Questions tagged [reference-request]

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

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Comparing fundamental groups of a complex orbifolds and their resolutions.

Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi_1(X)\cong \pi_1(\tilde X)$. ...
Dmitri Panov's user avatar
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17 votes
4 answers
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Good introduction to statistics from a algebraic point of view?

There are already lots of questions on this subject like Is there an introduction to probability theory from a structuralist/categorical perspective? Is there a combinatorial/topological treatment ...
doofin's user avatar
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17 votes
5 answers
2k views

Good source for representation of GL(n) over finite fields?

I'd like to gain some understanding of unitary representations of GL(n) over finite fields. Any good source would be appreciated. ======== edit ========= My original question was ambiguous. ...
user1258240's user avatar
16 votes
1 answer
4k views

Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum? $$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$ Additional information: Since $$ \sum_{\substack{p<n\\\text{...
Daniel Soltész's user avatar
16 votes
5 answers
3k views

Measure theory treatment geared toward the Riesz representation theorem

I'm looking for recommendations for books (or lecture notes) that develop measure theory in sufficient detail to state and prove the Riesz representation theorem (which is the characterization of the ...
Igor Khavkine's user avatar
15 votes
4 answers
3k views

Ordinary Generating Function for Bell Numbers

In the OEIS entry for Bell numbers, there appears a generating function $$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$ However, I could not locate any proof of ...
Amritanshu Prasad's user avatar
15 votes
0 answers
586 views

Precise form of the mean motion theorem

Consider an exponential polynomial $$f(t)=\sum_{k=1}^na_k\exp(i\lambda_kt),$$ where $a_k$ are complex and $\lambda_k, t$ real. The usual form of the Mean Motion Theorem says that the limit $$\lim_{t\...
Alexandre Eremenko's user avatar
15 votes
3 answers
974 views

Unit fraction, equally spaced denominators not integer

I've been looking at unit fractions, and found a paper by Erdős "Some properties of partial sums of the harmonic series" that proves a few things, and gives a reference for the following theorem: $$\...
mmm's user avatar
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15 votes
5 answers
3k views

Is it necessary that model of theory is a set?

From Model Theory article from wikipedia : "A theory is satisfiable if it has a model $ M\models T$ i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set $T$". ...
kakaz's user avatar
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15 votes
2 answers
709 views

$\mu$-presentable object as $\mu$-small colimit of $\lambda$-presentable objects

Remark 1.30 of Adámek and Rosický, Locally Presentable and Accessible Categories claims that in any locally $\lambda$-presentable category, each $\mu$-presentable object (for $\mu\ge\lambda$) can be ...
Reid Barton's user avatar
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15 votes
2 answers
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Obstruction theory for non-simple spaces

I'm looking for a good reference that has a detailed treatment of obstruction theory in the case where the target space is not simple. The specific situation I am interested in involves lifting a map ...
Evan Jenkins's user avatar
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15 votes
1 answer
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Multiplicative Structure of the Atiyah-Hirzebruch/Leray-Serre spectral sequence

This is related to this question (edit: now answered, see below). Is there a nice explanation of the multiplicative structure on the higher pages of that spectral sequence? I want to assume that $h$ ...
Sebastian Goette's user avatar
14 votes
3 answers
2k views

Convergence of L-series

I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function satisfies the ...
wood's user avatar
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14 votes
4 answers
2k views

Fourier decay rate of Cantor measures

For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
Syang Chen's user avatar
14 votes
2 answers
751 views

How big is the lattice of all functions?

Define the lattice $(\mathcal{L},\prec)$ as the set of all function $f:\mathbb{N}\rightarrow\mathbb{N}$ satisfying $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ ...
Jan-Christoph Schlage-Puchta's user avatar
14 votes
2 answers
941 views

"Fraïssé limits" without amalgamation

All structures are countable with countable signature. Given a structure $\mathcal{A}$, the age of $\mathcal{A}$, $Age(\mathcal{A})$, is the set of structures isomorphic to finitely-generated ...
Noah Schweber's user avatar
13 votes
1 answer
890 views

Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle? You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...
Alexey Ustinov's user avatar
13 votes
2 answers
1k views

What are Moschovakis cardinals?

The question is exactly that of the title: what are Moschovakis cardinals? Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency ...
Noah Schweber's user avatar
13 votes
1 answer
1k views

Reference for: CM Hilbert Modular forms arise from Hecke characters

For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
unramified's user avatar
13 votes
3 answers
1k views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...
Joonas Ilmavirta's user avatar
13 votes
6 answers
10k views

Frobenius number for three numbers

Given integers $a,b,c$ such that $\gcd(a,b,c) = 1$, it is well known that there exists only a finite set of numbers $n$ such that $n$ is not expressible as $ax+by+cz$ for non negative integers $x$,$y$,...
Jernej's user avatar
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13 votes
1 answer
696 views

Unpublished works of Woodin on SCH and Radin forcing

There are many unpublished results of Hugh Woodin on ''singular cardinals hypothesis'' and '' Radin forcing''. Some of his results are published later by others, but it seems that there are still many ...
Mohammad Golshani's user avatar
13 votes
5 answers
3k views

Reference request: Oldest calculus, real analysis books with exercises?

Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there. Edit. Unsolved exercises ...
12 votes
3 answers
752 views

Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?

The question is stated in the title of this post. It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
Iosif Pinelis's user avatar
12 votes
4 answers
1k views

What was Burroni's sketch for topological spaces?

