Desvl's blog

Irreducible Representations of SO(3) and the Laplacian

In this post, irreducible representations of $SO(3)$ are studied, with much more extensive applications of linear algebra.

2022-06-16 Algebra Representation Theory Linear Algebra Lie Groups

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Irreducible Representations of SO(3) and the Laplacian

Study Irreducible Representations of SU(2) Using Fourier Series

$SU(2)$ has a lot of interesting mathematical and physical properties. In this post we study its irreducible representations in a mathematician's way.

2022-05-08 Analysis Algebra Representation Theory Linear Algebra Fourier Series Lie Groups

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Study Irreducible Representations of SU(2) Using Fourier Series

The Group Algebra of A Finite Group and Maschke's Theorem

In this post we give several forms of Masher's theorem by studying group algebra, which eventually becomes a study of semisimple rings. One can consider this post a chaotic evil introduction to representation theory or something.

2022-01-18 Algebra Representation Theory Group Algebra Non-Commutative Ring

# Study new content # Serge Lang # Axiom of Choice # Division Ring # J-P Serre

The Group Algebra of A Finite Group and Maschke's Theorem

The ring of trigonometric polynomials with complex scalars

This post is a continuation of a previous post about the ring of trigonometric polynomials over the real field. Now we have jumped into the complex field, and the extension is not a trivial matter.

2021-12-29 Algebra Commutative Algebra Geometry Algebraic Geometry

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The ring of trigonometric polynomials with complex scalars

The ring of real trigonometric polynomials

The ring

Throughout we consider the polynomial ring

$R=\mathbb{R}[\cos{x},\sin{x}].$

This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory.

2021-12-24 Algebra Commutative Algebra

# Study new content # Noetherian

The ring of real trigonometric polynomials

The concept of generalised functions (distributions) and derivatives

In this post, we study the concept of generalised functions (a.k.a. distributions), and let's see how to evaluate the derivative no matter the function is differentiable or not.

2021-08-16 Analysis Functional Analysis

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The concept of generalised functions (distributions) and derivatives

The completeness of the quotient space (topological vector space)

2020-09-03 Analysis Functional Analysis Topological Vector Space

# Walter Rudin # Functional Analysis # Study new content

The completeness of the quotient space (topological vector space)

An Introduction to Quotient Space

2020-08-16 Analysis Functional Analysis Topological Vector Space

# Exercise solution # Walter Rudin # Functional Analysis # Study new content

An Introduction to Quotient Space