Desvl's blog

Mathematics. Articles in English (et en français dans le futur).

2025

05-11 Equivalent Conditions of Regular Local Rings of Dimension 1

2023

11-11 The Structure of SL_2(F_3) as a Semidirect Product

10-21 A Separable Extension Is Solvable by Radicals Iff It Is Solvable

09-18 Picard's Little Theorem and Twice-Punctured Plane

08-12 SL(2,R) As a Topological Space and Topological Group

07-17 Artin's Theorem of Induced Characters

05-27 Chinese Remainder Theorem in Several Scenarios of Ring Theory

04-06 Projective Representations of SO(3)

03-20 The Quadratic Reciprocity Law

02-13 Vague Convergence in Measure-theoretic Probability Theory - Equivalent Conditions

2022

12-23 The Pontryagin Dual group of Q_p

12-20 The Haar Measure on the Field of p-Adic Numbers

12-05 Every Regular Local Ring is Cohen-Macaulay

12-02 The abc Theorem of Polynomials

11-24 A Step-by-step of the Analytic Continuation of the Riemann Zeta Function

09-22 Properties of Cyclotomic Polynomials

09-04 Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem

08-21 Hensel's Lemma - A Fair Application of Newton's Method and 'Double Induction'

08-12 Irreducible Representations of GL_2(F_q)

07-26 Segre Embedding And Heights

06-30 The Calculus of Fields - Absolute Values

06-16 Irreducible Representations of SO(3) and the Laplacian

05-08 Study Irreducible Representations of SU(2) Using Fourier Series

05-06 The Fourier Transform of exp(-cx^2) and Its Convolution

04-15 A Detailed Proof of the Riemann Mapping Theorem

03-29 Examples in Galois Theory 3 - Polynomials of Prime Degree and Pairs of Nonreal Roots

03-27 Examples in Galois Theory 2 - Cubic Extensions

03-16 Examples in Galois Theory 1 - Complex Field is Algebraically Closed

01-18 The Group Algebra of A Finite Group and Maschke's Theorem

2021

12-29 The ring of trigonometric polynomials with complex scalars

12-24 The ring of real trigonometric polynomials

10-10 Characters in Analysis and Algebra

10-02 The Banach Algebra of Borel Measures on Euclidean Space

08-16 The concept of generalised functions (distributions) and derivatives

07-24 Curse of Knowledge

07-08 Elementary Properties of Cesàro Operator in L^2

05-26 Left Shift Semigroup and Its Infinitesimal Generator

05-10 Quasi-analytic Vectors and Hamburger Moment Problem (Operator Theory)

03-30 Quasi-analytic Classes

02-23 (Kind of) Missing Content in Your Linear Algebra Class (Still on Progress)

02-07 Dedekind Domain and Properties in an Elementary Approach

01-23 Several ways to prove Hardy's inequality

01-19 Tensor Product as a Universal Object (Category Theory & Module Theory)

01-15 Why Does a Vector Space Have a Basis (Module Theory)

01-14 Rings of Fractions and Localisation

01-10 The Grothendienck Group

2020

12-22 Study Vector Bundle in a Relatively Harder Way - Tangent Bundle

11-28 A Continuous Function Sending L^p Functions to L^1

11-27 Study Vector Bundle in a Relatively Harder Way - Definition

11-13 The Big Three Pt. 6 - Closed Graph Theorem with Applications

11-08 Partition of Unity on Different Manifolds (Part 1. Introduction)

10-18 Stirling公式的几种经典估计

10-15 A proof of the ordinary Gleason-Kahane-Żelazko theorem for complex functionals

10-01 The Big Three Pt. 5 - The Hahn-Banach Theorem (Dominated Extension)

09-27 A long exact sequence of cohomology groups (zig-zag and diagram-chasing)

09-24 The Fourier transform of sinx/x and (sinx/x)^2 and more

09-19 The Riesz-Markov-Kakutani Representation Theorem

09-12 The Big Three Pt. 4 - The Open Mapping Theorem (F-Space)

09-03 The completeness of the quotient space (topological vector space)

08-26 Introducing Riemann-Stieltjes Integral

08-18 Basic Facts of Semicontinuous Functions

08-16 An Introduction to Quotient Space

08-14 关于《第三次忍受》

08-12 The Big Three Pt. 3 - The Open Mapping Theorem (Banach Space)

08-01 Cauchy sequence in group theory

07-31 A Brief Introduction to Fréchet Derivative

07-21 An elementary introduction to profinite groups

07-16 线性ODE（四）——常系数高阶线性微分方程的普遍解法

07-04 A continuous function is 'mostly' nowhere differentiable

06-30 The Big Three Pt. 2 - The Banach-Steinhaus Theorem

06-06 The Big Three Pt. 1 - Baire Category Theorem Explained

06-01 Jordan标准形的求法——以一个四阶矩阵为例

05-21 矩阵Jordan标准形的原理

05-20 More properties of zeros of an entire function

05-17 Linear ODE but Quotient Spaces

05-16 Prove Picard's Existence and Uniqueness Theorem on Different Levels

05-12 线性ODE（三）——初探高阶线性方程

05-07 关于《有人弄乱了这束玫瑰花》

05-04 线性ODE（二）——用矩阵表示方程组

05-01 线性ODE（一）——线性方程的思想与一阶方程

04-18 高等数学入门——理解无穷小的阶（微积分的“科学计数法”）

03-29 The Lebesgue-Radon-Nikodym theorem and how von Neumann proved it

03-23 Counterexamples of Fubini's theorem

03-22 Fubini's theorem in Euclidean space (Understanding 'almost everywhere')

03-14 Topological properties of the zeros of a holomorphic function

02-08 准素分解与准素标准型

02-03 线性空间的循环分解

01-28 线性变换不变子空间的导子及其性质

01-22 线性空间的循环子空间与简单应用

01-17 抽象Lebesgue积分的构建

2019

10-31 欧拉反射公式和利用Fourier级数的证明

10-02 并不新颖的观点：不要追求生产力、效率了

09-07 高等数学入门：上下确界及其极限形式

08-24 调和级数发散的若干证明

08-19 洛必达法则的几种不同的证明

08-06 优美的Fourier级数（二）: 有界差分下的收敛(Jordan判别法)

08-05 优美的Fourier级数（一）: 绝对不只是求表达式的问题

08-04 压缩定理和其应用