Desvl's newsletter
By Desvl
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Blog Update - Hensel's Lemma
In this post, we study the so-called Hensel's lemma on local fields, and you will see how we find a root of a polynomial using Newton's iteration method. As a handful example, one can find a square root of an integer in a suitable field.
Blog Update - Irreducible Representations of GL_2(F_q)
Other than symmetric groups, alternating groups, and dihedral groups, there is another important class of finite groups: matrix groups over finite fields. In this post, we study the simplest non-trivial one: general linear group of finite field F_q of 2x2 mat…
Blog Update - Segre Embedding and Heights
Greetings, here are some introductory algebraic geometry content. We give a quick look at the Segre embedding and the concept of height (of an algebraic number or polynomial) and see how the Segre embedding can be used here.Enjoy!
Blog Update - Absolute Values on a Field (Number theory)
Greetings. In today's blog post we study absolute values on a field, not just Q or R or C. This will enable us to make use of calculus more easily in the future.Enjoy.
Blog update - Irreducible Representations of SO(3)
We found a way to characterise the irreducible representations of SO(3), through spherical harmonics. The way of doing it involves extensive applications of linear algebra so this is a good chance to attest to your skill in linear algebra if you want to.Enjoy!
Let's study irreducible representations of SU(2) in a mathematician's way
In this post, irreducible representations of SU(2) is studied. SU(2) plays an important role in physics, but instead of interpreting it physically, we explore the system purely mathematically. And you will have to recall many fundamental concepts in linear al…
How to compute the Fourier transform of exp(-cx^2)?
You can see the method in the following blog post. Trying to get the result straightforward is somewhat unrealistic, but here is a cool way. No contour integration, residue theorem is needed. We obtain our result by integration by parts and a simple different…
Blog Update - A Detailed Proof of the Riemann Mapping Theorem
The Riemann Mapping theorem is very important in complex analysis. But the proof is not easy. In the latest blog post we present a carefully explained proof. Also, it is a good chance to attest to the reader's skill in complex analysis. Hope it can help you i…
Blog Update - Polynomials of Prime Degree and Pairs of Nonreal Roots (Examples in Galois Theory)
Consider an irreducible polynomial over the rational field and of prime degree. What is its Galois group? We have one way to determine some of the polynomials if the number of nonreal roots are given. This example gathers many key facts of finite group theory…
Blog Update - Cubic Extension of A Field (Examples in Galois Theory)
In this issue we present a good example of Galois extension. Let K be a field whose characteristic is not 2 and 3. What is an irreducible polynomial over K? What is its Galois group? We will find out in this post. I hope this post can help you if you are stru…
Blog Update - The Fundamental Theorem of Algebra (Examples in Galois Theory)
You may also enjoy this video featuring a famous mathematician, David Eisenbud. You may know him (one day) about his Commutative Algebra textbook.
Blog update - Group Algebra and Maschke's Theorem
It has been quite some time before my last blog post. In this post we deal with some pure algebra matter. Group algebra, semisimple ring, representation theory. They are connected but not usually studied altogether. In this post I try to put them altogether t…
Blog update - The Ring of Trigonometric Polynomials with Complex Scalars
Finally there is another newsletter issue. I've been thinking about the content for a long time but I didn't start writing until the end of 2021. And this is only a fraction of what I have been thinking about. But I need some time to dive in further. Wish eve…
Blog update - The Banach Algebra of Borel Measures on Euclidean Space
It turns out this is not a post on multilinear algebra... Instead this is can be considered as some introductory abstract harmonic analysis. If you are interested in this topic I hope you can enjoy (or you can treat it as some exercise solutions).
Forecast of Future Issues - Multilinear Algebra
There will be a series of new contents on multilinear algebra later this year. Multilinear algebra is in an embarrassing position. It is not universally taught in linear algebra while it is required in other courses. It would not be too long, but I will cover…
Blog update - The concept of generalised function (distribution) and derivative
This is what to happen if we want to differentiate a function without considering whether it is differentiable. We study a Fields medal winning theory - distribution. There are two parts. Part 1 is prepared for everyone having finished calculus and linear alg…
Blog update - On the ring R[cos x, sin x]
If we study the ring R[cos x, sin x], we will get a lot of unexpected but satisfying results. Keep in mind that I assume the reader has finished Atiyah-MacDonald or equivalence before reading.
Blog update - Elementary Properties of Cesàro Operator in L^2
If you are interested in classic functional analysis then this blog post may interest you
Blog update - Left shift semigroup and further properties
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