# Questions tagged [reference-request]

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

12,408
questions

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### Does this theorem on tangential quadrilateral have a name?

Let $ABCD$ be a quadrilateral, $P$ be a point in the plane let $E$, $F$ be the projections of the incenters of triangles $\triangle CPB$, $\triangle BPA$ onto $PB$ respectively; Let $G$, $H$ be the ...

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77 views

### Orlik-Solomon algebra and hyperplane complements in positive characteristic

Let $k$ be an algebraically closed field of characteristic $p\geq 0$, $\underline H:=\{H_1,\dots, H_m\}$ a set of hyperplanes in $\mathbb A_k^n$ and $X:=\mathbb A^n-(\bigcup H_i)$.
Given a ring $R$ ...

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336 views

### Reference for the Brauer-Nesbitt theorem

In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two semisimple representations $\rho_1,\rho_2 : G\longrightarrow \operatorname{GL}_n(E)$ are ...

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36 views

### proper : (proper + $\omega^\omega$-bounding) = generic : x

If $P$ is a forcing notion, $A \subseteq P$ (usually an antichain), $q\in P$, then I write $A\cap q$ for the set of all conditions in $A$ which are compatible with $q$.
For a proper forcing notion $P$,...

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129 views

### Amenability, growth and asymptotic dimension

I recently found this question on MSE, relating growth of groups with whether they are amenable, elementary amenable or not. I would like to know if there is an extra relation to finite or infinite ...

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83 views

### Stability of super vector bundles

A super vector bundle is a $\mathbb{Z}_2$-graded bundle, see for example "Heat Kernels and Dirac Operators" of Berline-Vergne-Getzler section 1.3.
Does it exist an adapted notion of ...

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140 views

### What is this optimization problem called

Let $X$ be a set and $\mathcal{F}$ be a set of functions $f:X \to \Bbb{R}$ (for my purposes, it is fine to assume both sets are finite).
For a probability distribution $\mu$ on $\mathcal{F}$, we ...

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65 views

### A formula involving the heat kernel on the universal cover of a punctured plane

I am looking for the earliest reference to the following formula:
$$
\int_0^\infty\tilde{P}(1,e^{i\alpha},t)\frac{dt}{t}=\frac{1}{\pi \alpha^2},\quad \alpha>0,
$$
where $\tilde{P}(x,y,t)$ is the ...

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votes

**1**answer

296 views

### Product of vertex degrees of an edge in a planar graph

Let $G$ be a planar graph, which we may assume to be a triangulation, with vertex set $V$ and edge set $E$. Suppose the minimum vertex degree is at least 3, and suppose any two distinct edges share at ...

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175 views

### Unions=colimits in categories

The basic way to define a partial map $X\rightharpoonup Y$ in a category is as a span $X\hookleftarrow U\to Y$ in which the first map (the support) is mono and we call the second evaluation. These are ...

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votes

**1**answer

96 views

### Reference request: probabilistic models on climate (change)

I am looking for probabilistic models to address climate change. Are they known in the existing literature?
I have found the post Math behind climate modeling. concerning PDE models.
Many thanks for ...

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98 views

### A question on Gaussian small ball probability

Consider the random variable $$ G = \sum_{j=1}^{\infty} \lambda_j Z_j^2 $$
where $Z_j \sim_{\substack{i.i.d}} N(0,1)$ and $\lambda_j$ some non increasing sequence of positive numbers with $\sum_{j=1}^{...

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votes

**1**answer

376 views

### Constant term extraction using combinatorial Nullstellensatz

$\DeclareMathOperator\CT{CT}$Given a Laurent polynomial $g$, let $\CT(g)$ denote its constant term.
Consider the specific Laurent polynomial
$$f_n(x_1,\dots,x_r)=\left(1+\prod_{j=1}^r(1+x_j)+\prod_{j=...

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116 views

### Integers with exactly three factor pairs whose successors are relatively prime

I am interested in the following problem, and will appreciate pointers around how it can be solved – partially or fully – and/or indicators around whether it is even tractable:
Characterize $N \in \...

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**1**answer

148 views

### Reference for the Hodge diamond of the Iwasawa threefold

Let $X = G/\Gamma$ denote the Iwasawa threefold, where
$$G = \left\{\begin{pmatrix} 1 & z_1 & z_3\\ 0 & 1 & z_2\\ 0 & 0 & 1\end{pmatrix} : z_1, z_2, z_3 \in \mathbb{C} \right\},...

