# All Questions

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273answers
249k views

### Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
5answers
5k views

### Finding a 1-form adapted to a smooth flow

Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say ...
12answers
31k views

### Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain. I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...
107answers
78k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please. Motivation: I plan to use this list ...
63answers
29k views

### Important formulas in combinatorics

Motivation: The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...
6answers
6k views

2answers
4k views

### Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated 4/...
8answers
10k views

12answers
24k views

### How to solve $f(f(x)) = \cos(x)$?

I found the following equation on some web page I cannot remember, and found it interesting: $$f(f(x))=\cos(x)$$ Out of curiosity I tried to solve it, but realized that I do not have a clue how to ...
6answers
8k views

### Kahler differentials and Ordinary Differentials

What's the relationship between Kahler differentials and ordinary differential forms?
5answers
9k views

### What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
6answers
79k views

### Eigenvalues of matrix sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite? I am ...
2answers
8k views

### Does the curvature determine the metric?

Hello, I ask myself, whether the curvature determines the metric. Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that ...
3answers
17k views

### Which integers can be expressed as a sum of three cubes in infinitely many ways?

For fixed $n \in \mathbb{N}$ consider integer solutions to $$x^3+y^3+z^3=n \qquad (1)$$ If $n$ is a cube or twice a cube, identities exist. Elkies suggests no other polynomial identities are known. ...
7answers
5k views

### Asymptotic density of k-almost primes

Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is \pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\...

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