Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,544
questions
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Abelian subvarieties of abelian varieties --- reference request
This question may be too naive, in which case I apologise in
advance. Anyway, it is a well-known fact (see e.g. Milne's notes)
that any abelian variety A has only finitely many direct factors
up to ...
6
votes
3
answers
554
views
profinite spaces coming from profinite groups
This is probably well-known:
Does every nonempty profinite space occur as the underlying space of a profinite group? If not, which conditions have to be imposed?
- Is every profinite group ...
6
votes
1
answer
93
views
Topological entropy of semi-conjugated dynamical systems
Let $(X,T)$ and $(Y,G)$ be topological dynamical systems. If $(Y,G)$ is a factor of $(X,T)$ it is well known and easy to proof that $h(G)\le h(T)$ , where $h$ denotes the topological entropy. If the ...
6
votes
1
answer
207
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Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$
This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763.
It got upvotes, but no answers or comments, and so I ask it here.
Let $G$ ...
6
votes
1
answer
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Does $\overline{\mathbb{C}P^2 \setminus B^4}$ embed (smoothly/topologically) in $\mathbb R^5$?
Question: Does $\overline{\mathbb{CP}^2 \setminus B^4}$ (that is the closure of complex projective plane minus a 4-ball) embed (smoothly/topologically/piecewise linearly) in $\mathbb R^5$?
Background: ...
6
votes
1
answer
478
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A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
6
votes
1
answer
257
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A Sauer-Shelah-like lermma for prefix tree
I proved a variant of the Sauer-Shelah lemma and I was wondering if something like that is already known.
Let $S \subseteq \{0,1\}^n $. We say that a set of coordinates $K \subseteq [n]$ is shattered ...
6
votes
1
answer
428
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Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?
Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
6
votes
1
answer
340
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Are infinitary monads monadic?
As discussed here, Are monads monadic?, in "On the monadicity of finitary monads" by Steve Lack, the following is shown, the forgetful functor from $Mnd_f(C) \rightarrow Endo_f(C)$ is ...
6
votes
1
answer
165
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Norm of contragredient of unitary representations of compact quantum groups
Maybe the answer to the following question is known but I am unable to find it in the literature. Anyway, let me begin my question by fixing some notations and terms.
Let $G = (A, \Delta)$ be a ...
6
votes
1
answer
596
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The history and original paper of the Rosser–Iwaniec sieve
I'm trying to find Rosser's original paper where he introduces his eponymous sieve. I've already found https://arxiv.org/pdf/math/0505521 (where the reference isn't given, but where it is indicated ...
6
votes
1
answer
175
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Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity
I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$
works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...
6
votes
1
answer
419
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Bounding size of partial difference sets given size of partial sumsets
In this paper by Katz and Tao, the following bounds were established.
Let $A,B$ be finite subsets of an abelian group, with $|A|,|B|\le N$. We fix some $G \subset A\times B$. We define $C = \{a+b:(a,b)...
6
votes
1
answer
258
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Rickard's strengthening of Broué's abelian defect group conjecture and the lifting of some equivalences up to splendid derived equivalences
Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $K:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $G$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
6
votes
1
answer
564
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Does the purported proof of Rota's conjecture provide an algorithm for calculating the forbidden minors of matroids over arbitrary finite fields?
About six years ago there was a proof announced and later outlined in a notice from AMS. However right now I can only seem to find forbidden minor characterizations for matroids linearly ...
6
votes
1
answer
244
views
A unique equilibrium state which does not have Gibbs property
Let $T:\Sigma \rightarrow \Sigma$ be a topologically mixing subshift of finite type and let $f:\Sigma \rightarrow \mathbb{R}$ be a continuous functions over $(T, \Sigma)$. Assume that there is a ...
6
votes
1
answer
490
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Mori's cone theorem
I need the proof (reference) of Mori’s theorem about this implication :
Let $X$ be a projective complex manifold. If $X$ contains no rational curves, then $K_K$ is nef.
6
votes
1
answer
495
views
What is known about the functional square root of the Riemann zeta function?
Let us consider the Riemann zeta function $\zeta(s)$, where $s$ can take on values on the domain $\mathbb{R}_{>1}$:
$$\zeta(s) := \sum_{n=1}^{\infty} \frac{1}{n^{s}} .$$
I wonder what is known ...
6
votes
2
answers
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Fibre preserving maps of Borel constructions
Let $G$ be a discrete group with universal principal bundle $EG\to BG$, and let $X$ and $Y$ be left $G$-spaces. An equivariant map $\overline{f}:X\to Y$ induces a fibre-preserving map $f:EG\times_G X\...
6
votes
1
answer
238
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Arens regularity of Banach algebras
I was trying to learn the concept of Arens regularity of Banach algebras from T.W Palmers book -"Banach algebras and the general theory of $*$-algebras". There he have discussed the Arens regularity ...
6
votes
3
answers
599
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Electromagnetism as a $U(1)$-gauge theory
I would like to learn gauge theory, starting from the simplest case. I have heard that I should start with electromagnetism, which is just the $U(1)$-gauge theory. All the references I know are ...
6
votes
1
answer
241
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Concrete example to illustrate the theory about blocks of groups with cyclic defect groups
I'd like to to have a concrete example to illustrate the theory about blocks of groups with cyclic defect groups.
Thus, I am looking for a finite group $G$ and a prime $p$ dividing $|G|$ satisfying ...
6
votes
2
answers
270
views
RSK and crystal operators
Is there a good reference on how RSK (and the 3 other variants)
interact with crystal operators on the semi-standard tableaux $(P,Q)$ in the image?
That is, we have biwords, $W$ which are in ...
6
votes
1
answer
330
views
Can the number of solutions to a system of PDEs be bounded using the characteristic variety?
I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a ...
6
votes
2
answers
1k
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Functorial cones
This is probably more a reference request than a real question. I was studying dg-categories in order to understand how one can derive a functorial cone construction when a triangulated category (...
6
votes
1
answer
630
views
Is there an English translation of Laumon's proof of geometric Langlands for $\mathbb{G}_m$?
I'd like a detailed proof in English of Laumon's proof that the two Fourier-Mukai transforms taking the derived category of quasicoherent sheaves on $\mathbb{G}_m$-local systems of a curve $X$ to the ...
6
votes
1
answer
204
views
Is the projection onto the regular image an epimorphism?
Let $f:X\to Y$ be a morphism in a category $\mathcal{C}$.
Let $m:I\hookrightarrow Y$ be the regular image of $f$. This means that $f$ can be written as $f=m\circ e$, with $m$ regular mono (i.e. being ...
6
votes
1
answer
268
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Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials
$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}_n$ be the symmetric group, $\SYT_n$ be the set of standard young tableaux of size $n$.
For $u\in \...
6
votes
1
answer
241
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Nash embedding for complete manifold
I, ask my question as a comment in this post. Without answer I post a more detailed version.
I am looking for a reference about $C^\infty$ Nash isometric embedding for non compact manifold.
My ...
6
votes
1
answer
576
views
The Gauss Circle Problem asymptotic in dimension
The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"
For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\...
6
votes
1
answer
295
views
Citations graphs: what is known?
There has been much research related to web graphs and social graphs.
They can be thought of as a kind of random graphs, but the point is that
they are different from the well-known Erdős–Rényi model.
...
6
votes
2
answers
588
views
Interpolation space between $L^1\cap L^2$ and $L^1$
In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...
6
votes
1
answer
348
views
Homotopy cosheaf?
Let $C$ be a site, $\mathbf{S}$ some ($\infty$-? homotopy?) category of spaces.
Question. What do you call a (covariant!) functor $F:C\to \mathbf{S}$ enjoying the following property: for every ...
6
votes
2
answers
573
views
Online Interactive mathematics games for mathematicians or mathematicians-to-be!
I am aware that this is not a research level mathematics question. Also, it would not have a single answer (if any) and yet, I hoping that it will be considered as a community wiki question worthy of ...
6
votes
2
answers
627
views
Wiener Measure measure on functions?
I know that the Wiener measure for the Brownian motion $\{B_t\}_{t\ge 0}$ on the probability space $(\Omega, \mathscr{F},P)$ can be defined as $\mu=P\circ B^{-1}$ acting on the sigma-algebra generated ...
6
votes
1
answer
825
views
Rice's theorem in type theory
From the formula
$$\forall f\colon A\to A\,\exists x\colon A\,(f(x)=x)$$
we can get the scheme
$$\forall x\colon A\,(\phi(x)\vee\neg\phi(x))\Rightarrow\forall x\colon A\,\phi(x)\vee\forall x\colon A\,\...
6
votes
1
answer
479
views
Fourth cohomology of the modular group
Is $H^4(PSL(2,\mathbb{Z}),\mathbb{Z})$ known? I ask this in response to the recent calculation of the same cohomology group for $\mathrm{Co}_0$ and $\mathrm{Co}_1$.
6
votes
1
answer
965
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What did Zermelo say he was hoping for on the consistency of set theory?
Question.
What precise things are known about what Zermelo is hinting at in the below citation?
What are scholarly references on Zermelo's own attempts at proving consistency of his axioms? What did ...
6
votes
1
answer
677
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Thick subcategories
I hope this question is not too trivial for mathoverfolw.
Let $R$ be a commutative ring (with $0\neq 1$) and $D_{Perf}(R)$ the triangulated category of perfect complexes. Let $C$ be a thick ...
6
votes
1
answer
357
views
Evolving curves by Alexander Polden
I am writing a piece on curve shortening flow and lots of my sources have referenced Alexander Polden's honours thesis 'Evolving Curves' from the Australian National University. I have tried to find ...
6
votes
1
answer
251
views
Every PD group is $\pi_1$ of an aspherical manifold
It is conjectured that for a discrete, finitely presented group $G$ such that $BG$ satisfies Poincaré duality, there actually exists a closed manifold $M$ which is homotopy equivalent to $BG$.
This ...
6
votes
1
answer
881
views
Total space of canonical bundle as resolution of singularity
We know for $Y=\mathbb{P}^n$, the total space of the canonical sheaf $Tot(\omega_Y)$ is the resolution of $\mathbb{C}^{n+1}/\mathbb{Z}_{n+1}$ where the generator acts as scalar matrix of multiplying a ...
6
votes
2
answers
820
views
A dynamical system defined by the Riemann zeta function
Let $\zeta$ be the classical Riemann zeta function.
We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector ...
6
votes
1
answer
295
views
Reference request to proof that H$^2(\Gamma, \mathbb{Q}/\mathbb{Z}) = 0$
Does anyone maybe have a reference to the proof of the following result by Tate?
Let $\Gamma$ be the absolute Galois group of the rationals. Then the second cohomology group (for trivial $\Gamma$-...
6
votes
1
answer
364
views
Embedding a cancellative monoid into another in such a way that a prescribed element becomes left-invertible
Let $\mathbb A = (A, +_A)$ be a cancellative, but possibly non-commutative, monoid with identity $0$, and fix an element $x \in A$. Does there always exist a cancellative monoid $\mathbb B = (B, +)$ ...
6
votes
1
answer
268
views
Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?
Let
$$
M:=\{(P,\pi)\mid P\not\in\pi\}\subset\mathbb{P}^n\times\mathbb{P}^{n\,\ast}
$$
be the open and dense (and as such $2n$-dimensional) subset of non-incident point-hyperplane pairs. If $P=\mathbb{...
6
votes
1
answer
249
views
Fundamental class in $KO[1/2]$
Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...
6
votes
1
answer
1k
views
Derivatives of theta functions at zero
Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions
$$
\theta_s(z;\tau)_k=\sum_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, \...
6
votes
1
answer
192
views
Radon transform between affine grassmannians
Let $\overline{GR}(n,k)$ be the manifold of all affine k-dimensional subspaces in $R^n$, and let
$R:C^{\infty}_c(\overline{GR}(n,k))\to C^{\infty}_c(\overline{GR}(n,l))$, $0\le k<l\le n-1$, be the ...
6
votes
1
answer
1k
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Kullback Leibler "variance": does that divergence have a name?
If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence:
$$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$
and this ...