I have just finished the book “C Adams & R Franzosa – Introduction to Topology. Pure and Applied”. My aim is to reach to the level of the book “G E Bredon – Topology and Geometry”. Bredon’s book is not only too advanced to study after Adams’, but also I don’t think that it is […]

I tried to characterize mathematically what makes a function bell-shaped. I found the following: Definition. A $C^\infty$-function $f:\Bbb R\to\Bbb R$ is called bell-shaped if its $n$-th derivative has exactly $n$ zeros (counted with multiplicity) for all $n\in\Bbb N_0$. I do not claim that any intuitively bell-shaped curve is included (e.g. non-smooth functions), but I felt […]

I’ve heard that a resolvent is very useful in finding the roots of the polynomial. But I’m not sure what a resolvent even is. As much as I can figure out, it’s just another polynomial. But that makes no sense, because why would you want extra work? And how would you find the resolvent of […]

This question already has an answer here: Incredible frequency of careless mistakes 7 answers

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by $(e^{2i1\pi/n}, e^{2i2\pi/n},\dots, e^{2ik\pi/n},\dots,e^{2in\pi/n}, 0, 0, 0,\dots$). Now consider the set S of all such sequences, $$S=\left\{(e^{2i1\pi/n}, e^{2i2\pi/n},\dots, e^{2ik\pi/n},\dots,e^{2in\pi/n}, 0, 0, 0,\dots) : […]

For which odd integers $n>1$ is it true that $2n \choose r$ where $1 \le r \le n$ is odd only for $r=2$ ? I know that $2n \choose 2$ is odd if $n$ is odd but I want to find those odd $n$ for which the only value of $r$ between $1$ and $n$ […]

I was reading up on symmetric matrices and the textbook noted that the following is a remarkable theorem: A matrix $A$ is orthogonally diagonalizable iff $A$ is a symmetric matrix. This is because it is impossible to tell when a matrix is diagonalizable, or so it seems. I haven’t gotten to realize yet how important […]

I am interested in self-studying real analysis and I was wondering which textbook I should pick up. I have knowledge of all high school mathematics, I have read How to Prove It by Daniel J. Velleman (I did most of the excercises) and I have completed a computational calculus course which covered everything up to […]

I am a postgraduate student of mathematics from Slovenia (central Europe) with quite some experience in mathematics. While answering questions on this site, I often encounter the function $\sec(x)$ which is, as I understand, defined as $\sec(x) = \frac1{\cos x}$. During my studies, I never encountered this function. I am wondering two things: How widespread […]

I have been looking at various proofs of the IVT, and, perhaps the simplest I have encountered makes use of the Completeness Axiom for real numbers and Bolzano’s Theorem, which, honestly, I find a bit of an overkill. For an informal proof, we could write something like this: “If $f$ is continuous on $[a,b]$ then […]

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