📍🇪🇸 巴塞罗那-凯旋门

凯旋门？秋叶原！

还算这群西哥特蛮子有点品味😋

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Supervised and Unsupervised learning in machine learning

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AI: Deep Learning vs Machine Learning

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An introduction to different branches of mathematics

The note is mainly a sketch of basic knowledge concerning general topology, differential geometry, functional analysis, algebraic geometry, etc., starting from a discussion of Euclidean spaces. However, there maybe some mistakes in the note, so use at your own risk. For simplicity, some details are omitted and can be found in the references provided. Further materials concerning algebraic geometry, especially arithmetic algebraic geometry, can be referred to another note written by the author, N ...

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How to setup PrismJS and Autoloader plugin with Nuxt 3?

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How to find similar strings with python?

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Django use icontains with in for field lookup

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Mathjax loads slow. How to load local JS file with Nuxt?

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VSCode error: To enable project-wide JavaScript/TypeScript language features, exclude large folders...

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How to render math in Vue or Nuxt?

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Nginx is installed but command not found

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Proof of the isomorphism $\textrm{Hom}_{A-mod}(\textrm{Hom}_{A-mod}(A,A),A)\cong A$ for any ring $A$

Let $A$ be a commutative ring with identity. Then we have an isomorphism $$A \rightarrow \textrm{Hom}_{A-mod}(A,A), a \mapsto (x \mapsto ax)$$, whose inverse is $$\textrm{Hom}_{A-mod}(A,A) \rightarrow A, \varphi \mapsto \varphi(1).$$Then applying this isomorphism, we get $$\textrm{Hom}_{A-mod}( \textrm{Hom}_{A-mod}(A,A) ,A) \cong \textrm{Hom}_{A-mod}(A,A) \cong A.$$

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