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www.jeremykun.com | ||
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almostsuremath.com
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| | | | | The aim of this post is to motivate the idea of representing probability spaces as states on a commutative algebra. We will consider how this abstract construction relates directly to classical probabilities. In the standard axiomatization of probability theory, due to Kolmogorov, the central construct is a probability space $latex {(\Omega,\mathcal F,{\mathbb P})}&fg=000000$. This consists... | |
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xorshammer.com
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| | | | | There are a number of applications of logic to ordinary mathematics, with the most coming from (I believe) model theory. One of the easiest and most striking that I know is called Ax's Theorem. Ax's Theorem: For all polynomial functions $latex f\colon \mathbb{C}^n\to \mathbb{C}^n$, if $latex f$ is injective, then $latex f$ is surjective. Very... | |
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mattbaker.blog
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| | | | | In my previous post, I presented a proof of the existence portion of the structure theorem for finitely generated modules over a PID based on the Smith Normal Form of a matrix. In this post, I'd like to explain how the uniqueness portion of that theorem is actually a special case of a more general... | |
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algorithmsoup.wordpress.com
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| | | The ``probabilistic method'' is the art of applying probabilistic thinking to non-probabilistic problems. Applications of the probabilistic method often feel like magic. Here is my favorite example: Theorem (Erdös, 1965). Call a set $latex {X}&fg=000000$ sum-free if for all $latex {a, b \in X}&fg=000000$, we have $latex {a + b \not\in X}&fg=000000$. For any finite... | ||
