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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Reference for Understanding Shelah's Proof of Vaught's Conjecture for $\omega$-stable Theories

I'm looking for a source to help me better understand Shelah's proof of Vaught's Conjecture for $\omega$-stable Theories (https://shelah.logic.at/files/95409/158.pdf). An obvious candidate is Makkai's ...
Tesla Daybreak's user avatar
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Automorphism groups of subshifts and factor maps

Let $\pi : X \to Y$ be a factor map between subshifts over finite alphabets. Let $\operatorname{Aut}(X)$ and $\operatorname{Aut}(Y)$ stand for automorphism groups of these shifts. We say that $\varphi ...
Dominik Kwietniak's user avatar
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Reference request: Elliptic regularity estimate in domains with $C^{1,\alpha}$ boundary

I'm wondering if there is a reference for the following (or if it's not true). Let $\Omega$ be a bounded domain with $C^{1,\alpha}$ boundary, where $0<\alpha<1$. For the inhomogeneous Dirichlet ...
Benjamin Pineau's user avatar
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Have the affine simplicial line arrangments been enumerated?

I am looking for a classification (or attempt at enumeration) of affine simplicial line arrangements. A line arrangment is a family of straight lines in $\Bbb R^2$. It is simplicial if all regions are ...
M. Winter's user avatar
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Referring to the countability of $\Bbb Q$ as "Cantor's first diagonal argument"

I had a discussion with one of my students, who was convinced that they could prove something was countable using Cantor's diagonal argument. They were referring to (what I know as) Cantor's pairing ...
Tristan vd Vlugt's user avatar
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222 views

Group homomorphism from $\mathrm{GL}_p$ to $\mathrm{SL}_p$ in characteristic $p$

If $k$ is a commutative field of characteristic $p>0$, then the map $$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$ is a group ...
Tom De Medts's user avatar
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Circle numbers on edges of a graph

Let $k$ vertices in a graph be given. Some pairs of vertices are connected by an edge, each edge is labeled either $\{1,2\}$, $\{1,3\}$, or $\{2,3\}$. We can circle some of the numbers on the edges. ...
Karo's user avatar
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Mapping space between $n$-groupoids is an $n$-groupoid

Consider two simplicial sets $K$ and $L$. Their mapping space (or mapping complex) is the internal hom of simplicial sets, i.e. $\underline{\mathrm{Hom}}(K,L)$, where $$ \underline{\mathrm{Hom}}(K,L)...
SetR's user avatar
  • 81
6 votes
1 answer
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Simple proof of sharp constant in DKW inequality

The DKW inequality says that if $F_n$ is the empirical CDF corresponding to real-valued random variables $X_1, \dots, X_n$ distributed identically and independently from a distribution with CDF $F$, ...
Drew Brady's user avatar
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141 views

$\infty$-categorical enhancement of $\mathsf{D}_\mathsf{B}(\mathsf{A})$

In this question, it is asked why we like to consider $\mathsf{D}_\textrm{qc}(X)$ rather than $\mathsf{D}(\mathsf{QCoh}(X)).$ Professor Cisinski answers rather convincingly that the $\infty$-...
Stahl's user avatar
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Usefulness of total algebras and exotic generating series

In his first Algebra volume, Bourbaki [1] defines the structure of a “total algebra” i.e. the space of functions on a monoid $M$ (to a ring $k$) with the convolution product ( a function $f:\ M\to k$ ...
Duchamp Gérard H. E.'s user avatar
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583 views

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 ...
Georgonzola's user avatar
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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>...
Sam Sanders's user avatar
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Finiteness of wildly ramified cohomology

$\newcommand\p[1]{\left(#1\right)}\newcommand\Char{\operatorname{char}}\newcommand\Gal{\operatorname{Gal}}\newcommand\b[1]{\left\{#1\right\}}$ Let $K$ be a global field. All cohomology below is fppf-...
Niven's user avatar
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"Reference Request" for a lecture note by C. Skinner: Galois Representations, Iwasawa Theory, and Special Values of $L$-functions

This was originally posted on math.stackexchange as https://math.stackexchange.com/questions/4589793, where I was suggested to move it here. I'm searching a lecture note by C. Skinner named "...
Tongchen Xu's user avatar
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Every Polish space is the image of the Baire space by a continuous and closed map, reference

The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501) Every Polish space (i.e. every separable ...
Lorenzo's user avatar
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Heat Flows and spatial singularities

While working on an abstract problem, I came up with the following question: Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
Alexander Dobrick's user avatar
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Vector algebra in a Tarski space

By a Tarski space I understand a mathematical structure $(X,B,E)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and a 4-ary equidistance relation $E\subseteq X^2\times X^2$ ...
Taras Banakh's user avatar
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Have we discovered constructions for natural fractional dimensional spheres?

