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terrytao.wordpress.com | ||
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qchu.wordpress.com
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| | | | | (Part I of this post ishere) Let $latex p(n)$ denote the partition function, which describes the number of ways to write $latex n$ as a sum of positive integers, ignoring order. In 1918 Hardy and Ramanujan proved that $latex p(n)$ is given asymptotically by $latex \displaystyle p(n) \approx \frac{1}{4n \sqrt{3}} \exp \left( \pi \sqrt{ \frac{2n}{3}... | |
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francisbach.com
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| | | | | [AI summary] This text discusses the scaling laws of optimization in machine learning, focusing on asymptotic expansions for both strongly convex and non-strongly convex cases. It covers the derivation of performance bounds using techniques like Laplace's method and the behavior of random minimizers. The text also explains the 'weird' behavior observed in certain plots, where non-strongly convex bounds become tight under specific conditions. The analysis connects theoretical results to practical considerations in optimization algorithms. | |
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polymathprojects.org
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| | | | | It's probably time to refresh the previous thread for the "finding primes" project, and to summarise the current state of affairs. The current goal is to find a deterministic way to locate a prime in an interval $latex [z,2z]$ in time that breaks the "square root barrier" of $latex \sqrt(z)$ (or more precisely, $latex z^{1/2+o(1)}$).... | |
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susam.net
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| | | [AI summary] An article explains the mathematical relationship between fields and their trivial ideals within ring theory. | ||
