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fabricebaudoin.blog | ||
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thehighergeometer.wordpress.com
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| | | | | Here's a fun thing: if you want to generate a random finite $latex T_0$ space, instead select a random subset from $latex \mathbb{S}^n$, the $latex n$-fold power of the Sierpinski space $latex \mathbb{S}$, since every $latex T_0$ space embeds into some (arbitrary) product of copies of the Sierpinski space. (Recall that $latex \mathbb{S}$ has underlying... | |
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dominiczypen.wordpress.com
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| | | | | Let $latex \omega$ denote the first infinite cardinal - that is, the set of non-negative integers. Let $latex p_0 = 2$ be the smallest prime number, and let $latex (p_n)_{n\in\omega}$ enumerate all prime numbers in ascending order. Let $latex \mathcal{U}$ be a free ultrafilter on $latex \omega$. We consider the field $latex F = \big(\prod_{n\in\omega}\mathbb{Z}/p_n\mathbb{Z}\big)/{\mathcal... | |
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pfzhang.wordpress.com
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| | | | | Consider a smooth one-parameter family $latex {f_t}$ of diffeomorphisms on a manifold $latex M$. It is a flow if $latex f_0(x)=x$ and $latex f_{t}\circ f_{s}(x) = f_{s+t}(x)$ for every $latex x\in M$, , $latex t ,s \in \mathbb{R}$. Set $latex X(x)=\lim\limits_{h\to 0} \frac{1}{h}(f_h(x) - x)$. This generates a vector field $latex X: M \to TM$.... | |
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francisbach.com
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| | | [AI summary] The blog post discusses non-convex quadratic optimization problems and their solutions, including the use of strong duality, semidefinite programming (SDP) relaxations, and efficient algorithms. It highlights the importance of these problems in machine learning and optimization, particularly for non-convex problems where strong duality holds. The post also mentions the equivalence between certain non-convex problems and their convex relaxations, such as SDP, and provides examples of when these relaxations are tight or not. Key concepts include the role of eigenvalues in quadratic optimization, the use of Lagrange multipliers, and the application of methods like Newton-Raphson for solving these problems. The author also acknowledges contributions... | ||
