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.

Read moreStudy 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.

Read moreThe Fourier Transform of exp(-cx^2) and Its Convolution

We develop two almost straightforward way to compute the Fourier transform of $\exp(-cx^2)$, in the sense that any contour integration and the calculus of residues are not required at all. The first cool approach enables us to think about these elementary concepts much deeper, so I highly recommend to study this approach as long as you are familiar with ODE of first order.

Read moreA Detailed Proof of the Riemann Mapping Theorem

We offer a detailed proof of the Riemann mapping theorem, which states that every proper simply connected region is conformally equivalent to the open unit disc.

Read moreExamples in Galois Theory 3 - Polynomials of Prime Degree and Pairs of Nonreal Roots

In this episode we focus on the rational field. What can we know about the Galois group of an irreducible polynomial with prime degree? There is a method by counting the number of nonreal roots. From this, we obtain an algorithm to compute the Galois group.

Read moreExamples in Galois Theory 2 - Cubic Extensions

We study the Galois group of a cubic polynomial over a field with characteristic not equal to 2 and 3.

Read moreExamples in Galois Theory 1 - Complex Field is Algebraically Closed

We try to prove the fundamental theorem of algebra, that the complex field is algebraically closed, using as little analysis as possible. In other words, the following proof will be *almost* algebraic.

Read moreThe 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.

Read moreThe 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.

Read moreThe ring of real trigonometric polynomials

Throughout we consider the polynomial ring

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

Characters in Analysis and Algebra

In this post, we study the concept of character, what it is about in abstract harmonic analysis and how to use it Galois theory.

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