# 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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### Interpolation theorem for general rough paths

In Friz and Hairer's notes on rough paths, there is exercise 2.9 which is called the "interpolation theorem". It says that if you have a sequence of rough paths $\mathbf X^n=(X^n,\mathbb X^n)...

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### Extending diffeomorphisms

Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.
Question. Is it possible to ...

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### Bound for a sequence of vertices in a graph

I have come across the following problem. Let $d\in\mathbb{N}$. Let $G$ be a $k$-regular directed graph with $n$ vertices without parallel edges. For a vertex $v\in G$, let $e_v$ denote the union of $\...

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### Are dark numbers known? [closed]

The limit in analysis is approximated better and better by a sequence or series. Contrary to set theory.
Bertrand Russell wrote: "But nowadays the limit is defined quite differently, and the ...

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### A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori

$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...

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### Is there a name for the number $ n $ of points in general position s.t. $ \operatorname{Cox}(\operatorname{Bl}_{p_{1},\dots,p_{n}}(Z)) $ is not f.g.?

Let $ Z $ be a projective, normal, $ \mathbb{Q} $-factorial variety (so the Cox ring of $ Z $ is well defined). Is there a name in the literature for the minimal natural number $ n $ such that the ...

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### better correlations and a special flow

A Gauss map $T: [0,1]\to [0,1]: Tx=x^{-1}-[x^{-1}]$, and a suspension $S=\{(x,t) \in [0,1]\times [0, \infty): 0\le t < -\ln x\}/{\sim}$, where the equivalent relation $\sim$ is defined by $ (x, -\...

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### Is a Lie subgroup whose center is closed, a closed subgroup itself?

I want to show that a certain Lie subgroup (i.e. generated by the exponential of elements in some Lie subalgebra) of a Lie group is closed. My knowledge of the subject of Lie groups is rudimentary, ...

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### How to learn homotopy theory

I studied some basic algebraic topology (homotopy/homology/cohomology groups). When reading about the Dold-Thom theorem, the fancier and more recent sources sooner or later all started to use homotopy ...

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### Only finitely many rational curves in a complete linear system of a K3 surface

Consider a projective complex K3 surface $X$, then $\lvert D\rvert$ contains only finitely many rational curves for any divisor $D$ on $X$.
What is the original reference for this result?

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### Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...

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### Cohomology in a combinatorial way using ribbon graphs

I am interested in studying the cohomology of surfaces.
Let $S$ be a compact orientable connected surface. One possible way is to learn cohomology using differential forms.
Is it possible to approach ...

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### What integral formula is being used here?

I am trying to read the paper "Simple closed geodesics on convex surfaces" by E.Calabi and J. Cao and a certain passage is unclear for me. Before, let me contextualize and set up some ...

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### Knot Diffie–Hellman

Here's an idea for a knot-based Diffie–Hellman exchange:
Public: random (oriented) knot $P$.
Private: random (oriented) knots $A$ and $B$.
Exchange: Alice sends (randomized or canonical ...

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### Complete representation theory of $\mathrm{SL}(2,\mathbb R)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense ...

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### Books for learning branched coverings

I am self-studying branched coverings. I read it from B. Maskit's Kleinian groups book. I want some more references for reading branched covers. In particular, I want to understand how to create ...

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### Explicit proof of Quillen's connectivity theorem

Definition
Let $A$ be a commutative ring. An ideal $I \triangleleft A$ is called quasiregular if $I/I^2$ is flat over $A/I$ and there is a canonical isomorphism of algebras
$$
\Lambda_A I/I^2\...

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### What is `the ideal set theory' [closed]

This Q hardly has much sense but anyway, is there anything called the ideal set theory known?
Most assuredly this should have nothing to do with ideals (like the Frechet ideal etc.).

