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Results tagged with reference-request
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user 290
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 …
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{ …
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.
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 …
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 $a_i \in \mathbb{Z}$, …
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 …
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 …
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 …
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 …
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."
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 …
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 …
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, …
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 …