Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
1,800 questions
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Homotopy classes of maps
This is a reference request.
A theorem of Hurewicz (published in Beiträge zur Topologie der Deformationen. IV. Asphärische Räume, Proc. Akad. Wetensch. Amsterdam, volume 39, deel 2 (1936), 215-224, ...
- 31
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How tight is the bound $P(\|X\|^2 \ge t |\langle a,X\rangle|) \ge 1 - t\sqrt{\frac{2}{m-1}}$, where $X \sim N(0, I_m)$ and $\|a\| = 1$?
Let $X$ be a random vector in $\mathbb R^m$ with iid $N(0,1)$ coordinates and let $a$ be a fixed unit vector in $\mathbb R^m$. In another post (SE link here https://math.stackexchange.com/a/3792730/...
- 6,853
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References for the early history of Fontaine's tilting construction
Scholze attributes the tilting construction for perfectoid rings to Fontaine, who calls it "a classical construction in $p$-adic Hodge theory".
Would anyone happen to know an early reference where ...
- 4,164
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Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
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Regularity of Delaunay triangulation of a hypercube
First using a three dimensional unit cube as an example for the term "regularity", we can have two possible triangulations:
(A)
(B)
We say the lower triangulation is more "regular" than upper ...
- 339
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Earliest source for a Lie algebra construction
I am looking for the earliest reference to the fact that any associative algebra becomes a Lie algebra with bracket $AXB-BXA$, where $X$ is a fixed element of the algebra. This is observed in the ...
- 3,873
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Reference for instability of Newton basins of polynomials at "separation" of a multiple root
In a previous question on MO I mentioned that I had convinced myself of the following:
When $f_p(z) = z - p(z)/p'(z)$ and $p$ is a complex polynomial, the Julia set, $J(f_p)$ is not continuously ...
- 709
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A table for irreducible integral representation of finite cyclic groups
Is there such a table where the irreducible integral representations of finite cyclic groups
are listed?
Edited:
Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
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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 ...
- 2,720
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880
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Veblen function with uncountable ordinals & beyond
Disclaimer: I am not a professional mathematician.
Background: I have been researching large countable ordinals for awhile & I think the Veblen function is particularly eloquent. My understanding ...
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Euler characteristic - reference question
Let $X$ be an algebraic variety over $\mathbb C$ and let $\mathcal F$ be a constructible sheaf on $X$. It is well-known that the Euler characteristic of the cohomology of $\mathcal F$ is equal to the ...
- 7,037
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Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 10.5k
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References for Littelmann Path models and related combinatorial objects
I am looking for a reference (preferably textbook or lecture notes) having Littelmann path models and it's uses in representation theory, combinatorics. I am aware of the original papers by Littelmann....
- 427
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Simple representations of products of algebraic groups
I am looking for a reference for the following assertion that I believe to be true. All representations are assumed to be finite-dimensional.
Let $G_1$ and $G_2$ be affine algebraic group schemes ...
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References for constructible sheaves on complex analytic stacks
I'm looking for references on constructible sheaves and the six operation formalism on analytic stacks (stacks fibered over complex analytic spaces). Does anyone have some suggestions?
Basically I ...
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Exponential deconvolution using the first derivative
There is an interesting observation using the first derivative to deconvolve an exponentially modified Gaussian:
The animation is here at terpconnect.umd.edu.
The main idea is that if we have an ...
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Non-trivial examples of regular Lagrangian flow in BV case
What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant?
With concrete I mean that we can compute the flow ...
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References request: Auslander-Reiten theory of algebras like $B_{k,n}$
In the paper A categorification of Grassmannian cluster algebras, an algebra $B_{k,n}$ is defined as follows.
Denote by $C=(C_0, C_1)$ the circular graph with
vertex set $C_0=\mathbb{Z}_{n}$ clockwise ...
- 6,201
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Monotone version of one-dimensional Whitney extension theorem
Is there a version of the Whitney extension theorem that would extend a monotone $C^\infty$ function on a compact subset of $\mathbb R$ (satisfying the usual Whitney's compatibility conditions) to a ...
- 29.1k
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What does the regular representation of the coinvariant ring of a unitary reflection group look like?
Let $V$ be a complex vector space of finite dimension $n$ and let $W$ be a finite unitary reflection group. This is, $W$ is a subgroup of $GL(V)$ generated by reflections, i.e., elements $r \in GL(V)$ ...
- 3,271
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Amenability at infinity
I have a few questions about amenability at infinity for locally compact, second countable, Hausdorff topological groups. Recall that a locally compact group $G$ is said to be amenable at infinity if ...
- 1,702
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Reference for Connes Bourbaki membership or otherwise
Alain Connes being a leading French mathematician today one could ask whether he is a member of the Bourbaki group. Is there a published reference that would either refute or confirm this?
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A simple form of choice
While reviewing some categorical versions of the axiom of choice, it occurred to me that none of the formulations I'm aware of actually reflect how I use choice in practice: pronounce that we 'choose ...
- 10.1k
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Alternative proof of Liouville theorem for harmonic functions
From Prove Liouville theorem without using mean value property the following question arises:
To prove the Liouville theorem
If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 ...
- 217
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Reference request: English translation of Brieskorn 1970 paper
Is there any english (or french) translation of the following paper by Brieskorn (1970)?
