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terrytao.wordpress.com | ||
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extremal010101.wordpress.com
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| | | | | Suppose we want to understand under what conditions on $latex B$ we have $latex \begin{aligned} \mathbb{E} B(f(X), g(Y))\leq B(\mathbb{E}f(X), \mathbb{E} g(Y)) \end{aligned}$holds for all test functions, say real valued $latex f,g$, where $latex X, Y$ are some random variables (not necessarily all possible random variables!). If $latex X=Y$, i.e., $latex X$ and $latex Y$ are... | |
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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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a3nm.net
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| | | | | List of open questions | |
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math.andrej.com
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| | | [AI summary] The discussion revolves around the nuances of proof methods in constructive mathematics, particularly the distinction between proof by contradiction and proof by negation. Key points include the definition of irrational numbers without relying on the law of excluded middle, the use of contrapositive in proofs, and the limitations of certain classical theorems like the intermediate value theorem in constructive settings. The conversation also touches on the philosophical and practical implications of these proof methods in both classical and intuitionistic logic, as well as the role of type theory and univalent foundations in modern mathematical proofs. | ||
