep in a complex proof, but you hit a wall. Instead of giving up, you jot down a placeholder—“sorry, I’ll finish this later”—and move on. Now, picture AI stepping in to fill those gaps automatically. That’s the promise of SorryDB, a groundbreaking benchmark introduced in the paper “SorryDB: Can AI Provers Complete Real-World Lean Theorems?” (arXiv:2603.02668). This isn’t some abstract academic exercise; it’s a practical testbed pulling “sorry” statements from 78 real GitHub projects, challenging AI to prove theorems that actual mathematicians are working on. ...

Table of Contents Introduction Probability Fundamentals Conditional Probability and the Chain Rule Probability Distributions How LLMs Use Probability From Theory to Practice Common Misconceptions Conclusion Resources Introduction If you’ve ever wondered how ChatGPT, Claude, or other large language models generate coherent text that seems almost human-like, the answer lies in mathematics—specifically, probability theory. While the internal mechanics of these models involve complex neural networks and billions of parameters, at their core, they operate on a surprisingly elegant principle: predicting the next word by calculating probabilities. ...