Brieskorn, E., "Die Monodromie der Isolierten Singularitäten von Hyperflächen", Manuscripta Mathematica 2 (...
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Volume of PGL(2,F) \ PGL(2, A)
Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$?
This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. ...
- 11.2k
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528
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Karolyi's theorem for finite groups and its extensions
Suppose that $\mathbb A = (A, +)$ is a (possibly non-commutative) group, and denote by $p(\mathbb A)$ the minimum of $|S|$ as $S$ ranges in the set of non-trivial subgroups of $\mathbb A$, with the ...
- 10.5k
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References for the categories: DIFF PL LIP TOP
Is there any survey paper focusing on the study of DIFF PL LIP TOP categories?
- 1,101
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Reference request: complete, rigorous proof of compactification of moduli spaces of flow lines in Morse homology?
The result I'm looking for can be stated as follows (taken from Hutchings' notes):
Here the moduli spaces are referring to the spaces of flow lines of the negative gradient flow induced by the Morse ...
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Mixed (anisotropic) Sobolev spaces
Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, ...
- 23
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Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?
Let
$\Omega$ be a metric space,
$C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
$\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
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Reference request on a notion of independence for families of [real-valued] functions
This is basically another reference request.
Let $X$ be a set, and $\mathscr{F} = (f_i)_{i \in I}$ an indexed family of functions $X \to \bf R$. If $\preceq$ is a partial order on $I$, we say that ...
- 10.5k
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Entire composite square roots of functions of finite order
A composite square root of a function $g$ is a function $f$ such that $f(f(z)) = g(z)$. Not surprisingly, for arbitrary $g$ a function like this is hard to find. Specifically I am looking at functions ...
user78249
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Another combinatorial identity
Is it true that
$$\sum_{r=0}^p \sum_{i=0}^r a_{n,p,r,i}=0$$
for all natural $n$ and all natural $p\ge2n$, where
$$a_{n,p,r,i}:=\frac{(-1)^r (n+p-r-1)! (n p-i (r-i))}{i!(r-i)! (n-i)!
(p-r+i)! (n-r+i)! ...
- 128k
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Reference on generators of subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $...
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Seeking a bijective proof enumerating two partition sets: Part I
An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ of positive integers, for some $k\geq1$. Consider the following two sets of partitions of $n$. Fix a ...
- 43.2k
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679
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Reference on spectral fractional Laplacian
Are there Harnack type inequalities and Schauder type estimates for the spectral fractional Laplacian. References are welcome.
- 402
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627
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Numerically negative exceptional divisor on a surface.
Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...
- 2,215
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5
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611
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A good introduction to the study of the Thue Equation
Hi,
I am interested in studying the Thue equation, where we are concerned with a binary form $F(x,y) = a_0 x^r + a_1 x^{r-1}y + \cdots + a_r y^r$ and solutions of the form
$$F(x,y) = h$$
for some ...
- 26.9k
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Upper bound Wasserstein distance by $\chi^2$ distance
Given two random variables $X,Y$ which are both $\mathbb{N}$-valued and have the same expected value (which is some fixed positive constant), and denote their probability mass functions by ${\bf p} = (...
- 730
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570
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Semi-discrete Wasserstein distance to uniform
Does the $p$-Wasserstein distance have a simpler expression when applied to these two distributions :
A uniform distribution on $[0,1]^d$
A discrete distribution with $N$ equally-weighted point mass ...
- 686
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2
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$X$ Polish geodesic implies $(P_2(X), W_2)$ geodesic
If $X,d$ is a complete and separable space then the space of Borel probability measures with finite second moment on $X$ endowed with the Wasserstein distance $W_2$ is geodesic.
I am looking for a ...
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Bochner's theorem for measures of positive type
Is there a version of Bochner's theorem characterizing measures of positive type on a locally compact group? By a measure of positive type on the group $\Gamma$, I mean a measure $\mu$ satisfying $\...
- 195
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212
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Fourier mode decomposition and eigenvalues of Schroedinger operators with radial potential in N-dimensions
In the study the stability of minimal hipersurfaces $\Sigma \subset \mathbb{R}^{N+1}$ one is lead to study the Morse index of a Schroedinger operator $J := - \Delta_g + |A|^2$ (usually called Jacobi ...
- 141
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417
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Roots of determinant of matrix with polynomial entries
Let $p_1, p_2,\dots, p_n$ and $q_1,q_2,\dots,q_n$ be a collection of complex polynomials. Let $A$ be a $n \times n$ matrix satisfying
$$a_{ij} = \begin{cases} p_i(x) & \text{ if } i = j, \\ q_i(x)...
- 1,269
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First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
- 10.5k
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Sum of the digits in base $p+1$
Definition
Let $W$ be the function , defined as $W(a,b)=r$
given $a,b\in \mathbb{Z_+}$ and $a>1$
Take $m$ to be the integer s.t. $a^{m+1} \ge b > a^{m}$, i.e. $m = \...
- 599
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2
answers
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Place stabilizers for the absolute Galois Group
Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...
- 1,049
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2
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257
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Reference request on Min-Max theorem
Consider the following min-max problem
$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$
where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
user128095
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2
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If $(\mathbb M, \tau)$ is a topological monoid, is $\tau$ always induced by a [left] subinvariant semimetric?
Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).
Basically following some ideas of W. Lawvere (but not his ...
- 10.5k