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votes

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128 views

### Quiver and relations for ADE singularities in dimension one

Let $A$ be an ADE-hypersurface singularity in dimension one.
For example in Dynkin type $A_n$, A is given by $K[[x,y]]/(x^2+y^{n+1})$.
Then $A$ is CM-finite and let $M$ be the direct sum of all ...

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votes

**1**answer

45 views

### Bounding eigenvalue/eigenspace perturbations for hermitian matrices

Let $H$ be a Hermitian $n \times n$ matrix. Let $V$ be another such matrix.
For real $t$, let us consider the one-parameter family
$$ H(t) = H + t V$$
of Hermitian matrices.
Kato's perturbation theory ...

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vote

**1**answer

89 views

### Open images of submetrizable spaces

In [Tka] the author writes:
"Every topological space $X$ can be represented as an open continuous image of a completely regular submetrizable space $Y$ (in other words, $Y$ admits a continuous ...

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63 views

### Almost(?) elliptic operators

I would like some references concerning the following subject.
Suppose that $\Omega$ is a bounded subset in $\mathbb{R}^n$ with smooth boundary and consider the following PDE there stated
$$L(f)(x) = ...

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votes

**1**answer

171 views

### Moment integrals and determinants

Let $USp(2n)$ be the compact symplectic group of size $2n$, $dA$ its Haar measure
of total mass one, and $\det(1−A)$ being computed for the standard representation of
$A\in USp(2n)$ as a matrix of ...

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votes

**1**answer

203 views

### Abstraction logic

As part of my research on building an interactive theorem proving system, I have discovered a new logic that I call Abstraction logic. I have written up the details here: https://doi.org/10.47757/...

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114 views

### Comments and reference-request on books for KK-theory

I am looking for a good reference to learn Kasparov's KK-theory, where my motivation is to understand (and hopefully can do something about) the Atiyah-Singer index theorem in terms of KK-theory.
I ...

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195 views

### Stability of vector bundles and corresponding coherent sheaf

Let $j:Y\hookrightarrow X$ be an embedding of projective complex manifolds. Let $E\rightarrow Y$ be a vector bundle and $S=j_*E$ the corresponding coherent sheaf on $X$ (see Push forward of a Vector ...

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88 views

### Any concrete book for renormalization to recommend？

Any concrete book for renormalization to recommend？ concrete Enough，and simple enough， both in mathematics and physics.
Thanks in advance.

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182 views

### A map that names itself

Call the walking arrow $\Delta_{1}$, containing exactly one nontrivial 1-cell $[0<1] : 0 \to 1$. I am interested in a map $\Phi : \Delta_{1} \to \mathrm{Type}$, such that $\Phi [0<1] = \Phi$ (...

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391 views

### Quasi-compact surjective morphism of smooth k-schemes is flat

I have precedently posted the same question on Math.Stackexchange (https://math.stackexchange.com/questions/4277856/quasi-compact-surjective-morphism-of-smooth-k-schemes-is-flat), but to no avail; I ...

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62 views

### Top coefficient of the Lagrange polynomial as average of (n-1)-st derivative

Is there a formula expressing the top coeffient of the Lagrange interpolation polynomial for a function as an average of its ($n-1$)-st derivative (divided by $(n-1)!$)?
I am looking for a reference; ...

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1k views

### Reference request: Elementary proof of Lang's theorem

A few months ago, I read a nice elementary proof of Lang's theorem:
Theorem: Let $G$ be a connected linear algebraic group over $\overline{\mathbb{F}}_p$ and let $F : G \to G$ be a Frobenius map. Then ...

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votes

**1**answer

115 views

### Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation
\...

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252 views

### Number of rational points over finite fields mod $q$ is birational invariant

I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...

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955 views

### Are groups determined by their morphisms from solvable groups?

$\newcommand{\Grp}{\mathrm{Grp}}$Consider the category of groups $\Grp$, and within it we have the solvable groups $S$. Then any group $G$ determines the functor from solvable groups: $$h_G:=\text{hom}...

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135 views

### Has this form of the heat equation been solved for the radiation boundary condition

Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...

