정리. 길이 L, 중력가속도 g, 질량 m인 단진자의 운동방정식 $\ddot{\theta}=-\frac{g}{L}\sin\theta$이 주기 T를 가진다 하자. 이 때 최대 각도를 $\theta_0$라 써, $\lambda=(1-\cos\theta_0)/2$라 하자. 또, $u=(1-\cos\theta)/2$를 높이 함수라 하자. 그러면 시간 t가 구간 [0,T] 안에서 균등하게 임의로 골라질 때, 높이함수와 그 미분 $(u(t),\dot{u}(t))$은 타원곡선 $E_\lambda\colon y^2=4x(x-1)(x-\lambda)$의 (항등원 없는 연결성분 위로 제한된) Haar 측도를 따른다.
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Although Gödel’s proof of his incompleteness is renowned for the formal construction of the sentence “I cannot be proven,” there are ways to illuminate this within the framework of abstract nonsense. From the nLab point of view, this is a consequence of Lawvere’s fixed point theorem, as shown in Lawvere 1969 or Yanofsky 2003. However, since I am not entirely sure if I understood the work correctly, I will present my own re-invention of the wheel below.
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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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