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2022
The Calculus of Fields - Absolute ValuesIrreducible Representations of SO(3) and the LaplacianStudy Irreducible Representations of SU(2) Using Fourier SeriesThe Fourier transform of exp(-cx^2)A Detailed Proof of the Riemann Mapping TheoremExamples in Galois Theory 3 - Polynomials of Prime Degree and Pairs of Nonreal RootsExamples in Galois Theory 2 - Cubic ExtensionsExamples in Galois Theory 1 - Complex Field is Algebraically ClosedThe Group Algebra of A Finite Group and Maschke's Theorem
2021
The ring of trigonometric polynomials with complex scalarsImplicit Mapping Theorem of Complex Variables and Riemann SurfacesCharacters in Analysis and AlgebraThe Banach Algebra of Borel Measures on Euclidean SpaceThe concept of generalised functions (distributions) and derivativesCurse of KnowledgeThe ring of real trigonometric polynomialsQuasi-analytic Classes(Kind of) Missing Content in Your Linear Algebra Class (Still on Progress)Dedekind Domain and Properties in an Elementary ApproachSeveral ways to prove Hardy's inequalityTensor Product as a Universal Object (Category Theory & Module Theory)Rings of Fractions and LocalisationThe Grothendienck Group
2020
A Continuous Function Sending L^p Functions to L^1A Continuous Functional Sending L^p Functions to L^1The Big Three Pt. 6 - Closed Graph Theorem with ApplicationsPartition of Unity on Different Manifolds (Part 1. Introduction)Stirling 公式的几种经典估计A proof of the ordinary Gleason-Kahane-Żelazko theorem for complex functionalsThe Big Three Pt. 5 - The Hahn-Banach Theorem (Dominated Extension)A long exact sequence of cohomology groups (zig-zag and diagram-chasing)The Fourier transform of sinx/x and (sinx/x)^2 and moreThe Riesz-Markov-Kakutani Representation TheoremThe Big Three Pt. 4 - The Open Mapping Theorem (F-Space)Introducing Riemann-Stieltjes IntegralBasic Facts of Semicontinuous FunctionsAn Introduction to Quotient Space关于《第三次忍受》The Big Three Pt. 3 - The Open Mapping Theorem (Banach Space)Cauchy sequence in group theoryA brief introduction to Fréchet derivativeAn elementary introduction to profinite groups线性 ODE(四)——常系数高阶线性微分方程的普遍解法A continuous function is 'mostly' nowhere differentiableThe Big Three Pt. 2 - The Banach-Steinhaus TheoremThe Big Three Pt. 1 - Baire Category Theorem ExplainedJordan 标准形的求法——以一个四阶矩阵为例矩阵 Jordan 标准形的原理More properties of zeros of an entire functionLinear ODE but Quotient SpacesProve Picard's Existence and Uniqueness Theorem on Different Levels线性 ODE(三)——初探高阶线性方程关于《有人弄乱了这束玫瑰花》线性 ODE(二)——用矩阵表示方程组线性 ODE(一)——线性方程的思想与一阶方程高等数学入门——理解无穷小的阶(微积分的“科学计数法”)The Lebesgue-Radon-Nikodym theorem and how von Neumann proved itCounterexamples of Fubini's theoremFubini's theorem in Euclidean space (Understanding 'almost everywhere')Topological properties of the zeros of a holomorphic function准素分解与准素标准型线性空间的循环分解线性变换不变子空间的导子及其性质线性空间的循环子空间与简单应用抽象 Lebesgue 积分的构建
2019
并不新颖的观点:不要追求生产力、效率了高等数学入门:上下确界及其极限形式调和级数发散的若干证明洛必达法则的几种不同的证明优美的 Fourier 级数(二): 有界差分下的收敛(Jordan 判别法)优美的 Fourier 级数(一): 绝对不只是求表达式的问题压缩定理和其应用