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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Modulus of Continuity, Heat Flow, and Derivative Estimates

Given $f : \mathbf{R}^d \to \mathbf{R}$, define $P_t f$ by \begin{align} (P_t f)(x) = \mathbf{E} \left[ f (x + \sqrt{t} G) \right], \end{align} where $G \sim \mathcal{N} (0, I_d)$ is a standard ...
πr8's user avatar
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2 votes
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41 views

Cardinality of the set $\#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \}$

Let $\alpha \in I$ where $I$ is some closed interval that does not contain $0$. I am interested in upper bound for $$ M(\alpha) = \#\{ 1 \leq n \leq N: \| \alpha n^2/N \| < 1/N \} $$ where $N$ ...
Johnny T.'s user avatar
  • 3,585
2 votes
0 answers
68 views

Looking for a source on Conrad-Gabber's results about spreading out of rigid-analytic families

Brian Conrad and Ofer Gabber have some results that were announced 9 years ago here: https://www.ihes.fr/~abbes/Gabber/OferGabber.pdf and there's a talk by Gabber about them here: https://www.youtube....
Kim's user avatar
  • 4,104
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23 views

MDP Average Reward independent of Initial State

Consider a Markov Decision Process where the state space $S$ and the action space $A$ are continuous and compact. In state $s$, if action $a$ is chosen and the next state becomes $s'$, the ...
Euclid's user avatar
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5 votes
2 answers
425 views

A version of Hilbert's Nullstellensatz for real zeros

$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
Iosif Pinelis's user avatar
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0 answers
101 views

Atiyah sequence of a coherent sheaf

I want to show that if $X$ is a smooth complex projective variety, then analytification induces an equivalence of categories between algebraic integrable connections on $X$ and analytic integrable ...
Tanny Sieben's user avatar
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0 answers
35 views

Alexandrov's uniqueness theorem in Minkowski spacetime

Suppose $P$ is a convex polyhedron in $\mathbb{R}^{2,1}$. Each face of $P$ comes with induced metric tensor, if the face is space-like, then it is euclidean metric; every time-like face is isometric ...
Anton Petrunin's user avatar
4 votes
0 answers
231 views

Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?

I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$. Recently I was pointed to Katz and Mazur's book, ...
David Roberts's user avatar
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References for orbifold curves

I am looking for a good reference (if there is any) for the theory of orbifold curves from the perspective of stacks. By an orbifold curve I mean something like a $1$-dimensional irreducible Deligne-...
Alekos Robotis's user avatar
6 votes
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Equation in a nilpotent group

Let $G$ be a nilpotent group of class at most $r$ (that is, $\gamma^{r+1}G=1$). Let elements $g_1,\dotsc,g_n\in G$ be fixed. We are interested in the set $V\subseteq\mathbb Z^n$ of solutions $x=(x_1,\...
Semen Podkorytov's user avatar
-3 votes
0 answers
112 views

Friedrich Schur on the BCHD theorem (notes in English)

According to Sternberg in his book Lie Algebras, The formula [the BCHD formula] is named after three mathematicians, Campbell, Baker, and Hausdorff. But this is a misnomer. Substantially earlier than ...
Tom Copeland's user avatar
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Injection of $G(k)/Z(k)$ into $(G/Z)(k)$

In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...
Μάρκος Καραμέρης's user avatar
4 votes
1 answer
108 views

Legendre's Irrationality Condition for Generalized Continued Fractions

This MathOverflow post cites that Legendre allegedly showed that given $a_{i}\in\mathbb{Z}\setminus\left\{0\right\}, b_{i}\in\mathbb{Z}$, $$\cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}...
Vessel's user avatar
  • 145
2 votes
0 answers
120 views

GIT quotient and orbifolds

Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
Dr. Evil's user avatar
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3 votes
1 answer
210 views

Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?

It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$. Now, if $\varphi \in L^\infty (\mathbb ...
ashK's user avatar
  • 117
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0 answers
31 views

Reference request: injectivity of CWT, density of dilations and translations in $L^p$

Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on ...
Zhang Yuhan's user avatar
6 votes
1 answer
178 views

Attempts to define a matrix exponential over (as much as possible) general fields

Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as $$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$ where ...
rosan98's user avatar
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2 votes
0 answers
110 views

Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?

