# Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

14,437
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### Linear third order water wave pde admitting particular gamma factor solution. How do you understand evolution on vertical strip in complex plane?

I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...

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### An identity for characters of the symmetric group

I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...

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### Reference request: Discriminant of a $V_4$-extension of local fields is the product of discriminants of intermediate fields

Disclaimer - cross-posting: I already posted this question on MSE, here. In line with the accepted answer of this meta question, I am also asking it here, since it is a research-level question and it ...

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### Irrationality of cubic threefold (before Clemens and Griffiths)

I came across this notice, which seems to say Fano proved that a general cubic threefold is irrational back in 1940s. I'm interested in seeing this work, especially a proof without intermediate ...

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### The function $G(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k dy$ can be controlled when $|x|\rightarrow \infty$

In this paper, Lemma 6, Pinsky proves that $$H(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$$ attains its maximum in $x=0$ for $m<0$. This can also be proven using ...

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### Decrease of $(1/\zeta)^{(r)}(\sigma + i T)$ as $\sigma\to -\infty$?

What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$,
every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any ...

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### Some questions about the definition of Chern classes in Cheeger--Simons differential characters

In page 62 to 63 of the paper "Differential characters and geometric invariants" by Cheeger and Simons, they define, among other things, Chern classes taking values in differential ...

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### Are these finite semirings known?

I am trying to prove the properties below, and by doing this, I hope to find a way to speed up the computation of the below defined addition and multiplication. I am also interested if these finite ...

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### Reference needed for powers of semi-group generators

Let $\mathcal{L}$ be the infinitesimal generator of a Markov semi-group. I am looking for references that study powers of $\mathcal{L}$; i.e. $\mathcal{L}^n$, for $n\in\mathbb{N}$.
For example, if the ...

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### Wondering if Monsky-Washnitzer ever published a result claimed to be forthcoming in a later paper

At the very end of the paper Formal Cohomology I by Monsky and Washnitzer, they write the following:
"In some sense, the operator $\psi$ applied to a power series gives it "better
growth ...

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### Weighted least squares regression: Iterative modeling of variance

In chemical analysis, the instrument's signal are plotted as a function of chemical concentration. In general, higher the concentration higher is the response and the relationship is linear. At ...

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### On characters of the symmetric group: Part 2

This question is related to my earlier MO quest.
For an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\...

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### Number of divisors which are at most $n$

I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by
$$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...

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### Finding the DOI for a paper

How can we find the DOI of an old paper. For example, what are the DOI of the following papers?
Anderson, D.D. and Jayaram, C., 1995. Regular lattices. Studia Scientiarum Mathematicarum Hungarica, 30(...

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191
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### A probabilistic proof of van der Waerden theorem

Is there an elementary proof of van der Waerden's theorem on arithmetic progressions using probabilistic methods?

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### Reference request: number of antichains of a partially ordered set

Let $\mathbb{N}$ denote the set of all positive integers. For each $n \in \mathbb{N}$, define the set $$ P_n = \{ (a,b) \in \mathbb{N} \times \mathbb{N} : 1 \leq a \leq b \leq n \} $$ and consider the ...

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### Reference for a clear version of multigraded Serre-Grothendieck-Deligne correspondence local cohomology

The Grothendieck-Serre-Deligne correspondence states the following. Let $ R $ be a Noetherian, graded ring and let $ T $ be $ \operatorname{Proj}(R) $. If $ \mathcal{F} $ is a coherent sheaf on $ T $...

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### On Soergel's results concerning projectives modules in category $\mathcal{O}$

I am looking for a translated proof of two results of Soergel usually referred to as endomorphismensatz and struktursatz.
Both of those results were shown in the paper
Soergel, W. (1990). Kategorie 𝒪...

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407
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### On the existence of a real which is not set-generic over $L$

Recall that a real $r$ is set-generic over $L$ if there is a constructible forcing notion $\mathbb{P}$ and some $L$-generic filter $G\subset\mathbb{P}$ such that $r \in L[G]$.
I know that Jensen's ...

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### Factorization trees and (continued) fractions?

This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...

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### Defining ideal of a Schubert variety as a kernel

Consider the Plücker embedding of the variety of complete flags in $\mathbb C^n$: $$F_n\subset\mathbb P(\bigwedge\nolimits^1\mathbb C^n)\times\dots\times\mathbb P(\bigwedge\nolimits^{n-1}\mathbb C^n).$...

