Articles of soft question

To find all odd integers $n>1$ such that $2n \choose r$ , where $1 \le r \le n$ , is odd only for $r=2$

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 $n$ […]

Most important Linear Algebra theorems?

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 […]

Books for starting with analysis

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 […]

Definition of the $\sec$ function

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 […]

Is it mathematically wrong to prove the Intermediate Value Theorem informally?

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 […]

Riemann Integral as a limit of sum

One of the standard definitions of Riemann Integral is as follows: Let $f$ be bounded on $[a, b]$. For any partition $P = \{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\}$ of $[a, b]$ and any choice of points $t_{k} \in [x_{k – 1}, x_{k}]$ the sum $$S(P, f) = \sum_{k = 1}^{n}f(t_{k})(x_{k} – x_{k – 1})$$ is […]

Connections between SDE and PDE

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \frac{1}{2} \Delta u$ with initial data $u(0,x)=f(x)$, considered on the whole space. Here the solution is given by $u(t,x)=\mathbb{E}(f(x+W_t))$ where $W_t$ is a […]

What's so special about $e$?

This question already has an answer here: Intuitive Understanding of the constant “$e$” 18 answers

Applications of the p-adics

As I was preparing to spend a month studying p-adic analysis, I realized that I’ve never seen the theory of p-adic numbers applied in other branches of mathematics. I can certainly see that the field $\mathbb Q_p$ has many nice properties, e.g. $X\subset Q_p$ closed and bounded implies $X$ compact, $\sum\limits_{n=0}^\infty x_n$ exists iff $\lim\limits_{n\to\infty}x_n=0$, […]

Software for generating Cayley graphs of $\mathbb Z_n$?

Does it exist any program (for linux) which can generate a nice Cayley graph of any $\mathbb Z_n$? (If it’s possible to create such a graph at all, that is.) (where perhaps $n ≤ 100$ or something like that)