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### Topology of the moduli space of a 2-dim closed surface

Consider the moduli space $\cal{M}_{\Sigma_g}$ of a 2-dim closed surface $\Sigma_g$ of genus $g$. What is the topology of such a moduli space $\cal{M}_{\Sigma_g}$?
For example, what is $\pi_n ( \cal{M}...

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### Categories associated to digraphs

Let's take a directed graph, or a digraph, $G=(V,E)$ given by a finite set of vertices $V$ and a finite set of edges $E$. We can assume that pairs of vertices can have parallel edges between them and ...

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### "Higher" knot mutants

Mutation Wiki My related question 1 My related question 2
Top: How wiki describes mutation. Doesn't generalize well.
Bottom: How I think of it.
Now replace "four" in the Wiki text by "...

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### What is the website address that automatically shows related content instead of pointing to the paper or books [closed]

I was looking for vector bundles online, and I found a website that shows all the related results. By related results, I mean theorems/propositions/definitions, etc. It is not ProjectEuclid, which ...

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### Moduli all the way down

The notion of modulus of continuity is well-known from constructive mathematics, reverse mathematics, and computability theory. Intuitively, such a modulus is a function that returns the '$\delta>...

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### Multiple Wiener integral as Witt polynomial of Brownian motion

I know that if i have a Brownian motion $W_t$ the multiple Wiener integral
$\int_0^t \int_0^{\xi_1}...\int_0^{\xi_n} dW_{\xi_1}...dW_{\xi_n}$
can be expressed as $H_n(\int_0^t dW_s)$ where $H_n$ is ...

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### Identity involving Stirling number of the second kind

I'm looking for a citable reference for the following identity involving the Stirling numbers of the second kind $S(n, k)$ stated in Equation (27): For $n \geq 2$,
$$
\sum_{m=1}^n S(n, m) (-1)^m (m-1)!...

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### A Vandermonde like determinant with exponentials

Let $n\geq m$ be non negative integers, and consider a list of $(n+m+1)$ distinct numbers (complex or real). I am interested in getting a closed form formula for the following determinant: $\det\left[\...

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### Largest part and length of a partition in play

If $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq\lambda_k)\vdash n$ is an integer partition of $n$ then $\lambda_1$ is its largest part and $k$ is its length, $\ell(\lambda)$.
Define the statistic $...

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### Comparing cohomology of a total complex with the cohomology of semidirect product

$\DeclareMathOperator{\Tot}{Tot}$I have the following problem. Let $H$ and $G$ be groups such that $H$ acts on $G$, i.e., there exists a group homomorphism $H\to \mathrm{Aut}(G)$ and let $M$ be an ...

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### When is a bundle map between tangent bundles the differential of a function?

I previously asked this question on MSE but did not receive much of a response, so I'll attempt to post it here, edited with the comments received. I hope this question isn't too basic for MO ...

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### Further directions in representations of surface group into a Lie group

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$.
Now I am planning to ...

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### Fluctuation-dissipation theorem for Markov processes

In the context of particle systems of non-gradient types (see e.g. here, Step 2 on page 633), I recently encountered the concept of fluctuation-dissipation theorem (FDT). Since it is a major result in ...

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### Topology in non-mathematical literature

A great piece of knowledge that I heard from a talk of Robert Ghrist, is that one of the earliest instances of non-trivial manifolds (i.e. of dimension higher than 2) appears in Dante's Paradise, ...

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### References for group of invariance of the Painlevé property

I'm searching for some references about groups of invariance of the Painlevé property other than the book of Robert Conte or more generally birational transformation of Riemann surfaces.

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### Compatibility conditions for quadratic equations

In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$:
\begin{eqnarray}
0 &= A_1x^2 + B_1x + C_1 \\
&...

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### Reference request for recurrence relation of division polynomials

The recurrence relations for division polynomials of elliptic curves are well known:
$$\Psi_{2n} = \Psi_n \left( \Psi_{n+2} \Psi_{n-1}^2 - \Psi_{n-2} \Psi_{n+1}^2 \right) / \ 2y$$
$$\Psi_{2n+1} = \...