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**1**answer

122 views

### finitely generated C*-algebra as $C(X)$

In the question ($C(X)$ as finitely generated $C^*$-algebra), the answer show that spectrum of an abelian unital finitely generated C*-algebra is homeomorphic to compact subset of $\mathbb{C}^{n}$. I ...

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232 views

### Enumeration of dominated Dyck paths

Using horizontal steps $(1,0)$ and vertical steps $(0,-1)$, consider the lattice paths starting from $(0,q)$ and reaching $(p,0)$ with $p$ horizontal and $q$ vertical steps. The set of such paths $\...

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88 views

### Convergence of stationary distributions of a sequence of Markov Chains

I fairly new in the field of Stochastic Processes and Markov Chains so excuse my ignorance.
My question is: If we have a sequence of Markov chains such that each one has a stationary distribution $\pi^...

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**1**answer

202 views

### Covering the surface below a convex function

Is it possible to find the smallest positive real number $c$ (or at least the smallest positive integer $c$) such that the following result holds for all functions $f$ satisfying some conditions?
Let ...

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66 views

### On a core for Neumann Laplacians

Let $D \subset \mathbb{R}^d$ be a bounded smooth domain. We consider the Neumann semigroup $\{T_t\}_{t>0}$ on $C(\overline{D})$. In other words, $\{T_t\}_{t>0}$ is the semigroup of the normally ...

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75 views

### Tail bounds for random Gaussian chaos?

Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-...

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**1**answer

69 views

### Estimates on the discrepancy of random sequences

The discrepancy of a $[0,1]$-valued sequence $n \mapsto \alpha_n$ is the quantity $$D(N; \alpha) \stackrel{\text{def}}{=} \sup_{(a,b) \subset [0,1]} \left|\frac{\#\{1 \leq n \leq N : \alpha_n \in (a,b)...

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244 views

### Does ACA prove categoricity of the reals?

$\def\f#1{\text{#1}}$Does $\f{ACA}$ prove that any two internally complete ordered fields are isomorphic?
Here internal completeness is expressed roughly as "every sequence of reals with an upper ...

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814 views

### A quotient space of complex projective space

Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z_0:\dots:z_n]$ its points. If we glue $[z_0:\dots:z_n]$ and $[\overline{z_0}:\dots:\overline{z_n}]$ for any $[z_0:\...

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67 views

### Equivalence between smoothly regular and analytically regular

I think the following statement is true.
Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically ...

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89 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 ...

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236 views

### Geometry book recommendation

Context and mathematical maturity: I have knowledge of the usual engineering math courses, meaning differential+integral+vector calculus, linear algebra, probability and statistics, etc. and some pure ...

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**1**answer

2k views

### Reference request: a tale of two mathematicians

I've heard tell the following anecdote involving Pierre Gabriel and Jacques Tits at least twice in a lapse of four years or so:
When P. Gabriel presented the theorem in a conference [sometime around ...

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83 views

### Generalization of Killing field

Looking for a 3D Killing vector field $Y$ that generalises the following 2D Killing vector field $X=\big(x\log(x),-y\log(y)\big)$ for $x,y \in (0,1).$
Notice how $X$ preserves the Lorentzian metric $g=...

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306 views

### Learning roadmap to 'Differential cohomology in a cohesive $\infty$ topos'

I am very curious to study arXiv:1310.7930 (henceforth:DCCT) but am not sure if I have the pre-requisites. I am familiar with basic algebraic topology (singular cohomology, classifying spaces, ...

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**1**answer

107 views

### Definition of a $\psi$-Banach space [closed]

Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space ...

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50 views

### Reference request: Étale base change of differential-graded algebras

I've asked this question on Math.StackExchange, but didn't receive an answer, so I'd like to try my luck here.
I'm looking for a reference for the following fact, which I've recently stumbled upon:
...

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**1**answer

134 views

### Dual Banach space $X^*$ complemented in $\mathrm{Lip}_0(X)$?

$\DeclareMathOperator\Lip{Lip}$Let $X$ be a real Banach space. The dual $X^*$ is a closed subspace of $\Lip_0(X)$. ($\Lip_0(X)$ denotes the space of real-valued Lipschitz functions $f:X\to\mathbb{R}$ ...