We discuss a construction question that naturally arises from the Ax–Grothendieck theorem. In particular, this comes from a field automorphism between the complex numbers and the ultraproduct of prime algebraically closed fields of positive characteristics.
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A comment to my recent work, regarding an alternative model for describing the “holomorphic parameters” case.
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One way to see an elliptic curve is to view it as a smooth bidegree (2,2) curve in $\mathbb{P}^1\times\mathbb{P}^1$. This fact itself comes from the adjunction formula, but we suggest a way to derive a bidegree (2,2) formula from the Weierstrass equation. Based on that, we see how this connects with the tropicalized actions of Vieta involutions on elliptic curves.
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Theorem. For any irrational $\alpha>1$ whose continued fraction is $\alpha=[a_0;a_1,a_2,\ldots]$, the remainders $\alpha_i=[a_i;a_{i+1},a_{i+2},\ldots]$ have an amortized estimate $(\alpha_1\cdots\alpha_n)^{1/n} > \sqrt{2}$.
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Some trivia found while teaching Linear Algebra in the Fall 2022 quarter.
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A summary of the basics of the Ruelle transfer operator, along with some notes about how potentials work in the context of Szilard’s engine.
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Maxwell의 도깨비, Landauer principle 등을 보고 떠오른 잡상雜想.
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This note is to record some tales that may be adequate while introducing Intermediate Value Theorem (IVT) and Extreme Value Theorem (EVT) for continuous functions. Some ideas behind the proofs, a relation with long division algorithm, etc. are introduced.
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This post posts some rational approximation properties of logarithms $\log_pq$, where $p,q$ are multiplicatively independent natural numbers. This will be discussed as an immediate consequence of quantitative Baker’s theorem.
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전통적인 산술로 2차방정식의 수치해를 구하는 방법에 대한 소개 및 여기서 도출되는 몇 가지 ‘숫자 놀이’에 대한 글.
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This series of posts is aimed to be a rudiment of the Berkovich line, a locally compact space that “fills up” a non-archimedean field $K$. This post is the last of the series and continues from part 2.
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This series of posts is aimed to be a rudiment of the Berkovich line, a locally compact space that “fills up” a non-archimedean field $K$. This post continues from part 1.
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The idea of this post is to (1) introduce a formula to measure a distance on the upper half plane with the ‘standard’ hyperbolic metric, and (2) record some ideas of deriving this.
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This series of posts is aimed to be a rudiment of the Berkovich line, a locally compact space that “fills up” a non-archimedean field $K$. The materials here are more or less my summary of the lecture note by (M. Baker 2007), with some extra materials that I found meaningful to append.
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대한수학회 수학용어집에 따르면, homology, cohomology는 각각 음값을 그대로 쓴 ‘호몰로지’ ‘코호몰로지’라 번역하는 것이 표준이다. 다만, 보다 원어의 의미를 잘 생각해 보고자, 이 용어가 어떤 맥락에서 도입됐는지 등에 대해 알아 본 내용을 담아 보았다.
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This is a record of a talk that I made in the UChicago Math dept. Pedagogy seminar, in 2022-02-23.
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앞선 포스트에서는 부분분수 분해와 중국인의 나머지 정리 사이의 관계를 짚어보았으나, 막상 부분분수 분해를 중국인의 나머지 정리의 결과인 것으로 묘사한 바 있었다. 그러나 계산의 흐름을 보면 둘이 동치라 생각하는 편이 자연스러운데, 이 후속 포스트에서는 그 점을 명확히 하고자 한다.
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This is to record some Python macro that I have used to draw some TikZ figures:
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This ‘web log’ (etymology of ‘blog,’ as far as I know of) is intended to be a batch of notes that I make upon academic studies. Among the notes that I feel worth sharing, to be precise.
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