Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
64 questions from the last 30 days
23
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13
answers
5k
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Great theorems with elementary statements: 2026-onward
My 2021 book
Landscape of 21st Century Mathematics, Selected Advances, 2001–2020
collects great theorems with elementary statements published in 2001-2020. I now finishing the second edition of this ...
14
votes
1
answer
357
views
For which $q$ is the value of the chromatic polynomial of every planar graph at $q$ positive?
It is famous that all integers not less than 4 enjoy this property and bit less but also famous that so does $(5+\sqrt{5})/2$. I expect that this matter was studied a lot and ask about the state of ...
- 115k
12
votes
1
answer
458
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Reference request for a proof of Cayley's tree counting formula via the representation theory of the symmetric group
Inspired by a recent project Euler problem, I came up with a proof (sketched below) of Cayley's tree formula using the representation theory of $S_n$. I would like to ask for a reference in the ...
- 123
9
votes
1
answer
434
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Hoàng Xuân Sính's 2-group classification for 3 groups
I have a 3-category in which I am trying to compute the homotopy 3-type of its core. For each object $C \in \mathcal{C}$, I have computed the following groups:
$$\pi_1(\mathcal{C},C)=\{\text{1-...
5
votes
1
answer
377
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Selberg's result on primes in short intervals
A result of Selberg (A. Selberg. On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid., 47(6):87–105, 1943) says essentially
$$\int ...
- 1,656
3
votes
3
answers
338
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Weakening of the Idoneal Number condition
This is about the following Math.SE Q&A: "What is the largest integer $d$ such there is a congruence relation on primes p so that $x^2 + dy^2 = p^2$ has a non-zero integral solution?".
...
- 26.6k
6
votes
1
answer
394
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Square-free integers: direct approaches vs. sums of $\mu(n)$
Let $Q(x)$ denote the number of square-free positive integers $\leq x$. Let $R(x) = Q(x) - \frac{x}{\zeta(2)}$.
There is a long literature (starting with Axer (1911) and continuing, starting ca. 1980, ...
- 22k
8
votes
1
answer
462
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Variant of van der Corput's kth-derivative estimate
I ask this question as a non-expert in analytic number theory who needs to use a number theory result. Lemma 2.2 of this paper by Chan, Kumchev and Wierdl reads as follows:
Lemma 2: Let $k \ge 2$ be ...
- 13.1k
4
votes
1
answer
356
views
Ramanujan's work on the central factorial numbers
Background
The central factorial numbers are described on OEIS sequence A008955. Among the references, "Ramanujan's notebooks, part 1" (edited by Bruce Berndt) is listed. Upon checking this ...
- 5,065
5
votes
1
answer
345
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Minkowski sum of finite ellipses
Let $(E_n)_n$ be any finite collection of centred ellipses in $\mathbb{R}^2$. Suppose that $E_n$ are pairwise non-homothetic (i.e. there is no positive constant $c>0$ such that $E_n = c E_m$). Now ...
- 287
5
votes
1
answer
481
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Examples of "Cell divisions" in mathematics?
I stumbled upon a mathematical structure, which I would describe as a cell division from biology, while researching prime factorization trees:
The image show a cell division:
blue = Growth of classes,...
- 3,761
7
votes
1
answer
185
views
The relation between Temperley-Lieb algebra and representations of $U_q \mathfrak{sl}_2$
What is a good modern reference (or maybe this is known to someone here and lacks a good reference) which explains the relation between the Temperley-Lieb algebra and representations of $U_q(\mathfrak{...
- 3,464
6
votes
1
answer
508
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Which exact sections of Higher Topos Theory are required for “Higher Algebra”?
I am currently reading Lurie’s “Higher Topos Theory” as a prerequisite for his book “Higher Algebra”. At the start of HA he mentiones that an exposition to $\infty$-categories is required and cites ...
- 161
4
votes
1
answer
307
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Testing equal count between pairs of sets
For each fixed positive integer $N\in\mathbb{N}$, let's define two sets
\begin{align}
A_N:=&\{(a,b)\in\mathbb{N}^2: N=a(2b-1)+(2a-1)(b-1)\}, \\
B_N:=&\{(c,d)\in\mathbb{N}^2: N=c(2d-1)+(d-1)(d-...
- 43.9k
8
votes
1
answer
364
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Coefficient comparison between a double sum and a single sum
For a formal Laurent series $F(q)$, denote its coefficient of $q^j$ by $[q^j](F)$.
QUESTION. For integers $r\geq1$, is this true?
$$[q^{2r}]\sum_{n\geq1}\frac{q^n}{1-q^{2n}}\sum_{k=1}^n\frac{q^k}{1+q^...
- 43.9k