This elegant quote, which captures the essence of the mathematical fact that a 2-dimensional random walk is recurrent while in higher dimensions it is not, is attributed to Shizuo Kakutani. However, the original proof is due to George Pólya [1]. A -dimensional lattice is defined as a graph whose vertices are all points in with […]
ChatGPT, DeepSeek, and other large language models are revolutionizing the way we interact with information, automate tasks, and understand natural language. Furthermore, there is a lot of discussion around the dangers behind AI. I just came across a video in my feed featuring an interview with neural networks pioneer Geoff Hinton, who has received both […]
Rayleigh’s monotonicity principle states that if the resistances of a circuit of resistors are increased, the effective resistance between any two points can only increase. If they are decreased, they can only decrease. You can find a short proof in the Doyle-Snell book right here. One can use this intuitive principle to prove classic and […]
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