Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
1,711
questions
17
votes
4
answers
2k
views
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 $\pi_1(X)\cong \pi_1(\tilde X)$.
...
17
votes
4
answers
2k
views
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 ...
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. ...
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
$$ \sum_{\substack{p<n\\\text{...
16
votes
5
answers
3k
views
Measure theory treatment geared toward the Riesz representation theorem
I'm looking for recommendations for books (or lecture notes) that develop measure theory in sufficient detail to state and prove the Riesz representation theorem (which is the characterization of the ...
15
votes
4
answers
3k
views
Ordinary Generating Function for Bell Numbers
In the OEIS entry for Bell numbers, there appears a generating function
$$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$
However, I could not locate any proof of ...
15
votes
0
answers
586
views
Precise form of the mean motion theorem
Consider an exponential polynomial
$$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\...
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:
$$\...
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$". ...
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 ...
15
votes
2
answers
1k
views
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 ...
15
votes
1
answer
1k
views
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$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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$,...
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 ...
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,...
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'...
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 ...
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 ...
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 ...
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? ...
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 ...
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
...
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 ...
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, ...
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}$$
into fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\...
10
votes
2
answers
1k
views
Reference request: Oldest number theory books with (unsolved) exercises?
Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the ...
10
votes
1
answer
771
views
rational homotopy of a manifold
Given a finite dim rational homotopy type satisfying Poincaré duality,
what is the best reference to when it is the rational homotopy type of a fin dim manifold?
9
votes
2
answers
2k
views
Doubly-transitive groups
I want to know what all doubly-transitive groups look like. Do you know some good reference where I can read about it?
9
votes
1
answer
1k
views
Nonabelian $H^2$ and Galois descent
I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved.
Let $k$ be a perfect field and $k^s$ its fixed separable closure.
Let $X^s$ be a variety ...
9
votes
3
answers
2k
views
Generalizations and relative applications of Fekete's subadditive lemma
Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
9
votes
4
answers
1k
views
When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces)
The title has it all. I'm looking for a reference to the following:
Q. Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the ...
9
votes
1
answer
1k
views
Reference for push-pull formula in cohomology
I would like a precise reference for the following fact.
Assume that
$$
\begin{array}{ccc}
M\times_B N & \stackrel{f'}{\to} & N \newline
\quad\downarrow g' & & \quad\downarrow g \...
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 ...
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
$$\sum_{x=0}^{q-1} e_q(ax^2) = \left(\...
8
votes
1
answer
2k
views
Density of smooth functions on Hölder spaces
The following result is often cited without reference in the context of PDEs:
Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
7
votes
2
answers
1k
views
Surjective entire functions without critical points
It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk.
I am pretty sure that the answer to the ...
7
votes
4
answers
2k
views
Measure on real Grassmannians
OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix ...
7
votes
3
answers
2k
views
Convex hulls of families of probability measures
Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$.
In this paper for any family of probability ...
6
votes
1
answer
468
views
Sets of reals and absoluteness
Schoenfield's absoluteness states that if $\phi$ is $\Sigma^1_2$ then $V\models \phi$ iff $L\models \phi$. The set of reals in $L$ is $\Sigma^1_2$ and it is the largest countable $\Sigma^1_2$ set of ...
6
votes
3
answers
430
views
Non-polynomial splines, a non-linear problem
I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials.
To be specific, given a class of functions such as "decaying ...
5
votes
1
answer
338
views
Universal decay rate of the Fisher information along the heat flow
I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...