중국인의 나머지 정리 2
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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Posts by Tags
Adele
Algebra
Playing with the Ax–Grothendieck Theorem
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.
Published:
Algebraic Geometry
Playing with the Ax–Grothendieck Theorem
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.
Published:
Cone Model for Vieta Actions on Tropical Markov Cubics
A comment to my recent work, regarding an alternative model for describing the “holomorphic parameters” case.
Published:
Elliptic curves of bidegree (2,2)
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.
Published:
Analysis
Amortized Multiplicative Estimates of Remainders of Continued Fractions
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 $(\alpha_1\cdots\alpha_n)^{1/n} > \sqrt{2}$.
Published:
Discovering a nontriviality of the ε-δ definition, in a math way
This is a record of a talk that I made in the UChicago Math dept. Pedagogy seminar, in 2022-02-23.
Published:
Dedekind Completeness and Limits
Limit of a Sequence
Published:
Arithmetic
개평법
전통적인 산술로 2차방정식의 수치해를 구하는 방법에 대한 소개 및 여기서 도출되는 몇 가지 ‘숫자 놀이’에 대한 글.
Published:
Asian mathematics
개평법
전통적인 산술로 2차방정식의 수치해를 구하는 방법에 대한 소개 및 여기서 도출되는 몇 가지 ‘숫자 놀이’에 대한 글.
Published:
Berkovich Analytification
Berkovich Line, part 3
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.
Published:
Berkovich Line, part 2
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.
Published:
Berkovich Line, part 1
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.
Published:
Calculus
What to say for IVT or EVT classes
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.
Published:
Discovering a nontriviality of the ε-δ definition, in a math way
This is a record of a talk that I made in the UChicago Math dept. Pedagogy seminar, in 2022-02-23.
Published:
Dedekind Completeness and Limits
Limit of a Sequence
Published:
중국인의 나머지 정리
부분분수 분해
Published:
Partial Fractions and Interpolations
Partial Fraction Decompositions
Published:
Newton Method for Tangents
Evaluating Arctangents
Published:
Completeness Axiom
Dedekind Completeness and Limits
Limit of a Sequence
Published:
Complex Analysis
A Formula of Hyperbolic Distance
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.
Published:
Complex Geometry
Elliptic curves of bidegree (2,2)
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.
Published:
Continued Fractions
Amortized Multiplicative Estimates of Remainders of Continued Fractions
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 $(\alpha_1\cdots\alpha_n)^{1/n} > \sqrt{2}$.
Published:
Diophantine approximation
Quantitative Baker Theorem and Rational Approximation of logarithms
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.
Published:
Dynamics
Cone Model for Vieta Actions on Tropical Markov Cubics
A comment to my recent work, regarding an alternative model for describing the “holomorphic parameters” case.
Published:
Topological Pressure, Szilard Engine
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.
Published:
Maxwell의 도깨비 관련 잡상
Maxwell의 도깨비, Landauer principle 등을 보고 떠오른 잡상雜想.
Published:
Elliptic Curve
Elliptic curves of bidegree (2,2)
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.
Published:
English
An Abstract Nonsense for Gödel Incompleteness
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.
Published:
Playing with the Ax–Grothendieck Theorem
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.
Published:
Cone Model for Vieta Actions on Tropical Markov Cubics
A comment to my recent work, regarding an alternative model for describing the “holomorphic parameters” case.
Published:
Elliptic curves of bidegree (2,2)
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.
Published:
Amortized Multiplicative Estimates of Remainders of Continued Fractions
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 $(\alpha_1\cdots\alpha_n)^{1/n} > \sqrt{2}$.
Published:
Miscellaneous Linear Algebra Things
Some trivia found while teaching Linear Algebra in the Fall 2022 quarter.
Published:
Topological Pressure, Szilard Engine
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.
Published:
What to say for IVT or EVT classes
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.
Published:
Quantitative Baker Theorem and Rational Approximation of logarithms
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.
Published:
Berkovich Line, part 3
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.
Published:
Berkovich Line, part 2
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.
Published:
A Formula of Hyperbolic Distance
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.
Published:
Berkovich Line, part 1
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.
Published:
Discovering a nontriviality of the ε-δ definition, in a math way
This is a record of a talk that I made in the UChicago Math dept. Pedagogy seminar, in 2022-02-23.
Published:
Dedekind Completeness and Limits
Limit of a Sequence
Published:
Partial Fractions and Interpolations
Partial Fraction Decompositions
Published:
Some Code Fragments
This is to record some Python macro that I have used to draw some TikZ figures:
Published:
Newton Method for Tangents
Evaluating Arctangents
Published:
How this blog works
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.
Published:
Extreme Value Theorem
What to say for IVT or EVT classes
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.