The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm $$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$ The space $L^2([a,b]\times S^2)$ ...
Laithy's user avatar
  • 885
4 votes
0 answers
86 views

Symmetric functions and pattern avoidance

It is known that the number of $k$-regular simple graphs with vertices labeled by $1,2,\dots,n$ can be expressed as the coefficient of $x_1^k \dots x_n^k$ in a symmetric function, which is $$ \prod_{1\...
minhtoan's user avatar
  • 1,454
16 votes
3 answers
859 views

Conjectures in the representation theory of the symmetric group

Question: What are current open conjectures about the representation theory of the symmetric group? I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
Mare's user avatar
  • 26k
1 vote
0 answers
35 views

Absolute irreducibility implies free action on framed universal deformation ring

Let $\overline{\rho}: G\longrightarrow \text{GL}_n(\mathbb{F}_p)$ be a residual representation of the Galois group of a number field. Let $R_{\overline{\rho}}^{\square,\text{univ}}$ be the universal ...
kindasorta's user avatar
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6 votes
1 answer
832 views

What is this huge generalization of the Modularity Theorem?

A friend of mine wrote: The point is of course that the Modularity Theorem (as I stated it) is/should be really just a special case of some much bigger theorem which sets up a bijection between ...
John Baez's user avatar
  • 21.5k
3 votes
3 answers
178 views

Existence of an extension operator $E: W_0^{s,p}(\Omega)\rightarrow W^{s,p}(\mathbb{R}^d)$?

If $\Omega$ is an open Lipschitz domain with bounded boundary then their exists a continuous operator $E: W^{s,p}(\Omega) \rightarrow W^{s,p}(\mathbb{R}^d)$ where $s\in (0,1)$ such that $Eu=u$ on $\...
Perelman's user avatar
  • 143
1 vote
0 answers
62 views

Reference request: finding entries that prevent matrix from being correlation matrix

I am currently doing some research with a quantitative finance firm and my supervisor has raised an interesting question that shows up a lot with their clients: quite often, clients will want to do ...
Martin Skilleter's user avatar
0 votes
1 answer
97 views

Hansel's simple proof of the Skolem Mahler Lech theorem

In his paper 'A simple proof of the Skolem Mahler Lech theorem' Hansel gives a proof of the theorem in the case that the coefficients of the rational series belong to $\mathbb{Q}$. He claims that the ...
trinket34's user avatar
4 votes
0 answers
56 views

Techniques to estimate PDE which are elliptic in some directions and degenerate in others

I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
Aidan Backus's user avatar
2 votes
0 answers
60 views

Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles

Let $m \in \mathbb{Z}^n$ be a partition of $2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of $m$ are ...
Matthias's user avatar
  • 203
3 votes
0 answers
74 views
+200

Representations of a reductive Lie group vie central character and K-types

Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-...
Nandor's user avatar
  • 119
3 votes
0 answers
83 views

$\mathbb{Z}_p$-points of a $\mathbb{Z}_p$-model of a reductive linear algebraic $\overline{\mathbb{Q}}_p$-group

Let $G$ be a (connected) reductive linear algebraic group over $\overline{\mathbb{Q}}_p$. By definition, this means that $G$ is a closed subgroup of some $\mathrm{GL}_n$. We can always find a ...
Otto's user avatar
  • 31
6 votes
1 answer
166 views

Strengthening of a classical set mapping theorem of Lázár

We are writing a paper in which we need to use the following theorem to prove that a certain poset satisfies c.c.c. As far as I remember I learned this theorem from András Hajnal. Theorem 1: If $\...
Lajos Soukup's user avatar
  • 1,477
2 votes
0 answers
79 views

Equivariant disk theorem in dimension 2

All groups I'll consider are finite. An important part of equivariant differential topology in dimension 3 is the equivariant disk theorem, which says that for a $G$-action on a compact $3$-manifold ...
Evan Scott's user avatar
6 votes
1 answer
197 views

Analytic maps on Banach spaces: analyticity upgrade

Consider the following problem. Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and $$ f:U\to G $$ an analytic map, such ...
Lorenzo Pompili's user avatar
5 votes
1 answer
462 views

Hilbert's and Gödel's expanded definition of "Recursive Function"

There is a very interesting comment in this post: I must also make one terminological caveat: Hilbert, and later Godel, used the phrase "recursive function" in a way very different from the ...
Mike Battaglia's user avatar
0 votes
0 answers
39 views

Reference request: "doubly empirical" measure associated to a random measure

I am considering the following type of situation. Suppose we have a random probability measure, by which I mean a probability measure on a space of probability measures atop some Polish space $X$. In ...
pseudocydonia's user avatar
4 votes
0 answers
128 views

Is there a statement in Presburger arithmetic about primes this simple heuristic fails for?

