## Top new questions this week:

### Cardioid in coffee mug?

I've been learning about polar curves in my Calc class and the other day I saw this suspiciously $r=1-\cos \theta$ looking thing in my coffee cup (well actually $r=1-\sin \theta$ if we're being ...

(calculus) (polar-coordinates)

### Why are we justified in using the real numbers to do geometry?

Context: I'm taking a course in geometry (we see affine, projective, inversive, etc, geometries) in which our basic structure is a vector space, usually $\mathbb{R}^2$. It is very convenient, and also ...

(geometry) (analytic-geometry) (foundations)

### What does it mean when two Groups are isomorphic?

I'm not asking for the formal definition I know it. An isomorphism is a bijective homomorphism. In my book it's indicated many times when two groups are isomorphic, and I don't understand what's the ...

(abstract-algebra) (group-theory) (soft-question) (group-isomorphism)

### This infinitely nested root gives me two answers $\sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{...}}}}}}$

I am trying to evaluate $$\sqrt{4+\sqrt{8+\sqrt{32+\sqrt{512+\sqrt{\frac{512^2}{2}+\sqrt{...}}}}}}$$ where those numbers inside roots are $$a_{n+1}=\frac{a_n^2}{2}$$ And I found two ways to solve ...

(sequences-and-series)

### Are there $n$ groups of order $n$ for some $n>1$?

Denote $N(n)$ : the number of groups with order $n$. Can $N(n)=n$ hold for some $n>1$ ? I checked the OEIS-sequence as well as the squarefree numbers $n$ in the range $[2,10^6]$ and found no ...

(group-theory) (number-theory) (finite-groups)

### Can you determine a differential equation from its solutions?

A linear first-order differential equation has two solutions: $$y_1(x)=x^2 \\y_2(x)=\frac{1}{x}$$ Determine the differential equation I did some research and I think I can use the wronskian ...

(differential-equations)

### What is the motivation behind the study of sequences?

I was discussing some ideas with my professor and he always says that before you work on something in mathematics, you need to know the motivation for studying/working on it. A better way to put this ...

(sequences-and-series) (analysis) (motivation)

## Greatest hits from previous weeks:

### If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, ...

(probability) (statistics) (experimental-mathematics)

### A "simple" 3rd grade problem...or is it?

So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long ...

(algebra-precalculus) (recreational-mathematics)

### Using multiple integrals for tough single integrals

I'm just getting started on double integrals, and I recently saw the super cool way to use double integrals to arrive at $$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$ So, I am wondering if ...

(integration) (multivariable-calculus) (big-list)

### Galois' theory: fixed subfield formula.

In a homework dealing with Galois' theory, I am asked to prove the following standard statement, known as the fixed subfield formula: Theorem. Let $L$ be a field and $G$ be a finite subgroup of ...

(linear-algebra) (galois-theory) (extension-field)
Let's say I have a piecewise continuous function which has the fourier series $\sum_n\ c_{n}e^{inx}$ and I assume that $\sum_n\ n|c_{n}|$ converges, then I know the following holds: The fourier ...