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### The minimal surface operator in a Riemannian metric

Let $\Omega \subset \mathbf{R}^n$ be a domain, and let the cylinder $\Omega \times \mathbf{R}$ above it be endowed with a Riemannian metric $g$. (Note this is not assumed invariant in the vertical ...

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### A question about the square root error of one dimensional random walks

Consider a one dimensional random walk, in which the probability of moving left along a line is $q=1/2$ and the probability of moving right is $p=1/2$. The square root error $\langle d_N \rangle$, ...

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### Name for the cell poset of the staircase partition

Is there a standard name for the cell poset of the staircase partition $(n,n-1,\dots,1)$, where, in English notation, a cell covers the adjacent cell in the row above and in the column to the right? ...

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### A variation of domino tiling problem with fusions

I know several specific variations of the domino tiling problem has been determined to be decidable or undecidable, such as the seed domino problem. I have a variation which I have not been able to ...

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### Comparing spectral radius of two graphs using the entry of Perron vector

Suppose we have a graph $G$.
Let $A$ be the adjacency matrix of $G$ and $x$ be the corresponding Perron vector.
Let $x = (x_1,x_2,\cdots,x_n)^t$, where $x_i$ corresponds to the vertex $i \in V(G)$.
We ...

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### SDE with non-degenerate diffusion visits every point

I am asking an extension of the question here for SDEs of the Ito form.
Consider the SDE $dX_t =\sigma(X_t) dW_t$, where $W$ is a $d$-dimensional Brownian motion and $\sigma:\mathbb{R}^n\to \mathbb{R}...

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### For curves $C$ of genus $1$, the period (or index?) of $C$ is greater than $1$ iff $C(k)$ is empty

As the title suggests, does anyone have a reference for the proof of this fact? Actually, I can't remember where I've seen it before, or if I even remembered the statement correctly. Here are some ...

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### Yang-Mills algebra and lower central series of surface groups

Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area.
First, in "...

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### When is Euclidean distortion finitely determined?

The Euclidean distortion of a metric space $X$, denoted $c_2(X)$, is the infimum of $c$ for which there exists a map $f\colon X\to\ell^2$ such that
$$d_X(x,y) \leq \|f(x)-f(y)\|_{\ell^2} \leq c\cdot ...

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### What's the minimum ratio of positive cells such that the player has a positive probability to reach the boundary of a large random map?

A map with $(2n-1)$ rows and $(2n-1)$ columns of square cells is generated randomly as follows: the value of each cell is chosen from {1, -1} randomly and independently with probability {$p$, $1-p$}. ...

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### Intersection theory on schemes with Gorenstein singularities

Is there a good reference/book on Intersection theory on schemes with Gorenstein singularities? Does the construction of the intersection of cycles discussed in Fulton's book also hold for schemes ...

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### Classifying indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$

I'm now interested in classifying the indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$ : the group ring of $\mathbb{Z}/p\mathbb{Z}$ over the ring $\mathbb{Z}/p^{2}\...

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### Mumford's computation of the determinant of cohomology of a relative curve

In Integral Grothendieck-Riemann-Roch theorem, Pappas mentions that Mumford computed the determinant of cohomology of $f:X\to S$ a relative curve integrally, and thus proved an integral version of GRR ...

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### Reference for Varopoulos isoperimetric inequality with multiplicity

The Varopoulos isoperimetric inequality for a bounded domain $D$ in a nilpotent group $\Gamma$ of growth $n$ reads
$$
\# D \le \mathrm{const} \cdot (\#\partial D)^{n/(n-1)}
$$
See Ch. 6.E+ in Gromov's ...

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### Calculating degree via homotopy

I'm looking for a reference for the following:
Suppose that $f_1,f_2\colon S^n\rightarrow S^n$ are smooth maps. Let $i\colon S^n\rightarrow \mathbb{R}^{n+1}$ be the inclusion, and suppose that $F\...