 ## Top new questions this week:

### What is the minimal density of a set A such that A+A = N?

Thinking about the four square theorem and related questions, I found myself wondering: What is the minimal density of a set $A \subset \{0, 1, 2, ... \}$ such that $A + A = \mathbb{N}$? What I know: ... asked by Zur Luria Score of 22 answered by Emil Jeřábek Score of 38

### Recognizing free groups

While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...

gr.group-theory geometric-group-theory free-groups asked by ThorbenK Score of 14 answered by Carl-Fredrik Nyberg Brodda Score of 21

### Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Euler proved $$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$ where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...

nt.number-theory ca.classical-analysis-and-odes analytic-number-theory l-functions dirichlet-series asked by Nomas2 Score of 14 answered by GH from MO Score of 16

### Request for explicit character tables of conjectured, non-existent finite simple groups

In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...

reference-request gr.group-theory finite-groups ho.history-overview characters asked by Sebastien Palcoux Score of 12 answered by Dave Benson Score of 18

### How exactly are realizability and the Curry-Howard correspondence related?

Consider, on the one hand: the Curry-Howard correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$-calculus, and on the other, propositions and ...

lo.logic soft-question computability-theory type-theory realizability asked by Gro-Tsen Score of 12 answered by Andrej Bauer Score of 8

### Is the Tukey order well-founded

Consider the Tukey order restricted to directed orders of the form $(U,\supseteq)$, where $U$ is an ultrafilter on $\omega$. It is defined as follows: For two ultrafilters $U,W$ on $\omega$, we say ...

set-theory order-theory ultrafilters asked by Tom Benhamou Score of 11 answered by Gabe Goldberg Score of 10

### Can a nowhere locally Hölder function be differentiable almost everywhere?

Fix $0 < \alpha < 1$. Suppose $f$ is nowhere locally $\alpha$-Hölder continuous - that is, it is not $\alpha$-Hölder on any open subinterval of $\mathbb R$. Is it possible for $f$ to be ...

real-analysis asked by Nate River Score of 10 answered by user479223 Score of 12

## Greatest hits from previous weeks:

### Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

books ca.classical-analysis-and-odes big-list textbook-recommendation real-analysis asked by Ryan Score of 38 answered by lhf Score of 32

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...

soft-question ho.history-overview big-list mathematics-education online-resources asked by alex Score of 420 answered by Charles Siegel Score of 69

### "Philosophical" meaning of the Yoneda Lemma

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward. Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...

ct.category-theory intuition yoneda-lemma asked by Sam Derbyshire Score of 148 answered by Urs Schreiber Score of 102

### Why does the monster group exist?

Recently, I was watching an interview that John Conway did with Numberphile. By the end of the video, Brady Haran asked John: If you were to come back a hundred years after your death, what problem ...

gr.group-theory finite-groups monster asked by Leibniz's Alien Score of 37

### Which popular games are the most mathematical?

I consider a game to be mathematical if there is interesting mathematics (to a mathematician) involved in the game's structure, optimal strategies, practical strategies, analysis of the game ...

big-list soft-question recreational-mathematics game-theory asked by Douglas Zare Score of 115 answered by aorq Score of 96

### Theorem versus Proposition

As a non-native English speaker (and writer) I always had the problem of understanding the distinction between a 'Theorem' and a 'Proposition'. When writing papers, I tend to name only the main result(...

soft-question mathematical-writing asked by MRA Score of 64 answered by user1835 Score of 94

### Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...

reference-request big-list examples intuition alternative-proof asked by Mariano Suárez-Álvarez Score of 397 answered by Mariano Suárez-Álvarez Score of 552

## Can you answer these questions?

### Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$

Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...

measure-theory hilbert-spaces lebesgue-measure continuity positivity asked by Daniel Goc Score of 1

The Nielson realization theorem for a surface says that every finite subgroup of the mapping class group is realized by a finite subgroup of homeomorphisms on the surface. Furthermore, for a genus $g \... mapping-class-groups equivariant-cohomology equivariant-homotopy asked by Noah Wisdom Score of 2 ### Roth's Theorem Variations? One can motivate Roth's theorem as follows. On$[0,1]$consider the function$f$that takes$x$to the cardinality of the set {$p/q : |x - p/q | < 1/q^{2 + \epsilon}$} . Now one can see that$\...

nt.number-theory asked by David Feldman Score of 1
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