Published:
Graphics
Some Code Fragments
This is to record some Python macro that I have used to draw some TikZ figures:
Published:
History of Mathematics
「호몰로지」 용어에 대한 일고(一考)
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.
Published:
History of mathematics
개평법
전통적인 산술로 2차방정식의 수치해를 구하는 방법에 대한 소개 및 여기서 도출되는 몇 가지 ‘숫자 놀이’에 대한 글.
Published:
Hyperbolic geometry
A Formula of Hyperbolic Distance
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.
Published:
Intermediate Value Theorem
What to say for IVT or EVT classes
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.
Published:
Korean
Maxwell의 도깨비 관련 잡상
Maxwell의 도깨비, Landauer principle 등을 보고 떠오른 잡상雜想.
Published:
개평법
전통적인 산술로 2차방정식의 수치해를 구하는 방법에 대한 소개 및 여기서 도출되는 몇 가지 ‘숫자 놀이’에 대한 글.
Published:
「호몰로지」 용어에 대한 일고(一考)
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.
Published:
중국인의 나머지 정리 2
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.
Published:
중국인의 나머지 정리
부분분수 분해
Published:
Lagrange Polynomials
Partial Fractions and Interpolations
Partial Fraction Decompositions
Published:
Landauer Principle
Maxwell의 도깨비 관련 잡상
Maxwell의 도깨비, Landauer principle 등을 보고 떠오른 잡상雜想.
Published:
Lindenbaum Algebra
An Abstract Nonsense for Gödel Incompleteness
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.
Published:
Linear Algebra
Miscellaneous Linear Algebra Things
Some trivia found while teaching Linear Algebra in the Fall 2022 quarter.
Published:
Mathematical Logic
An Abstract Nonsense for Gödel Incompleteness
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.
Published:
Mathematical Pedagogy
What to say for IVT or EVT classes
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.
Published:
Maxwell's Demon
Maxwell의 도깨비 관련 잡상
Maxwell의 도깨비, Landauer principle 등을 보고 떠오른 잡상雜想.
Published:
Model Theory
Playing with the Ax–Grothendieck Theorem
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.
Published:
Newton's method
Newton Method for Tangents
Evaluating Arctangents
Published:
Non-archimedean Field
Berkovich Line, part 3
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.
Published:
Berkovich Line, part 2
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.
Published:
Berkovich Line, part 1
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.
Published:
Number Theory
중국인의 나머지 정리 2
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.
Published:
중국인의 나머지 정리
부분분수 분해
Published:
Number theory
Quantitative Baker Theorem and Rational Approximation of logarithms
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.
Published:
Numerical algorithm
개평법
전통적인 산술로 2차방정식의 수치해를 구하는 방법에 대한 소개 및 여기서 도출되는 몇 가지 ‘숫자 놀이’에 대한 글.
Published:
Order Completeness
Dedekind Completeness and Limits
Limit of a Sequence
Published:
Partial Fractions
중국인의 나머지 정리 2
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.
Published:
중국인의 나머지 정리
부분분수 분해
Published:
Partial Fractions and Interpolations
Partial Fraction Decompositions
Published:
Pedagogy of Limits
Discovering a nontriviality of the ε-δ definition, in a math way
This is a record of a talk that I made in the UChicago Math dept. Pedagogy seminar, in 2022-02-23.
Published:
Python
Some Code Fragments
This is to record some Python macro that I have used to draw some TikZ figures:
Published:
Szilard's engine
Topological Pressure, Szilard Engine
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.
Published:
Thermodynamics
Maxwell의 도깨비 관련 잡상
Maxwell의 도깨비, Landauer principle 등을 보고 떠오른 잡상雜想.
Published:
Topological pressure
Topological Pressure, Szilard Engine
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.
Published:
Topology
「호몰로지」 용어에 대한 일고(一考)
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.
Published:
Transcendental number
Quantitative Baker Theorem and Rational Approximation of logarithms
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.
Published:
Transfer Operator
Topological Pressure, Szilard Engine
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.
Published:
Translation
「호몰로지」 용어에 대한 일고(一考)
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.
Published:
Tropical Geometry
Berkovich Line, part 3
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.
Published:
Berkovich Line, part 2
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.
Published:
Berkovich Line, part 1
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.
Published:
Tropicalization
Cone Model for Vieta Actions on Tropical Markov Cubics
A comment to my recent work, regarding an alternative model for describing the “holomorphic parameters” case.
Published:
Elliptic curves of bidegree (2,2)
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.
Published:
Ultraproducts
Discovering a nontriviality of the ε-δ definition, in a math way
This is a record of a talk that I made in the UChicago Math dept. Pedagogy seminar, in 2022-02-23.
Published:
Upper half plane
A Formula of Hyperbolic Distance
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.
Published:
announcement
How this blog works
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.
Published:
p-adic Numbers
중국인의 나머지 정리 2
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.
Published: