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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2 votes
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Any connection between extension of algebraic structure and forcing of set theory?

Any connection between extension of algebraic structure and forcing of set theory? And more, are there any approach from one of the two to other field to solve problem?
7 votes
2 answers
218 views

Complete surfaces in $M_g$

Let $M_g$ be the moduli space of genus g curves. In Zaal's paper ("A complete Surface in $M_6$ in Characteristic $> 2$"), the author mentioned that there is a known construction of ...
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Mathematical characterization of gravitational geons as reference request, and their properties as main question

I've edited (ten days ago) a question on Physics Stack Exchange, this Mathematical characterization of gravitational geons, post with identifier 726281 the users of the site were kind adding in the ...
10 votes
0 answers
81 views

V-categories enriched in a monoidal V-category

In an email to the categories mailing list dated 21 August 2003, Street writes: Max reminded me of his old result (not in the LaJolla Proceedings, but known soon after) that a monoidal V-category is ...
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2 votes
1 answer
218 views

Growth rate of an outer automorphism of a free product

$\DeclareMathOperator\Out{Out}$Let $G=G_1\ast\cdots\ast G_k\ast F_p$ be a Grushko decomposition of a finitely generated group $G$, $\mathcal{O}$ the outer space relative to this decomposition, $[\phi]\...
6 votes
2 answers
309 views

Dual surfaces of a first cohomology class of a 3-manifold

Let $M$ be closed 3-manifold and $\alpha\in H^1(M;\mathbb Z_2)$ an arbitrary element. (In my case we know that $M$ is non-orientable and $\alpha^3=0$.) It is well known that there is a closed 2-...
4 votes
1 answer
110 views

Intersection form of $2n$-manifold for odd $n$

Let $M$ be closed orientable $2n$-manifold, where $n$ is odd. It is well known that the $\mathbb Z$-module $H^\bullet(M;\mathbb Z)$ has graded-commutative multiplication and $H^{2n}(M;\mathbb Z)\simeq\...
9 votes
0 answers
219 views

Is “simplicity is elementary” still hard? (Felgner’s 1990 theorem on simple groups, and subsequent work)

I came across a reference in this MathOverflow answer to an intriguing result of Ulrich Felgner [1]: among finite non-Abelian groups, the property of being simple is first-order definable. According ...
4 votes
0 answers
107 views

On skew monoid rings and skew ordered series rings

To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...
2 votes
1 answer
144 views

Triangle equality for cosine similarity in high dimensions

I'm trying to understand whether I can use the following equality in my application -- for $u,v,w \in \mathbb{R}^d$: $$\cos(u,w)\approx \cos(u,v)\cos(v,w)$$ Where $\cos(x,y)$ gives cosine of the angle ...
3 votes
1 answer
98 views

Reference request for equivalent formulations of being absolutely indecomposable

I would like to ask the following question. I am searching for a reference for the following statement: Suppose $k$ is a perfect field. Let $A$ be a (symmetric) $k$-algebra and let $M$ be a finitely ...
3 votes
1 answer
321 views

Equivalence between two fractional Sobolev spaces

For $s \in (0,1)$, we consider the spectral fractional Laplacian \begin{align} (-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k \end{align} where \begin{align*} \begin{cases} ...
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On the upper bound estimation of $D(N)$ in Chen Jingrun's theorem

What are the current research results on the estimation of the upper bound of $D(N)$ in Chen Jingrun's theorem? Including but not limited to Chen Jingrun's improvement 7.8342 and Wu Jie's improvement ...
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7 votes
1 answer
286 views

Combinatorial reciprocity for symmetric functions

I am wondering whether a certain instance of combinatorial reciprocity (in the sense of Stanley's classic paper "Combinatorial Reciprocity Theorems"), concerning symmetric functions, is ...
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4 votes
1 answer
323 views

Nonsmooth version of Hopf boundary point lemma

Let $$ Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u $$ be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite. Here I'm only considering smooth coefficients, and the domain $\...
2 votes
0 answers
60 views

Formulas for special elements of the nil-Hecke ring

Kostant and Kumar introduced the nil-Hecke ring for a crystallographic Coxeter group, which we will take to be $S_\infty$, which is the ring generated as a left module over the polynomial ring $\...
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3 votes
0 answers
124 views

Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces

Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
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2 votes
2 answers
264 views

Reference for zero sum estimates of Dirichlet L functions

Let $\chi$ be a primitive character mod $p$ (prime) and $\rho = \beta + i \gamma$ be a non-trivial zero of $L(s, \chi)$. I am reading a paper by Ihara and Murty where they use following estimate : $\...
4 votes
2 answers
454 views

Kolmogorov's approach to probability theory

Question: Did Kolmogorov develop a set of axioms for probability theory motivated by Algorithmic Information Theory in the 1960s? Context: In 1965, Andrey Kolmogorov considered three approaches to ...
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3 votes
0 answers
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Ergodic diffeomorphisms of the circle

From the paper Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a ...
2 votes
1 answer
182 views

On the stack of semistable curves

This is a question related to Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology? Let $\mathcal C\rightarrow \mathcal M^{ss}_g$ be the universal curve over the ...
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4 votes
1 answer
178 views

On a result of Cartan for homogeneous manifolds arising from a quotient of discrete subgroups

I'm not sure if this is completely relevant to MO, let me know if this would be better on MSE. I have been told today by a professor of mine that the following is a classic result of Cartan. Suppose $...
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5 votes
0 answers
178 views

Reference on "infinite dimensional Lie algebras" from a mathematical physics point of view

It happens that I stumbled on a class of infinite dimensional Lie algebras that are not Kac-Moody algebras and for which I was not really prepared for. I know some general results on infinite ...
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4 votes
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Maximal function estimate for differential quotient of function satisfying $\nabla f \in BMO$

For a function $f \in W^{1,p}(\mathbb R^N)$, it is well-known that there exists a constant $C_N$ (dependent on $N$) such that $$ |f(x)-f(y)| \le C_N|x-y|(\mathcal M|\nabla f|(x) + \mathcal M|\nabla f|(...
4 votes
2 answers
543 views

Sum of many squares modulo $n$

Let $n$ be a positive integer and $0 \leq i < n$. Define $$ N(i) = \# \left\{ (x_1,\dots, x_s) \in [1, n]^s: x_1^2 +\dots + x_s^2 \equiv i \mod n \right\}. $$ I am looking for a reference for ...
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0 votes
0 answers
50 views

Continuity of "inversion operator" between function spaces

Question: When is the operation of inversion continuous as a map between spaces of invertible functions? Let $\mathcal{F}$ be a function space such that $f\in\mathcal{F}\implies$ $f$ is invertible and ...
5 votes
0 answers
159 views

Finite atlas on a smooth manifold

If a smooth manifold admits a finite atlas, then for many technical purposes it is as good as compact. I was surprised to learn that every connected smooth manifold actually has a finite atlas. This ...
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3 votes
0 answers
153 views

What does a character of a scheme mean?

Here is a soft question I met in the book Introduction to Grothendieck Duality Theory by Altman and Kleiman. In Chapter I the proposition 2.1 uses a term called "a character of $X$" where $X$...
1 vote
1 answer
169 views

Bridges between geometry and combinatorics

Geometry and combinatorics are two different branches of mathematics. Does there exist any connection between them? In many cases, mathematicians solve some geometric problems by reducing them to a ...
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1 vote
0 answers
37 views

Reference for rigidity of higher rank action

I heard some results about the rigidity of higher rank action and it looks very interesting. I would like to know if there are any good survey of paper to get started in this field. Thank you in ...
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0 votes
0 answers
121 views

Is there an analogy between convergence and homotopical triviality?

Is there an analogy (formal or intuitive) between the notions of convergence and contractibility ? Is the notion of convergence an instance of homotopical triviality in some formalism, or should be ? ...
3 votes
1 answer
130 views

$l$-adic sheaf associated to an algebraic representation of $\mathrm{GSp}_{4}(\mathbb{Q})$

Let $Y (N) $ be the moduli scheme of dimension two principally polarized Abelian schemes with level $N$. It is claimed in "G.Laumon - Fonctions zeta des variétés de Siegel" (Lemma 4.1) that ...
5 votes
1 answer
58 views

Do $F$-traces of simple modules at $p'$-classes uniquely determine the module?

Let $G$ be a finite group and let $p$ be a prime number such that $p\mid |G|$. Let $\text{IBr}(G)$ denote the set of irreducible Brauer characters of $G$ for the prime $p$. Assume $\mathbb{F}_{q}$ is ...
2 votes
1 answer
286 views

The stack of equivariant local system is quasi-smooth

Let $G$ be a (connected ?) algebraic group and $X$ a smooth, projective, and connected algebraic curve, both over an algebraically closed field $k$ of characteristic $0$. My questions are then as ...
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5 votes
2 answers
375 views

Explicit description of the right adjoint

Let $C$ be a diagram. Consider a functor $F: C \to \mathbb{E}_{\infty}(Sp)$ from the diagram to the category of $\mathbb{E}_{\infty}$-rings in spectra. Let $R$ be the limit of this diagram. Given the ...
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1 vote
0 answers
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Reference request: arithmetical implications of an ambient Galois extension

This is a cross-post of a question I asked on StackExchange. See there for further details. Let $L/K$ be a Galois extension of algebraic number fields of finite degree over $\mathbb{Q}$, with group $G$...
0 votes
0 answers
60 views

Reference request for Poincare-Hopf theorem in a compact submanifold

I recently read the following question about the Poincare-Hopf theorem in a compact submanifold. All the answers were very satisfactory to me. Is there any reference where I can look for more details ...
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4 votes
0 answers
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Earliest known proof of "Any degree one self-map of an orientable connected finite-type non-compact surface is homotopic to a homeomorphism"

I attended a talk where the speaker said the following is due to Nielsen. I searched here and there but couldn't find the corresponding paper, if any. So, what is the earliest known proof of the ...
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2 votes
1 answer
143 views

What fraction of the values of a quadratic polynomial can be prime?

I have an explicit, monic quadratic polynomial $P(x)$ and an integer $m$. Can I bound the number of prime values in $P(0), P(1), \ldots, P(m)$? A reference would be appreciated, if available. An ...
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2 votes
2 answers
441 views

How to use that the Hessian is negative definite in this proof

Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...
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4 votes
2 answers
495 views

Is this exact sequence known?

$\newcommand{\Tors}{{\rm Tors}} \newcommand{\tf}{{\rm\, t.f.}} \newcommand{\Gt}{{\Gamma\!,\,\Tors}} \newcommand{\Gtf}{{\Gamma\!,\tf}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\Z}{{\mathbb Z}} \...
1 vote
0 answers
83 views

Schwarz space on the upper half-plane

Let us think of the Schwartz space $\mathcal{S}(\mathbb{R}^2_+)$ on the upper half-plane $\mathbb{R}^2_+=\mathbb{R}\times(0,+\infty)$ defined as $$ \mathcal{S}(\mathbb{R}^2_+)=\left\{f\in C^\infty(\...
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5 votes
1 answer
214 views

All-set-homogeneous spaces

This is a follow-up to the question of Joseph O'Rourke Which metric spaces have this superposition property? A metric space $X$ will be called all-set-homogeneous if for any subset $A\subset X$ any ...
2 votes
0 answers
36 views

Coordinate free supersymmetric sigma model Lagrangian

I would like to know if there is a coordinate free version of the Lagrangian of the supersymmetric sigma model on a $2$-dimensional spacetime, with target space a Kähler manifold. The action for this ...
1 vote
0 answers
77 views

About the classification of simply connected homogeneous 3-manifolds

I've read somewhere (but cannot locate the source) that the following classification holds: simply connected homogeneous 3-manifolds are either isometric to $S^2 \times \mathbb{R}$ or to a metric Lie ...
  • 1,401
3 votes
0 answers
58 views

What is the minimum length of a $k$-permutation-avoiding word on $n$ letters?

Let $w$ be a word over the alphabet $[n] := \{1, \dots, n\}$. For a fixed $S \subseteq [n]$, let $w_S$ be the word obtained from $w$ by deleting all entries not in $S$, then removing (all but one ...
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6 votes
1 answer
177 views

Reference request: generalized randomness

There is a plethora of notions of randomness for Cantor space ($\phantom{}^\omega 2$) (Schnorr randomness, Martin-Löf randomness, weak 2-randomness, the various forms of higher randomness such as $\...
1 vote
1 answer
134 views

Distribution of $\alpha n^2/q$ modulo $1$?

Let $0 \neq \alpha \in [0,1]$ and $q$ a positive integer. Let $||.||$ denote the distance to the closest integer and define $$ N_i(q) = \sum_{ \substack{ -q/2 \leq n \leq q/2 \\ \frac{i}{q} \leq || \...
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0 votes
0 answers
70 views

Décalage and the simplicial path object

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\const{const}$Let $[+1]:\Delta\to\Delta$ be the décalage endomorphism sending $n\mapsto n+1$ adding a new minimal element, i.e. $f:n\to m$ is sent to $...
6 votes
0 answers
222 views

On improvements of the GPY sieve

When $\chi_\mathbb P(n)$ denotes the characteristic function of primes and $\mathcal H=\{h_1,h_2,\dots,h_k\}$ is some admissible $k$-tuple, the GPY sieve can be formulated as follows: $$ S(x)=\sum_{x&...
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