# Questions tagged [reference-request]

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

10,929
questions

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### Who classified varieties that are commutative groups?

Who are the authors of the theorems asserting that connected varieties/manifolds which are abelian groups are isomorphic to ${\bf R}^k \times {\bf T}^n$?
In the smooth setting, I guess this is due to ...

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**1**answer

115 views

### Are “strongly finite dimensional” homotopy invariant sheaves with transfers (locally) constant?

Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\...

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86 views

### Regarding extensions of finite groups by Tori

I know how to prove the following result. However, my proof is a little bit long and complicated and only uses fairly low tech results in group cohomology. It would be nice if I could find a citation ...

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### On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber

This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory.
Allow me to first give a minor introduction.
Let $(...

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572 views

### If $M$ and $N$ are closed and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true that $M$ and $N$ are diffeomorphic?

If $M$ and $N$ are closed smooth manifolds, and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true that $M$ and $N$ are diffeomorphic?

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291 views

### Group structure for distributive lattices

On the (finite) Boolean lattice there is a group structure given by the symmetric difference and this group is an elementary abelian 2-group.
Question: Does there exist a natural group structure on ...

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**1**answer

60 views

### Density function approximation with respect to $L^1$ distance

Given iid samples $X_1,...,X_N$ drawn from some unknown distribution with not necessarily continuous density function $f(x)$ are there any theorems/papers where based on the data $X_1,...,X_N$ an ...

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86 views

### Average value of a fractional part of a function

Let $f(x): \mathbb{R} \to \mathbb{R}_{\geq 0}$ be a smooth function. I am interested in estimating
sums of the form
$$
\sum_{ A < n \leq B } \{ f(n)\}
$$
where $\{ c \}$ denotes the fractional part ...

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144 views

### Dirichlet's unit theorem in finite characteristic

I'm looking for a source of the following analog of Dirichlet's unit theorem for finite characteristic fields:
Let $\mathbb{F}_p$ be a finite field and denote $K=\mathbb{F}_p(x)$ to be the field of ...

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vote

**1**answer

108 views

### Polynomial isometries of $\mathbb{A}^2_\mathbb{C}$

I have the following question, which I'm sure must be explored somewhere.
Consider a group of polynomial automorphisms of $\mathbb{A}^2_\mathbb{C}$ preserving a standard hermitian metric. Is there any ...

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74 views

### versal deformation ring of a p-divisible group with some tensors

I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with ...

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253 views

### References for quivers and derived categories of coherent sheaves for a string theory student

I'm a student mostly from physics knowledge hoping to learn about the math involved the string theory paper Topological Quiver Matrix Models and Quantum Foam.
Context: The topological string theory ...

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votes

**1**answer

62 views

### Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks

A version of Brauer's second main theorem is as follows:
Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$.
If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...

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299 views

### A modern reference to the Zsygmondy Theorem

I need to cite the classical Zsigmondy Theorem, which was proved in 1892.
Is there any modern reference to this theorem?
I mean some standard textbook in Number Theory containing this theorem together ...

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votes

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848 views

### A generalization of metric spaces

Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i....

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26 views

### Necessity of conditions $N$ odd, square-free and $\chi$ quadratic in Kohnen's plus space - modular forms of half-integral weight

Kohnen introduced the "plus" space as a subspace of the space of modular forms of half integral weight, first in his 1980 paper and then generalized the work in a later 1982 paper. Why is ...

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90 views

### Lower bound for diagonal Ramsey numbers —- reference request

Using the first moment method, in 1947 Erd\H{o}s gave a lower bound on the diagonal Ramsey numbers $R(k,k)$:
$$
R(k,k) \geq (1+o(1))\frac{k}{e\sqrt{2}} 2^{k/2}.
$$
In 1975 Spenser used the Lov\’asz ...

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180 views

### How to describe the compact real forms of the exceptional Lie groups as matrix groups?

I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe ...

