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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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22 votes
11 answers
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Maxwell's equations and differential forms

Is there a textbook that explains Maxwell's equations in differential forms? What I understood so far is that the $E$ and $B$ fields can be assembled to a 2-form $F$, and Maxwell's equations can be ...
21 votes
2 answers
2k views

Applications of number theory in dynamical systems

I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics. ...
J W's user avatar
  • 760
21 votes
5 answers
1k views

Is there a midsphere theorem for 4-polytopes?

The (remarkable) midsphere theorem says that each combinatorial type of convex polyhedron may be realized by one all of whose edges are tangent to a sphere (and the realization is unique if the center ...
Joseph O'Rourke's user avatar
20 votes
1 answer
2k views

Does every compact metric space have a canonical probability measure?

Edit: Shortly after this post it was rightly pointed out by @AntonPetrunin that the measure $\mu$ may not be unique. @R W then showed how one can construct a metric space where the limiting measure is ...
M. Kelly's user avatar
  • 203
19 votes
2 answers
2k views

Who first proved the generalization of Bertrand's postulate to (2n,3n) and (3n,4n)?

In Wikipedia's page for Bertrand's postulate, it is said that its (2n,3n) version was proved by El Bachraoui in 2006. Seems likely that it was first proved way before than that! Can anyone point to ...
Jose Brox's user avatar
  • 2,992
19 votes
2 answers
2k views

Dual versions of "folding" symmetric ADE Dynkin diagrams?

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE diagrams ...
Jim Humphreys's user avatar
18 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
18 votes
2 answers
840 views

Reference to a conjecture on unit vectors in Euclidean space

I have heard that there exists the following conjecture (if I am not mistaken). Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists another unit vector ...
asv's user avatar
  • 21.8k
18 votes
1 answer
5k views

Best strategy for small resolutions

I would like to know if there is a standard technique to check if a singular variety admits a small resolution. What are the main references for these types of questions? I am mostly interested in ...
JME's user avatar
  • 3,022
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 $\pi_1(X)\cong \pi_1(\tilde X)$. ...
Dmitri Panov's user avatar
  • 28.9k
17 votes
4 answers
2k views

Eulerian number identity

The Eulerian number $A(n,m)$ is defined as the number of permutations $\sigma \in S_n$ having precisely $m$ descents, i.e. indices $i$ such that $\sigma(i)>\sigma(i+1)$. The wikipedia entry on ...
M Mueger's user avatar
  • 615
17 votes
2 answers
1k views

Jon Beck's untitled manuscript containing the "tripleability theorem" (i.e. the monadicity theorem)

Many papers refer to an untitled manuscript of Jon Beck (Cornell, 1966) for the origin of the monadicity theorem (originally called a "tripleability theorem"). An early proof is in Manes's ...
varkor's user avatar
  • 10.6k
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 ...
doofin's user avatar
  • 283
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
2 answers
590 views

Can you perturb an inscribed polytope so all its edges grow?

Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point. My question is the following: Let $P, P'$ be two non-...
Miek Messerschmidt's user avatar
16 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
  • 1,626
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
2 answers
2k 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 ...
Evan Jenkins's user avatar
  • 7,237
15 votes
2 answers
754 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
  • 25.2k
15 votes
0 answers
602 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
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
3 answers
1k 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
  • 305
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 $h$ ...
Sebastian Goette's user avatar
14 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
  • 3,463
14 votes
2 answers
761 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
2k 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
14 votes
4 answers
3k 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
1k 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
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
  • 2,810
14 votes
1 answer
1k views

What are the applications of Dowker's theorem?

Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$: a simplex in $K$ is empty or consists of finitely ...
Francis Snapper's user avatar
13 votes
3 answers
678 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
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
2 answers
1k views

Categories on which one can determine all model structures?

Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
Tim Campion's user avatar
  • 63.9k
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
1 answer
921 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
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
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
  • 5,822
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
1 answer
883 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
  • 3,979
12 votes
1 answer
720 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
12 votes
3 answers
784 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 Carlson's user avatar
11 votes
1 answer
804 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
  • 3,880
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
626 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
10 votes
2 answers
3k 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
10 votes
1 answer
833 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
0 answers
350 views

How are the hypergeometric motives of WZ-Pairs connected?

If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
Jorge Zuniga's user avatar
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