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### Smooth map between oriented manifolds

Let $f: M\rightarrow N$ be a smooth map between smooth closed oriented connected manifolds of same dimension.
Question: is it true that $f$ is smoothly homotopic to some smooth map $g: M\rightarrow N$...

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### On characters of the symmetric group: Part 1

Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\...

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### Are covering families of localizations stable under pushouts?

For a commutative ring $A$, we call a finite family of localizations $A \to A_{S_i}$ (where $S_i$ are some subsets of $A$) a covering if the canonical morphism $A \to \prod A_{S_i}$ is an effective ...

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### Smallest prime factor of numbers

The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...

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### Explicit character tables of non-existent finite simple groups

In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...

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### Closed formula for number of ones in a proper factor tree

Edit [2023 Dec 7]: One of my specific wonders, along with that of students, is around when a recursive formula might have – or be expected to have – an explicit or closed formula. What is the ...

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### Reference request: unfolding of Integral representation of an L-function

Are there any text or papers that thoroughly address unfolding of integral representation of an L-function such as D. Ginzburg's On Spin L-function for Orthogonal Groups page 762-763 and page 774 (or ...

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### Variants of the Bonk-Schramm embedding

Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...

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### On dense subspaces of $L^p$-spaces of finitely additive measures

Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for ...

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### Classifying spaces beyond CW complexes

We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...

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### When does an iteration not add functions $\eta\to V$ at the final stage?

I am interested in better understanding the following property:
Let us say that an iteration of forcings $\langle\mathbb{P}_\alpha,\dot{\mathbb{Q}}_\beta\mid\alpha\leq\gamma,\beta<\gamma\rangle$ is ...

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### Self adjoint operators from energy functionals

It is known that the equation
$$
\Delta f = 0
$$
on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet ...

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### Pfaffian elements and anomalies

If $X$ is a compact even dimensional spin manifold, then we have a family of chiral Dirac operators parametrized by $Met(X)$, the (infinite dimensional) manifold of Riemannian metrics on $X$. This is ...

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### Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?

I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.
Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...

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### List of equivalent conditions for the invariant subalgebra to be polynomial

Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by ...

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### Grothendieck purity for Brauer groups of stacks

Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...

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### The origin of a planar graph theorem of Steinitz and Rademacher

The subsequent statements are extracted from the article titled 'Generating r-regular graphs' (https://doi.org/10.1016/S0166-218X(02)00593-0).
A well-known classical theorem of Steinitz and ...

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### Does the interior of Pascal's triangle contain three consecutive integers?

This question defeated Math SE, so I am posting it here.
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...

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### Abel–Plana formula with fractional offset

The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...

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### Are “most” bounded derivatives not Riemann integrable?

Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...

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### Hereditarily just-infinite pro-$2$ groups

An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...

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### Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?

Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...

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### What do $\gamma$-radonifying operators radonify?

In the second volume of their Analysis in Banach Spaces, Hytönen et al. introduce the notion of $\gamma$-radonifying operator more or less as follow.
Let $(\gamma_j)_{j\in\mathbf N}$ be a sequence of ...

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### Reference for application of local cohomology to complex manifolds with points removed

Reference request - I am looking at Dolbeault cohomology on compact complex manifolds (not Riemann surfaces) with points removed. I have been told that the key to doing this is to look at Local ...

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### A result on symmetric closed monoidal categories

In this discussion from the categories mailing there is mention of the following result by Robin Houston, supposedly proved in 2006:
Theorem. Let $\mathcal{C}$ be a symmetric closed monoidal category,...

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### Fractional Bochner spaces $H^s((0,\infty); V)$

I'm looking for "intrinsic" definitions of fractional Bochner spaces such as $H^s((0,\infty); V)$ for a Banach space $V$ and exponent $s \in (0,1)$. I have seen these spaces being defined as ...

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### Injective hulls of metric spaces

In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...

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### Reference request for dinatural transformations arising from free Cartesian closed categories

Let $g_n$ be a discrete graph with $n$ nodes and $\operatorname{F}$ the free functor of the adjunction between the category of graphs and the category of Cartesian closed categories and functors, as ...

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### Pontryagin product on the homology of cyclic groups

Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...