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100 views

### Existence of a special function

Consider a $C^2$ bounded domain $D$ of $\mathbb{R}^d$. Let $b \subset \partial D$ a non-empty part of the boundary. Let $n(x)$ be the unit outward vector on $\partial D$.
Is there any smooth function $...

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49 views

### Reference request: associative subalgebras of Cayley algebras are at most 4-dimensional

By a Cayley algebra I mean an 8-dimensional algebra (over an arbitrary field) formed in the Cayley-Dickson process. (They are also called octonion algebras, but I prefer to reserve the term octonion ...

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41 views

### Parabolic inductions for p-adic reductive groups

So I wish to ask for articles/comments surveying conjectures and theorems about parabolic induction for p-adic (non-archimedean case) reductive groups, and how local Langlands behaves under such. That ...

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1k views

### Summing infinitely many infinitesimally small variables makes sense in algebra

There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra:
Consider the ring of ...

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484 views

### Lattices on classical combinatorial families

I am asking for examples of lattices defined on classical combinatorial families, such as Permutations, Catalan objects, set partitions or integer partitions, graphs.
I am mosty interested in lattices ...

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**3**answers

275 views

### Number of permutations with longest increasing subsequences of length at most $n$

Is there a known expression for, or a nontrivial upper bound on, the number of permutations in $S_k$ with longest increasing subsequence of length at most $n$?
Let $l(\sigma)$ denote the length of the ...

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60 views

### Is there a name for and/or reasonably nice characterisation of “mixingly physical” measures?

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a diffeomorphism, and let $\mu$ be a probability measure on $M$ with compact support.
As stated in the ...

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**1**answer

302 views

### GCH implies acceptability

I have been studying the concept of acceptability, particularly in its relation to GCH.
There are many versions of it in the sources I have found, with some slight variations, and some of them are ...

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19 views

### Maximum nonintersecting interval pick

This surely has been solved in the context of scheduling already! (Shall I ask on some computer SE instead?)
Assume we have a set of closed "intervals" on $\mathbb Z$ ($\mathbb R$ isn't ...

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259 views

### Are Chow groups invariant under universal homeomorphisms?

Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...

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48 views

### First order asymptotics for maxima of stationary Gaussians with vanishing covariance

Let $G$ be a centered stationary Gaussian process indexed by the integer lattice $\mathbb Z^d$. A straightforward Borel-Cantelli argument shows that
$$\limsup_{m\to\infty}\frac{1}{\sqrt{\log m}}\left(\...

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120 views

### is there a `nice` lattice on the set of unlabelled graphs with $n$ vertices?

It is easy to endow the set of vertex-labelled graphs with $n$ vertices with a lattice structure: take the union and the intersection of the edge set as meet and join respectively.
However, I wonder ...

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**1**answer

114 views

### Exit time estimate for a simple continuous-time random walk

Let $S = (S_t, t \geq 0)$ be a simple one-dimensional continuous-time random walk with total jump rate one, $S_0 = 0$. Denote by $T_k$ the time when $S$ exits the interval $I_k = [-k,k] \cap \...

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86 views

### On a transcendental number defined as a variation involving the Lambert $W$ function in the nested square root representation of the golden ratio

Define the real number $\xi$ satisfying
$$\xi=\sqrt{1+W\left(1+\sqrt{1+W(1+\ldots)}\right)}\tag{1}$$
where $W(x)$ denotes the main branch of the Lambert $W$ function, as reference I add that Wikipedia ...

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145 views

### Classification of finite simple groups with abelian Sylow 2-subgroups

In this MathSE question,
classification of finite simple groups with Abelian Sylow 2-subgroups,
credit is rightly given to John Walter. But in the introduction to his paper, Walter explicitly states ...

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227 views

### Algebraic geometry additionally equipped with field automorphism operation

I am looking for some facts on theory, which is essentially algebraic geometry but with field automorphisms added as 'basic' operations. (Precisely, I mean universal algebraic geometry for (universal) ...

