How to read a book in mathematics?
How is it that you read a mathematics book?Do you keep a notebook of definitions? What about theorems?Do you do all the exercises? Focus on or ignore the proofs?I have been reading Munkres, Artin,...
View ArticleInteresting ways to write 2023 [closed]
The year 2023 is near and today I found this nice way to write that number:$\displaystyle\color{blue}{\pi}\left(\frac{(\pi !)!-\lceil\pi\rceil\pi !}{\pi^{\sqrt{\pi}}-\pi...
View Article"Dimension" of $\mathbb{R}^3$
Suppose I have two linearly independent vectors $u$ and $v$ in $\mathbb{R}^3$. Taking the cross product $u \times v$ gives me another linearly independent vector, which we call $w$. Then the collection...
View ArticleIs it right to think of the grothendieck group $K(A)$ as the categorification...
Grothendieck $K$-groups make sense for exact categories I think, but let $k$ be a field and let's assume $A$ is a $k$-linear abelian category, which are supposed to 'categorify' vector spaces over $k$....
View ArticleNext book in learning Algebraic Number Theory
I have just finished the book Introductory Algebraic Number Theory by Kenneth S. Williams and Saban Alaca. My aim is to reach graduate level to do research, especially in one or more of the following...
View ArticleElementary results from Algebraic Number Theory
The purpose of this question is to motivate me to study algebraic number theory.Let me explain. My motivation for studying number theory is to learn about beautiful results with simple, accessible...
View ArticleSoft question: advice or hints on which specialty courses to take
So I'm a first-year math undergraduate right now. My university is technical (European uni, 3-year-program, don't know if it matters here), so its math program does not award me a Bachelor of Science...
View ArticleBooks for "hard" mathematical analysis
I really enjoy mathematical analysis, particularly the type that involves extensive estimates and calculations, often referred to as "hard" analysis in contrast to "soft" analysis .I am wondering if...
View ArticleWhy didn't Munkres write this Proposition? ("Analysis on Manifolds" by James...
I am reading "Analysis on Manifolds" by James R. Munkres.Theorem 11.3. Let $Q$ be a rectangle in $\mathbb{R}^n$; let $f:Q\to\mathbb{R}$; assume $f$ is integrable over $Q$.(a) If $f$ vanishes except on...
View ArticleIs there any Richard Feynman parallel in mathematics world?
This is a soft question. I was studying the Red Book by Richard Feynman which consists of his lectures which he delivered in 60's. We all know that what a brilliant teacher an explainer he was....
View ArticleHow does Stewart & Tall's Complex Analysis compare to more commonly used texts?
I am trying to find a good complex analysis text for self-study and I was wondering what people think of Ian Stewart and David Tall's Complex Analysis. I was trying to find some reviews on this forum...
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Shape with dimension based on zooming level
This question was inspired by a section in the 3blue1brown video on fractal dimension, where it is described how the box-counting dimension can change based on zoom level.I have no idea which topic...
View ArticleDifference between $R(z,\omega)dz$ and $R(z,\omega)$
I heard a class about Riemann's work on Abelian integral. However, he consistently used different notations for the integrand $R(z,\omega)dz$ and the irrational formula $R(z, \omega)$. Are integrands...
View Article2$?">Which sets of functions differentiate in a "cyclic ring" of order $N>2$?
When we learn about differentiation we often simultaneously learn about the exponential function and the trigonometric functions.Differentiation is linear, and so we can represent it using the language...
View ArticleSurprising applications of topology
Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The...
View ArticleComplex differential forms in Riemann's era
I took a class on Riemann's work on Abelian integrals $\int R(z,\omega)dz$. In that class, I learned that Riemann distinguished between $R(z,\omega)$ and $R(z,\omega)dz$ and pursued their research in...
View ArticleHow much of mathematics is computation and how much of it is creativity?...
Likewise the title, I am interested to know about the nature of mathematics. More specifically, I want to know just how much we can churn out of formulas and vice versa without any reasonable amount of...
View ArticleIs the question of Lebesgue measurability of projective sets “leaning” one...
A lot of independence results (Continuum Hypothesis, existence of Suslin trees, etc.) can seem (to my mind anyway) a bit “esoteric”, i.e. not impacting in an obvious way to “normal” mathematics (areas...
View ArticleBooks with SAGE portions
I recently finished working through Adventures in Group Theory and really appreciated the use of SageMath it employs. I considered myself moderately proficient with Sage, but I found working through...
View ArticleFor the periodic sequence, is there always an algebraic closed form?
This question is a generalized form of the problem I asked before:Algebraic Closed Form for $\sum_{n=1}^{k}\left( n- 3 \lfloor \frac{n-1}{3} \rfloor\right)$Let, look at this periodic...
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