I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
Sidharth Ghoshal's user avatar
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130 views

A theorem by R.L. Moore

The following result is due to R.L. Moore. Let $K\subseteq\mathbb C$ be compact. Suppose that $K$ is connected, and that $\mathbb C\setminus K$ is connected. Then $\partial K$ is connected. Does ...
ray's user avatar
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On improvements of the GPY sieve

When $\chi_\mathbb P(n)$ denotes the characteristic function of primes and $\mathcal H=\{h_1,h_2,\dots,h_k\}$ is some admissible $k$-tuple, the GPY sieve can be formulated as follows: $$ S(x)=\sum_{x&...
TravorLZH's user avatar
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406 views

“Cohomological equation” in dynamical systems

Let $$\dot{x}=Ax+v_r(x)+v_{r+1}(x)+ \dots$$ with $x \in \mathbb{C}^n$ and $v_r: \mathbb{C}^n \to \mathbb{C}^n$ a homogenous, polynomial function of order $r.$ Then, being able to find a suitable $h$ ...
display llvll's user avatar
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178 views

Does combinatorial deleted product become equivalent to the topological deleted product after enough subdivision?

Suppose $X$ is a topological space. Define the (topological) $n$-fold deleted product of $X$ to be the space or ordered $n$-tuples of pairwise distinct points in $X$. $$F(X, n):= \{(x_1, \ldots, x_n)\...
Gregory Arone's user avatar
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364 views

Is this just a numerical accident or what?

In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation $$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m} =\prod_{...
T. Amdeberhan's user avatar
6 votes
0 answers
160 views

Symplectic cohomology of $T^* \mathbb{CP}^2$

I'm looking for an explanation for why the symplectic cohomology $SH^*(T^* \mathbb{CP}^2,\mathbb{Z})$ is 2-torsion (I heard this in passing; perhaps it's not even true!). By a clever argument that I ...
inkievoyd's user avatar
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Recent literature on the gaps of reals on $L$ or other inner models?

I'm doing my Bachelor's Thesis on Gödel's constructible universe $L$. I'm interested in the gaps without new reals (sets of natural numbers) in the hierarchy, as presented in Gaps in the constructible ...
Martín S's user avatar
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Existing literature on logics "describing their own equivalence notions"

Say that a regular logic $\mathcal{L}$ is self-equivalence-describing (SED) iff for every finite language $\Sigma$ there is a larger language $\Sigma'$ containing at least $\Sigma$ and two new unary ...
Noah Schweber's user avatar
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0 answers
196 views

Consistency strength of Sy Friedman's result about admissibility spectrum

A result by Sy Friedman in his book "fine structure and class forcing", is that, assume $0^\sharp$ exists, there exists a real number R such that the ordinals admissible in R (called $\...
Reflecting_Ordinal's user avatar
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0 answers
392 views

Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)

For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as ${\displaystyle \eta (q) =q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$ By an $\eta$-quotient ...
Davood Khajehpour's user avatar
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0 answers
114 views

Original reference for the correspondence between commutative algebraic theories and commutative monads

Commutative algebraic theories were introduced by Linton in the 1966 paper Autonomous Equational Categories. Commutative monads were introduced by Kock in the 1970 paper Monads on symmetric monoidal ...
varkor's user avatar
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6 votes
0 answers
166 views

Higher homotopy groups of an orbifold

Given an orbifold $\mathcal{O}$, I have seen many ways to define the orbifold fundamental group: Thinking of $\mathcal{O}$ as a groupoid $\mathcal{G}$, $\pi_1^{orb}(\mathcal{O})$ can be defined as ...
CuriousUser's user avatar
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6 votes
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115 views

Premeasurability of affiliated operators for type $\textrm{III}$ von Neumann algebras

$\DeclareMathOperator\dom{dom}$If $M\subset B(H)$ is a semifinite von Neumann algebra with faithful, normal, semifinite trace $\tau$, then a closed operator $T:H\rightarrow H$ intertwining the action ...
Jon Bannon's user avatar
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6 votes
0 answers
118 views

Explicit homotopy for Hochschild chains from natural isomorphism

Let $A,B$ be $k$-linear (possibly, dg-)categories, let $f,g:A\to B$ be two linear functors, and let $T:f\Rightarrow g$ be a natural isomorphism. If one denotes by $C_\bullet(A,A)$ the standard ...
DamienC's user avatar
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6 votes
0 answers
154 views

$\mathfrak{sl}_2(\mathbb{Z})$'s properties as a Lie algebra over a ring

I was wondering what was known about $\mathfrak{sl}_2(\mathbb{Z})$ as a Lie algebra; in particular, what is known about its representation theory? I know of some texts which treat Lie algebras ...
Catherine Li's user avatar
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0 answers
139 views

Dickson's conjecture for Beatty sequences

A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided ...
Joshua Stucky's user avatar
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0 answers
544 views

What are the topics in noncommutative algebraic geometry?

