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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8 votes
1 answer
2k views

Density of smooth functions on Hölder spaces

The following result is often cited without reference in the context of PDEs: Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
6 votes
2 answers
2k views

Wasserstein distance and the Kantorovich-Rubinstein duality

The only few references I could find on this topic are either amateur blog posts (http://n.ethz.ch/~gbasso/download/A%20Hitchhikers%20guide%20to%20Wasserstein/A%20Hitchhikers%20guide%20to%...
5 votes
1 answer
145 views

Equivalence generated by Jacobian minors

Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim_k g$ if at each point in the domain, the determinants of all $k \times k$ ...
9 votes
3 answers
705 views

Homotopy group action and equivariant cohomology theories

Many of the introductory notes on generalized equivariant cohomology theories assume that one is working over the category of $G$-spaces or $G$-spectra. However, one thing that concerns me is that the ...
4 votes
1 answer
106 views

$AC^p$ curves and pointwise metric speed in abstract metric spaces?

For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if $$ d(\gamma(s)...
2 votes
0 answers
95 views

Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?

Nowadays there are many papers on the number theory using heuristics. I have read some of them. But I have no clear understanding of the Bayesian Probability(subjective probability). The concept of ...
4 votes
2 answers
522 views

Convergence of conditional measures for a convergent sequence of probabilities whose projection is constant

Setting Suppose $\mu_n$ is a sequence of probability measures on $[0,1]\times [0,1]$ converging to a limit probability $\mu$ meaning that $$ \lim_{n\to+\infty}\int f(x,y)d\mu_n(x,y) = \int f(x,y)d\mu(...
5 votes
1 answer
240 views

Permanent of a Kronecker product of matrices

It is well known that $\det(A \otimes B) = \det(A)^m \det(B)^n$ when $A$ and $B$ are square matrices of size $n$ and $m$ where $\otimes$ denotes the Kronecker product. Question: Is there a similar ...
3 votes
1 answer
170 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 $\...
7 votes
2 answers
655 views

Élie Cartan's paper "Les groupes réels simples, finis et continus" of 1914

Question 1. Does Élie Cartan's paper Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355 contain a classification of $\Bbb C$-linear involutions of simple ...
2 votes
0 answers
126 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 ...
8 votes
1 answer
609 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 ...
5 votes
1 answer
354 views

Minimum cardinality of a cofinal collection of countable subsets of a set

Setup Let $X$ be a set of cardinality $\kappa\geq \aleph_0$. Edit: Based on Todd Eisworth's suggestion: What is the minimum cardinality of a collection $\hat{X}$ of countable subsets of $X$ such that ...
32 votes
23 answers
29k views

Textbook recommendations for undergraduate proof-writing class

I am teaching the proof-writing class (for the 3rd time) in the Fall and plan to buck the party line and use a different text than the default Bond and Keane. My parameters are as follows: Logic, ...
6 votes
0 answers
295 views

When did the main conjecture in Vinogradov's mean value theorem first appear in literature?

Recently I was asked about the history of Vinogradov's mean value theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$...
12 votes
1 answer
349 views

Rational homotopy invariance of algebraic $K$-theory

Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra $$ K(...
0 votes
1 answer
134 views

Faithful representation of group of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, "Theory of groups of finite order". The group ($\mathbb{Z}_{p^{2}}\rtimes \mathbb{Z}_{p^{}}) ...
3 votes
2 answers
2k views

Properties of the total variation norm on space of totally finite measure (from Bogachev)

Let $(X,d)$ be a metric space, $\mathcal{B}$ the Borel $\sigma$-algebra on $X$, and $\mathcal{M}(X)$ the space of totally finite measures on $\mathcal{B}$. Let $\|\mu\|_{TV}$ be the total variation ...
3 votes
1 answer
227 views

Reference request: Variational techniques for complex "iterated" Lagrangians

I am interested in solving variational problems of the form $$ \min_u \int \Big\{L(x,y,u(x,y)) + \phi\Big(\int J(z,y,u(z,y))\,dz\Big)\Big\} p(x,y)\,dx\,dy. $$ for some known, smooth functions $L,J,\...
5 votes
2 answers
344 views

Frégier and Frégier's Theorem

A curious and interesting gem is Frégier's theorem, quoted here from David Wells: Choose any point $P$ on a conic, and make it the vertex of a right angle which rotates about $P$. Then the ...
6 votes
2 answers
835 views

"Well-known fact" that every irreducible 3-manifold with non-empty boundary has an incompressible surface

I have seen in several sources that this results holds, however none of them included the proof. Does anyone know where I can find one? Also, it would be great if someone could provide me with a ...
18 votes
2 answers
936 views

Comparing the Edelman-Greene bijection to David Little's bijection

In their 1987 paper "Balanced Tableaux", Edelman and Greene construct a bijection between standard young tableaux with staircase shape $(n-1,n-2, \dots , 1)$ and reduced decompositions of ...
2 votes
0 answers
355 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 ...
4 votes
1 answer
233 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 ...
1 vote
0 answers
105 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)...
6 votes
2 answers
394 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 ...
4 votes
1 answer
194 views

Simple trace formula with different spectral footprint?

A standard idea when dealing with the Arthur-Selberg trace formula (or a relative trace formula, for that matter) is to impose local conditions on the test function $f=\prod_vf_v$ to obtain a simple ...
0 votes
1 answer
73 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 ...
1 vote
2 answers
247 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 ...
31 votes
1 answer
2k 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 + \...
1 vote
1 answer
134 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 ...
1 vote
0 answers
72 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 ...
3 votes
1 answer
89 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 $\mu$ ...
5 votes
1 answer
255 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 ...
4 votes
2 answers
522 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 ...
10 votes
3 answers
1k 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 ...
3 votes
1 answer
116 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 $...
5 votes
0 answers
73 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 ...
0 votes
0 answers
106 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 ...
46 votes
1 answer
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 ...
5 votes
1 answer
383 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 ...
1 vote
1 answer
317 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 \...
2 votes
0 answers
111 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 ...
4 votes
2 answers
416 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 ...
6 votes
0 answers
332 views

Independence of characters with respect to polynomials

I came across the following property : Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors, $\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\mathcal{...
0 votes
0 answers
30 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 ...
3 votes
1 answer
756 views

Solving multilinear equations

Let $N=\{1,2,\ldots,n\}$. Suppose we are given $n$ equations, with each equation taking the form $\sum_{A\subseteq N}\left(c_A \prod_{i\in A}x_i \right) = 0$, where each $c_A$ is a real number ...
2 votes
1 answer
71 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(\...
7 votes
3 answers
1k 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? ...
5 votes
2 answers
525 views

Properties of measures that are not countably additive but have countably additive null ideals

This is a very naive question, maybe more of a reference request than anything else. Let $(X, \mathcal X)$ be a measurable space. If $m$ is a real-valued function on $\mathcal X$, we say that $m$ has ...

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