Articles of soft question

Hillary Clinton's Iowa Caucus Coin Toss Wins and Bayesian Inference

In yesterday’s Iowa Caucus, Hillary Clinton beat Bernie Sanders in six out of six tied counties by a coin-toss*. I believe we would have heard the uproar about it by now if this was somehow rigged in her favor, but I wanted to calculate the odds of this happening, assuming she really was that lucky, […]

(Soft) What maths should I concentrate on at 16-18 years old?

Some background information first of all: I’m 16 now and just started studying mathematics intensely. I live in the UK and my goal is to eventually become very good at advanced mathematics (graduate level and up) but that is still far in the future. I’m about a year or slightly more ahead from the expected […]

Applications of the formula expressing roots of a general cubic polynomial

I know that the mathematics related to finding the general formula by expressing the roots of a third (and fourth) degree polynomial by means of radicals has had an impressive impact on mathematics (complex numbers and group theory, just to say). However: are there any useful application of these formulas? My impression is that calculus […]

Exponential fields as structures with three binary operations.

The exponential rings and fields are usually studied as structures with two binary operations $(+,\cdot)$ and one unary operation $\exp(x)$ defined on a set $K$. Why not consider the exponential as a binary operation $\star : K\times K\rightarrow K\;,\quad x\star y=x^y$ and search for properties of a structure with three binary operations with suitable axioms? […]

New Elementary Function?

In the February 2000 issue of FOCUS magazine, a short article suggests that the Lambert W function could be introduced into curriculum as a new elementary function saying: “… a case can be made for according it equal respect with the traditional transcendentals of calculus.” As the inverse of $xe^x$, Lambert W is easy to […]

Sequence Notation — Which brackets to use?

I’m teaching sequences at the moment. I’ve always put sequences in round brackets, for example $(1,2,3,4,5)$ is a sequence whose first member is $1$, whose second member is $2$, and so on. I’ve also always used round brackets to define a sequence in the following way: “Consider the sequence $(a_n)$ where $a_k = k^2+1$ for […]

What's the deal with integration?

So at uni we learned tricks and techniques for integration until cows came home. But to what end? Any/All definite integrals can be evaluated using numerical methods. Most integrals in application can not even be evaluated in elementary terms anyway. So is integration like calligraphy?, where it is pretty to do/look at but a printer […]

Which are the mathematical problems in non-standard analysis? (If any)

I would like to learn non-standard analysis, at least the basics of it. I will make use of this book: Elementary Calculus: An Infinitesimal Approach (Dover Books on Mathematics), by H. Jerome Keisler. Before anything else, please let me take up some links that DO NOT have a fully fleshed out answer to my question, […]

How to write well in analysis (calculus)?

This is kind of a subjective question, I know; often I find myself failing exams and homeworks because of the way i write down proofs. Either I don’t know how to start, or somehow the main point of the proof is lost. I’ve noticed that in many books there’s an “style” but doesn’t matter how […]

Usefulness of induced representations.

I am learning representation theory from Serre’s book by myself. Currently I am reading about induced representations, but I don’t understand the importance. The concept looks strange and the definition appears quite complicated compared to the topics discussed before it. Can someone briefly exposit its importance?