Preface: I know very little about noncommutative algebra and noncommutative geometry, so please feel free to make improvement suggestions for my question. Also, to my knowledge there are several ...
6 votes
0 answers
312 views

Reference request: colored Motzkin path interpretation of Catalan numbers

Recall that a Dyck path of length $2n$ is a lattice path in $\mathbb{Z}^2$ from $(0,0)$ to $(2n,0)$ consisting of $n$ up steps $U=(1,1)$ and $n$ down steps $D=(1,-1)$ which never goes below the $x$-...
Sam Hopkins's user avatar
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6 votes
0 answers
208 views

Borel equivariant cohomology operations

Fix a group $G$. For an abelian coefficient group $A$ let $H^*_G(-;A):=H^*(EG\times_G-;A)$ be the Borel cohomology with $A$ coefficents. This is a functor from $G$-spaces to graded abelian groups ...
Mark Grant's user avatar
6 votes
0 answers
114 views

Determine the location of the boundary where the heat changes fastest

I am motivated by the following question: Given a uniform heat source in a convex domain, and suppose that the outside temperature is equal to $0$, can we determine where the long-time temperature ...
student's user avatar
  • 1,320
6 votes
0 answers
210 views

Optimal configurations on the flat torus

I'm studying (with my colleagues R.Piergallini and S.Isola) configurations of points on the flat torus which minimize an attractive or repulsive potential depending on the distance. Two model cases ...
Alessandro Della Corte's user avatar
6 votes
0 answers
206 views

Resolution graph of higher dimensional ADE singularities

I am looking for different configurations of the exceptional divisors arising from blowing up a higher dimensional ADE singularity (see p. 240 of this article of Bruns for a description of such ...
user43198's user avatar
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6 votes
0 answers
187 views

Base-change for simplicial spaces

Base-change for simplicial spaces Let us say that a map of simplicial spaces $X_* \to Y_*$ is a base-change if for all $n$ the canonical map $$ X_n \to (X_0)^{n+1} \times_{(Y_0)^{n+1}}^h Y_n $$ is ...
Jan Steinebrunner's user avatar
6 votes
0 answers
204 views

Parameter independence of Stanley's "content formula". Why?

For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively. R. Stanley remarked following ...
T. Amdeberhan's user avatar
6 votes
1 answer
582 views

How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?

$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
matt stokes's user avatar
6 votes
0 answers
147 views

Reference for "$\mathrm{PFA}$ implies $L(\mathbb{R}) \cap \bigcup_{1 \leq k < \omega} \mathcal{P}(\mathbb{R}^k)$ is productive"

The preprint of the recent result of Aspero and Schindler, "Martin's Maximum$^{++}$ implies Woodin's Axiom $(*)$", mentions productive pointclasses, and states that "$\mathrm{PFA}$ ...
Zoorado's user avatar
  • 1,215
6 votes
0 answers
228 views

Bigraded endomorphisms of the motivic sphere over a field

In An introduction to $\mathbb A^1$-homotopy theory ([1]) and On the motivic $\pi_0$ of the sphere spectrum ([2]) Morel describes a computation of $\bigoplus_{n\in \mathbb Z} [S^0, \mathbb G_m^{\wedge ...
Maxime Ramzi's user avatar
  • 13.3k
6 votes
1 answer
277 views

Proof of Denjoy-Riesz Theorem and Moore's Generalization?

The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set,...
John Samples's user avatar
6 votes
0 answers
268 views

Some characterization of Mahlo cardinals

Following the well-known characterization of supercompact cardinals by Magidor, in our paper we have defined the notion of a $\kappa$-Magidor model, for supercompact cardinal $\kappa$. I defined a ...
Rahman. M's user avatar
  • 2,341
6 votes
0 answers
187 views

Looking for a combinatorial proof for an identity involving $q$-Catalan triangles

Let $C_n=\frac1{n+1}\binom{2n}n$ be the Catalan numbers. Following my earlier post on MO, one fine colleague asked me if there is a $q$-analogue of the identity formed by the so-called Shapiro's ...
T. Amdeberhan's user avatar
6 votes
0 answers
311 views

Does this plane geometry theorem have a name (well-known)?

Consider three circles $(O_1)$, $(O_2)$, $(O_3)$. Denote the homothetic center of $\{$$(O_1)$, $(O_2)$$\}$ by $A$, the homothetic center of $\{$$(O_2)$, $(O_3)$$\}$ by $B$. Let $C$, $D$ be two points ...
Đào Thanh Oai's user avatar

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