Onto group actions and symmetries.
I was trying to explain informally to someone how group actions encode symmetries of an object $X$. In general the data of a group action is a group homomorphism $f:G \to \text{Aut(X)}$, and just take...
View ArticleBook series like AMS' Student Mathematical Library?
I had the joy of discovering AMS'Student Mathematical Library book series today, and I have been pleasantly surprised by how enticing some of the titles seem: exciting and expositionary, a perfect...
View ArticleWhy use the word axiom rather than property?
In the definition of a vector space, one often uses the word axiom to say that anything that satisfies the given axioms is a vector space. But the word axiom usually refers to a statement that is...
View Article99.99999999999% of chance something will happen, and it didn't. [closed]
Suppose someone said there was a 99.99999999999% chance something would happen in a given specific day, and then we waited and it did not happen in that day after all. We cannot conclude he was wrong,...
View ArticleAdvice and supplementary sources (videos, books) to tackle rudin's real and...
I just started grad school and I have a mandatory class in Calculus and Probability.Professor's plan is to use baby rudin (Principles of Mathematical Analysis) and papa rudin (Real and Complex...
View ArticleHow to do math research? [closed]
When doing research related to math, should I first start by reading papers, doing a literature review, and trying to understand the literature first, or should I start working with my proposed...
View ArticleWhat exactly is a mathematical statement?
A mathematical statement, so I have heard, is a statement that is either true or false but not both. My initial source for this is not a book on the philosophy of mathematics; instead, it is a...
View ArticleProofs of the Volume of a Sphere.
I was asked to explain why the volume of a sphere is $\frac{4}{3}\pi r^3$ to a student that does not have the knowledge of calculus. In doing so I thought of an argument and I cannot seem to find that...
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Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition...
Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition behind)In this other question I did about Bump functions, other user told in an answer that these kind of functions "tends...
View ArticleWhat Are Some Series One Could Do a Whole Project On? [closed]
I’m in the planning process for an 8-month project on mathematical series, their theories, and proofs behind them. I want to know what kinds of series I could make a detailed thesis and paper on with...
View ArticleThe number of combinations that a regular icosahedron can have for rigid motion
I read Lectures on the Ikosahedron and the Solution of Equations of the Fifth Degree, by Felix Klein pp.26-30 If you read this part, you can see the paragraph like below.We then have the proposition,...
View ArticleWhat makes $\frac{1}{z}$ special in complex residues?
I'm going back over some introductory complex analysis, and I'm trying to find an intuitive notion of what makes the $\frac{1}{z}$ term special when calculating complex closed line integrals of...
View ArticleWhich subsets of the symmetric group $S_n$ can be generated by $log_2(n)$...
I am interested in the properties of all members of the symmetric group $S_n$, where $n$ is a power of 2 ($n = 2^l$ for $l \in \mathbb{N}$), which can be generated by $l=log_2(n)$ levels of pairwise...
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An approximation to $\ln(1+e^x)$ and how to use it for splitting the...
Trying to split the logarithm of the sum of two exponential functions (this question), I found the following approximation for the Softplus function$f(x)=\ln(1+e^x)$:$$\ln(1+e^x) \approx \begin{cases}...
View ArticleWhere do Mathematicians Get Inspiration for Pi Formulas?
Question:Where do people get their inspirations for $\pi$ formulas?Where do they begin with these ideas?Equations such as$$\dfrac 2\pi=1-5\left(\dfrac 12\right)^3+9\left(\dfrac...
View ArticleWhat is the importance of the Collatz conjecture?
I have been fascinated by the Collatz problem since I first heard about it in high school.Take any natural number $n$. If $n$ is even, divide it by $2$ to get $n / 2$, if $n$ is odd multiply it by $3$...
View ArticleResources to Practice Counting Problems in Group Theory
I'm studying elementary group theory in one of my courses. As a class, we have finished almost all of the exercises in Chapter 2 of Artin's Algebra. I noticed that I have substantially greater...
View ArticleVery good linear algebra book.
I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...),...
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How are demons relevant to the Grothendieck-Riemann-Roch theorem?
The following illustration appears on the Wikipedia page for the Grothendieck-Riemann-Roch theorem, as Grothendieck's comment on the theorem.Why is the Grothendieck–Riemann–Roch theorem in hell?
View ArticleChoosing a Research Focus in Probability Theory with a Theoretical Physics...
Even if this seems like an opinion-based question, please don't delete it. It might be very helpful for a lot of other people.I’m currently reading Probability and Measure Theory by Robert Ash. Since I...
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