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This tag is used if a reference is needed in a paper or textbook on a specific result.

3 votes

Integer numbers of the form $m = x^n + y^n$

This is not an answer but a series of comments. Here's an obstacle stopping you from straightforwardly generalizing the classical based on the Gaussian integers to, say, sums of two cubes. The prob …
Qiaochu Yuan's user avatar
18 votes
Accepted

Wrong-way Frobenius reciprocity for finite groups representations

The point is that there are two ways to describe the restriction functor. The first way is as $\text{Hom}_{\mathbb{C}[G]}(\mathbb{C}[G], -)$, thinking of $\mathbb{C}[G]$ as a $(\mathbb{C}[G], \mathbb{ …
Qiaochu Yuan's user avatar
5 votes
Accepted

Transcendence of $\log 2$

This follows from the Lindemann-Weierstrass theorem. There is a sketch of a proof and several references at the Wikipedia article.
Qiaochu Yuan's user avatar
1 vote

reference request: variations on Pascal's triangle

You are just counting walks of various lengths starting from the leftmost vertex of a path graph $P_{N+1}$ with $N+1$ vertices. It is possible to explicitly write down the eigenvectors and eigenvalues …
Qiaochu Yuan's user avatar
8 votes
Accepted

Reference request : an elementary product-sum formula for binomial coefficients

$\binom{X}{a}\binom{X}{b}$ is the number of ways to choose a subset of size $a$ and a subset of size $b$ from a set of size $X$. The union of these two subsets is a subset of size anywhere from $\text …
5 votes
1 answer
448 views

Does this problem have a name? [Ducci Sequences]

Let $a_1, ... a_n$ be real numbers. Consider the operation which replaces these numbers with $|a_1 - a_2|, |a_2 - a_3|, ... |a_n - a_1|$, and iterate. Under the assumption that $a_i \in \mathbb{Z}$, …
Qiaochu Yuan's user avatar
6 votes

Formula for the entry of a matrix power

I agree with LSpice that I don't think this really needs a proof or a citation, but in combinatorics this sort of thing is often called "the transfer matrix method" and accordingly it is stated in com …
Qiaochu Yuan's user avatar
12 votes

The free group $F_2$ has index 12 in SL(2,$\mathbb{Z}$)

Maybe this works and maybe it doesn't: the index of the congruence subgroup $\Gamma(2)$ in $\text{SL}_2(\mathbb{Z})$ is $|\text{SL}_2(\mathbb{F}_2)| = 6$, and $\Gamma(2)$ contains a free subgroup of i …
Qiaochu Yuan's user avatar
3 votes

Additive functors to abelian groups: "additional structure" and functors induced by "additiv...

You want to look up Morita theory for enriched categories. By a "linear category" I will mean an $\text{Ab}$-enriched category. Write $\widehat{A}$ for the category of presheaves of abelian groups o …
Qiaochu Yuan's user avatar
5 votes
Accepted

Seeking more information regarding the "rigoidal category" of $\mathbb{N}$-graded sets

If $M$ is any monoidal category, the presheaf category $[M^{op}, \text{Set}]$ inherits a monoidal structure given by Day convolution. It is uniquely determined by the condition that it restricts to th …
Qiaochu Yuan's user avatar
13 votes
Accepted

Reference for "multi-monoidal categories"

Look at Section 3 of Leinster's Higher Operads, Higher Categories, where the term used is "unbiased monoidal category."
Qiaochu Yuan's user avatar
3 votes
Accepted

Exact sequences of pointed sets - two definitions

The basic reason for the appearance of 1) in the long exact sequence in homotopy is that it is exactly the kind of exactness you get if you apply $\pi_0$ to a fiber sequence $F \to E \to B$ of pointed …
Qiaochu Yuan's user avatar
8 votes

Inverse problem of Chern Classes

Suppose $M$ is a closed oriented $2n$-manifold which admits a complex spin ($\text{Spin}^c$) structure, which means that its second Stiefel-Whitney class $w_2(M) \in H^2(M, \mathbb{Z}_2)$ is the $\bmo …
Qiaochu Yuan's user avatar
6 votes
Accepted

Finitely generated subrings of $\mathbb{R}$ are finitely approximable

$A$ is a finitely generated integral domain. By the Nullstellensatz, its Jacobson radical vanishes (because its nilradical vanishes), meaning every nonzero element $a \in A$ avoids some maximal ideal, …
Qiaochu Yuan's user avatar
1 vote

Deformations of a complex trivial up to quasi-isomorphism

In general we can consider a differential $d_t$ which depends polynomially on $t$ and whose value at $t = 0$ is our original differential. Simple examples show that we can pick up extra cohomology at …
Qiaochu Yuan's user avatar

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