In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni'...
Kevin Arlin's user avatar
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12 votes
1 answer
842 views

The dance marathon problem

In his book, "The Strange Logic of Random Graphs", Joel Spencer describes the "Dance Marathon" problem: Imagine $n$ couples at a Dance Marathon. Each dance each couple remains ...
Bill Bradley's user avatar
  • 3,809
12 votes
1 answer
2k views

Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ...
user avatar
12 votes
3 answers
656 views

IBN for algebraic theories

Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
Martin Brandenburg's user avatar
11 votes
2 answers
1k views

Yang–Baxter explanation

What are the most simple examples which can explain the meaning of Yang–Baxter equation? Is there any way to explain this mysterious object to a person who is not a professional in quantum groups? ...
Alexey Ustinov's user avatar
11 votes
1 answer
576 views

Formula for $U(N)$ integration wanted

Before you jump on the "duplicate" buttom, let me say that I do not want to hear about Weingarten calculus and I do not want to see a character of the symmetric group. What I would like is a formula ...
Abdelmalek Abdesselam's user avatar
11 votes
2 answers
1k views

Most dense subset of numbers that avoids arbitrarily long arithmetic progressions

The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression. I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid ...
Joseph O'Rourke's user avatar
11 votes
11 answers
1k views

Lattices on classical combinatorial families

I am asking for examples of lattices defined on classical combinatorial families, such as Permutations, Catalan objects, set partitions or integer partitions, graphs. I am mosty interested in lattices ...
Martin Rubey's user avatar
  • 5,533
10 votes
2 answers
780 views

Category theory from MK class theory perspective?

I'm looking for a text that treats category theory from the perspective of MK class theory. MK is already very well-designed and equipped for the type of abstraction that occurs in category theory, ...
Alec Rhea's user avatar
  • 8,977
10 votes
1 answer
822 views

This is not a dyadic cosine-product

The double-angle formula, $\sin2x=2\sin x\cos x$, turns the scary-looking integral $$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{2^k}$$ into fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\...
T. Amdeberhan's user avatar
10 votes
2 answers
1k views

Reference request: Oldest number theory books with (unsolved) exercises?

Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the ...
10 votes
1 answer
771 views

rational homotopy of a manifold

Given a finite dim rational homotopy type satisfying Poincaré duality, what is the best reference to when it is the rational homotopy type of a fin dim manifold?
Jim Stasheff's user avatar
  • 3,840
9 votes
2 answers
2k views

Doubly-transitive groups

I want to know what all doubly-transitive groups look like. Do you know some good reference where I can read about it?
Klim Efremenko's user avatar
9 votes
1 answer
1k views

Nonabelian $H^2$ and Galois descent

I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved. Let $k$ be a perfect field and $k^s$ its fixed separable closure. Let $X^s$ be a variety ...
Mikhail Borovoi's user avatar
9 votes
3 answers
2k views

Generalizations and relative applications of Fekete's subadditive lemma

Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
Salvo Tringali's user avatar
9 votes
4 answers
1k views

When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces)

The title has it all. I'm looking for a reference to the following: Q. Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the ...
Salvo Tringali's user avatar
9 votes
1 answer
1k views

Reference for push-pull formula in cohomology

I would like a precise reference for the following fact. Assume that $$ \begin{array}{ccc} M\times_B N & \stackrel{f'}{\to} & N \newline \quad\downarrow g' & & \quad\downarrow g \...
Mark Grant's user avatar
8 votes
2 answers
805 views

Reference for homotopy colimit = total complex

I'm looking for a reference for the following fact: take a simplicial chain complex $ X:\Delta^{op}\to Ch_{\geq 0}(\mathcal A)$ for $\mathcal A$ a nice abelian category (say, cocomplete with enough ...
Maxime Ramzi's user avatar
  • 13.3k
8 votes
2 answers
616 views

The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$

I'm playing with exponential sums... If $q$ is an odd prime and $a$ an integer such that $q \nmid a$, then the following formula for the Gaussian sum is known $$\sum_{x=0}^{q-1} e_q(ax^2) = \left(\...
user avatar
8 votes
1 answer
2k views

Density of smooth functions on Hölder spaces

The following result is often cited without reference in the context of PDEs: Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
Nautilus's user avatar
  • 717
7 votes
2 answers
1k views

Surjective entire functions without critical points

It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk. I am pretty sure that the answer to the ...
Moishe Kohan's user avatar
  • 9,664
7 votes
4 answers
2k views

Measure on real Grassmannians

OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix ...
Thierry Zell's user avatar
  • 4,536
7 votes
3 answers
2k views

Convex hulls of families of probability measures

Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$. In this paper for any family of probability ...
SBF's user avatar
  • 1,605
6 votes
1 answer
468 views

Sets of reals and absoluteness

Schoenfield's absoluteness states that if $\phi$ is $\Sigma^1_2$ then $V\models \phi$ iff $L\models \phi$. The set of reals in $L$ is $\Sigma^1_2$ and it is the largest countable $\Sigma^1_2$ set of ...
Rachid Atmai's user avatar
  • 3,756
6 votes
3 answers
430 views

Non-polynomial splines, a non-linear problem

I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials. To be specific, given a class of functions such as "decaying ...
gmvh's user avatar
  • 2,758
5 votes
1 answer
338 views

Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation $$ \partial_t u=\Delta u $$ ...
leo monsaingeon's user avatar

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