I came up with the following conjecture while thinking about ways to formulate some heuristics about primes: Conjecture: Given a statement $s$ in Presburger arithmetic, using an additional unary ...
Command Master's user avatar
-1 votes
0 answers
52 views

Define a bounded variation function on Cantor sets

Can we define a bounded variance function on a fractal set $X$, like a Cantor set? Is it the same as the usual one : $$\|f\|=:\sup_n \left\{ \sum_{t_1<\cdots < t_n}|f(t_i)-f(t_{i+1})\mid t_i \in ...
user525625's user avatar
4 votes
1 answer
107 views

Second cohomology group of the contact Lie algebra $K_3$

Let $F$ be a field of characteristic zero and, for all $n>0$, consider the contact Lie algebra $K_{2n+1}$. It follows from Corollary 3 of the paper [V. Guillemin - S. Shnider: Some stable results ...
Rocky Smith's user avatar
-1 votes
0 answers
79 views

Regularity of Radon-Nikodym derivative

It is well known that given two measures $\nu$ and $\mu$ in, say, $\Omega \subset \mathbb{R}^n$ open and bounded, if $\nu \ll \mu$ then $\exists f \in L^1 ( \Omega, d \mu)$ such that $\nu= f \mu$, and ...
tommy1996q's user avatar
5 votes
0 answers
139 views

New investigations on Homotopical Algebraic Contexts

Homotopical algebraic context are models that allows Toën and Vezzosi to do derived geometry. It have been defined in their seminal paper Homotopical Algebraic Geometry II. These are general abstract ...
Nikola Tomić's user avatar
4 votes
1 answer
223 views

What is the convergence rate of this "infinite monkey"-type probability?

Cross-posted from Math Stack Exchange, where it hasn’t received an answer yet: Let $S$ be a finite set and $n,m\in\mathbb N$. Consider the process $R=(R_i)_{i\in\mathbb N}$ where all $R_i$ are iid ...
Joseph Expo's user avatar
1 vote
0 answers
16 views

Can all real positive semidefinite Hankel matrices be decomposed into sum of rank 1 real positive semidefinite Hankel matrices?

Denote the set of real positive semidefinite $d\times d$ Hankel matrices as $\mathcal{S}$. Can we always decompose one $S\in \mathcal{S}$ into sum of rank $1$ $S_i\in \mathcal{S}$, i.e., $S=\sum_i S_i$...
zzy's user avatar
  • 11
3 votes
1 answer
139 views

When the fundamental group of subgraph of groups embeds?

Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
tomasz's user avatar
  • 1,216
2 votes
0 answers
51 views

Nielsen–Thurston classification and configuration spaces

Viewing the $n$-strand braid group as the mapping class group of an $n$-punctured disk, braids can be classified as periodic, reducible, or pseudo-Anosov. The same group is also the fundamental group ...
Ben Knudsen's user avatar
0 votes
1 answer
121 views

Is the space $C_0^{k}(\Omega)$ a Montel space?

I asked this question in the MathStackExchange, but I think I'm not get any answer. I'm trying to find a reference for the following result: Theorem: Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ ...
Math's user avatar
  • 481
3 votes
0 answers
161 views

A relative Abel-Jacobi map on cycle classes

I have a question about relativizing a classical cohomological construction that I think should be easy for someone well versed in such manipulations. Background: Suppose $X$ is a smooth projective ...
Asvin's user avatar
  • 7,646
4 votes
0 answers
76 views

Expansion of Schubert polynomials into standard elementary monomials

I have an explicit formula for expressing any Schubert polynomial in terms of standard elementary monomials that may or may not be cancelation-free. I haven't determined this yet, but it seems likely ...
Matt Samuel's user avatar
  • 2,038
0 votes
1 answer
48 views

$L^\infty$ estimate for elliptic PDE with mixed boundary conditions

Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint. Consider the problem $$\Delta u = f \quad\text{in $\...
BBB's user avatar
  • 31
6 votes
1 answer
165 views

Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces

I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
Steve's user avatar
  • 61
6 votes
1 answer
374 views

Second-order ordinal definability

As is familiar, a set $S$ is ordinal definable ($S \in \mathsf{OD}$) just in case there exists a formula $\varphi(x, \vec{z})$ of the first-order language of set theory with parameters $\vec{z} = z_1, ...
Beau Madison Mount's user avatar
2 votes
1 answer
162 views

Reference request: Best book to cite on a property of the family of Cauchy distributions

Kai Lai Chung once began a section of a textbook on probability by writing "Everybody knows" that $$ e^x = \sum_{n\,=\,0}^\infty \frac{x^n}{n!}. $$ (with those quotation marks). Other ...
Michael Hardy's user avatar

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