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grossack.site | ||
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almostsuremath.com
Continuing on from the previous post, I look at cases where the abstract concept of states on algebras correspond to classical probability measures. Up until now, we have considered commutative real algebras but, before going further, it will help to look instead at algebras over the complex numbers $latex {{\mathbb C}}&fg=000000$. In the commutative case,... |
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www.jeremykun.com
The First Isomorphism Theorem The meat of our last primer was a proof that quotient groups are well-defined. One important result that helps us compute groups is a very easy consequence of this well-definition. Recall that if $ G,H$ are groups and $ \varphi: G \to H$ is a group homomorphism, then the image of $ \varphi$ is a subgroup of $ H$. Also the kernel of $ \varphi$ is the normal subgroup of $ G$ consisting of the elements which are mapped to the identity under $ \varphi$. |
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leanprover-community.github.io
[AI summary] This blog post discusses the formalization of topology and related concepts in the Lean theorem prover, covering typeclasses, filters, compactness, Hausdorff spaces, and file organization within the mathlib library. |
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www.jeremykun.com
For fixed integers $ r > 0$, and odd $ g$, a Moore graph is an $ r$-regular graph of girth $ g$ which has the minimum number of vertices $ n$ among all such graphs with the same regularity and girth. (Recall, A the girth of a graph is the length of its shortest cycle, and it's regular if all its vertices have the same degree) Problem (Hoffman-Singleton): Find a useful constraint on the relationship between $ n$ and $ r$ for Moore graphs of girth $ 5$ and degree $ r$. |
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