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139 views

### A group action on another group action quotient: how to best describe the resulting structure and does it have a name?

Suppose I have an action $\alpha:G\times X\to X$ of a group $G$ on a set $X$ and, on top of that, an action $\beta:H\times(X/G)\to(X/G)$ of another group $H$ on the set of $G$-orbits.
Is there a nice ...

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282 views

### Bass-Serre theory textbook

I am a PhD freshman working on topological graph theory and geometric group theory. I would like to learn some Bass-Serre theory. What do you think is the best introductory textbook in this topic? ...

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731 views

### Is this formal noncommutative power series identity known?

I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series
$$ 1 + \...

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32 views

### Terminology: “transformed Brownian motion”

I cam across this article studies Markov processes which are functions of a Brownian motion. I general, if we relax the markov requirement, are such processes studied? If so do they have a name?
To ...

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158 views

### Generalized differential geometry based on Penrose's abstract tensor systems?

Penrose graphical notation has been an important precursor of string diagrams for monoidal categories. It was introduced in Penrose's paper Applications of negative dimensional tensors with intended ...

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48 views

### The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool

In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...

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**1**answer

96 views

### On $(\prod_{\substack{1\leq s\leq X\\s\text{ semiprime}}}s)(\sum_{\substack{1\leq s\leq X\\s\text{ semiprime}}}\frac{1}{s})$ as $X\to\infty$

Few weeks ago an user from Mathematics Stack Exchange answered my question On an inequality that involves products and sums related to the sequence of semiprimes (asked May 26). It seems that for ...

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143 views

### Intrinsic characterisation of a class of rings

This may be well known, but I was unable to find an answer browsing literature. Let us temporarily call a commutative (unital) ring $R$ an O-ring if there exists an integer $n \ge 1$, a local field of ...

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**1**answer

149 views

### Inequality for difference of consecutive atom probabilities for binomial distribution

Edit: This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and migrated to its ...

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79 views

### Simplicial matrices and the nerves of weak n-categories II, III, and IV

Duskin introduced his nerve functor (see the nLab or Kerodon) in the paper
Duskin, John W. Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories. [Link].
While three ...

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113 views

### An unpublished result of H. Hamm

Let $f$ be a polynomial on $\mathbb{C}^n$. Denote
$$X_{R,p} = \{|x|<R\}\cap \{|f(x)|<p\}.$$
In "On the polynomials of I. N. Bernstein" Malgrange writes that H. Hamm proved that $f^{-1}(...

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### Is there a better reference for existence/regularity for parabolic PDEs (and systems) than the book of Ladyzenskaja, Solonnikov, Uralceva?

The book of Ladyzenskaja, Solonnikov, Uralceva contains almost everything most people need yet the typesetting and notation is disgusting to the eye. Is there any better text that covers the same type ...

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100 views

### A maximization problem with permutations

Consider a partition $f:S_n\rightarrow [n]$ of $S_n$ into $n$ parts. Denote the permutations that map $j$ to $k$ by $s(j,k)$. Set $S(f):=\Sigma_{1\leq i,j\leq n}max_{1\leq k\leq n}|f^{-1}(i)\cap s(j,k)...

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222 views

### Induced map in K-theory by a “trivial” bimodule

Let $R$ be a ring (not necessary commutative) and let $P_{\bullet}$ be a perfect $R$-bimodule (chain complex). I will denote the category of perfect right $R$-chain complexes by $\textbf{Perf}(R)$. ...

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284 views

### From Zurab's integral representation for the Apéry's constant to almost impossible integrals

I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...

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**1**answer

100 views

### English translation of a Russian paper by Gordin and Lifšic

Unfortunately I can't read Russian, I was wondering if there is an English translation of this paper
“The central limit theorem for stationary Markov processes”, Dokl. Akad. Nauk SSSR, 239:4 (1978), ...