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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3
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0answers
71 views

Identity involving a quadratic term inside the Pochhammer symbol

This identity came up in my research: $$ \sum_{m=1}^n m^2 \frac{(\frac{xy}n + m-1)_{2m-1} (n+m-1)_{2m-1}}{(x+m)_{2m+1} (y+m)_{2m+1}} = \frac{n^2}{(x^2-n^2) (y^2 - n^2)}. $$ Here $n$ is a fixed ...
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21 views

Smooth approximation in Sobolev spaces for surfaces with boundary

Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$ with closure $\overline{\mathbb{D}}$, and let $\varphi:\partial \mathbb{D}\to \partial \mathbb{D}$ be any continuous homeomorphism. Let $\mu$ be a ...
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42 views

Are metrics of the form $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat?

Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric on $S^2$)...
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73 views

Literature on analogous arithmetic function of logarithm function

In number theoretical estimations, often we take the logarithms of a natural number to express it properly. A perfect example of this is the von-Mangoldt function. I am looking for an analogous ...
2
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0answers
65 views

Blowups of log del Pezzo surfaces at smooth points

It follows from a result of Küchle that the blowup of a smooth del Pezzo surface will again be del Pezzo, provided that the inequality $-K^2>0$ remains true after blowing-up. Let's say a surface is ...
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158 views

Künneth theorem in étale cohomology

I am searching for an account of the Künneth theorem in étale cohomogy. Does the Künneth theorem in étale cohomology also follow from the 6-functor formalism or some other formalism? It would be nice ...
6
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2answers
196 views

Terminology for a set that does not surject onto $\omega$ (in ZF)

Short question: Is there a standard term for a set $F$ such that there does not exist a surjection $F \twoheadrightarrow \omega$ (in the context of ZF)? More detailed version: Consider the following ...
3
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1answer
49 views

Proof of Levinson-Durbin algorithm

Is there any article or reference book with a full proof of the Levinson-Durbin algorithm used for solving linear system with a Toeplitz matrix ?
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189 views

Examples or references for this claim about elliptic Calabi-Yau threefolds

In this article (page 2) , the authors say: "it is expected, based on known examples, that Calabi–Yau threefolds of large Picard rank are always elliptically fibered, perhaps after flopping a ...
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2answers
159 views

Compressible Ebin-Marsden?

In Ebin and Marsden's paper Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, there is a footnote on the first page indicating that non-homogeneous cases and the case of ...
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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 ...
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2answers
99 views

On an angle distribution of a random linear subspace of a given dimension

$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
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120 views

Key ideas behind p-adic Baker's theorem

I'm trying to understand Kunrui Yu's series of papers [1 2 3] on lower bounds of linear forms of p-adic logarithms (i.e., p-adic Baker's theorem). I know the proof of the usual Baker's theorem through ...
7
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149 views

In need of help with parsing non-Archimedean function theory

My current work revolves around studying functions from the $p$-adic integers to the $q$-adic rationals, where $p$ and $q$ are distinct primes ("$(p,q)$-adic functions", as I call them). I'...
2
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1answer
79 views

Subset which maximizes $\frac{\int_E\min(p(x), q(x))}{\int_E\max(p(x), q(x))}$?

Let $p(x), q(x)$ be two p.d.f.s of distributions on $\mathbb{R}$. I am interested in finding the subset $E$ that maximizes the quantity $$\frac{\int_{E}\min(p(x),q(x))\mathrm{d}x}{\int_{E}\max(p(x),q(...
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2answers
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The source of the Integral

Wolfram alpha calculates the integral $$\int\limits_0^\infty \frac{x^2\ln{x}}{e^x-1}dx=2\zeta^\prime(3)+3\zeta(3)-2\gamma\zeta(3).$$ However, I need to cite the source of this identity (the table of ...
3
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1answer
99 views

Getting the "salient" geometric objects out of an abstract congruence group

I'm not entirely sure what I'm trying to ask. According to my understanding of the Erlangen programme, each "geometry" (in the sense of Euclidean or hyperbolic or elliptic geometry) is ...
5
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1answer
336 views

Reference request: Generic k3 surface has Picard number 1

I keep running into the statement that "the generic k3 surface has Picard rank 1". For instance the answer of this question (end) and this paper (following Example 1.1) or this paper (proof ...
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109 views

Checking elementary proofs with proof checkers

I am not sure if this is the right place to post this, but I have seen discussions related to proof checking here, so let me post it. If there is better place for it, please give me a hint as to where ...
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74 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 ...
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262 views

Is there any literature on $\sum_{i=1}^{k} \left[ {k \atop i} \right] \zeta(i+1) $?

As per these questions, I'm trying to evaluate $$\sum_{n=2}^{\infty} \big{(} \zeta(n)^{2}-1 \big{)} = 1+ \sum_{m=2}^{\infty} \frac{H_{-\frac{1}{m}}}{m}. $$ Here, $H_{x}$ is a generalized Harmonic ...
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1answer
148 views

Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?

Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$. Also $\lambda_n$ is given as a sum over the non ...
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83 views

Translating set-theoretic concepts to polymorphic type theory or beyond

I've been trying to read Coquand's "An Analysis of Girard's Paradox" lately. I've noticed that he gets a type-theoretic variant of Burali-Forti's paradox once he extends Church's system with ...
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27 views

selection theory for normal non-paracompact domains?

Are there theorems in selection theory without either paracompactness or convexity assumptions ? That is, a theorem that claims existence of selections for any (perfectly or hereditary) normal spaces, ...
2
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0answers
175 views

Ambiguity about the exact definition of coefficients of modular forms

You can see the parts after my questions in the boxes. I received the answer to my first question in the comments. I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
3
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0answers
55 views

Every Borel linearly independent set has Borel linear hull (reference?)

I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone. Theorem. The linear hull of any linearly independent Borel set in a Polish ...
2
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0answers
58 views

Non-associative Clifford algebra

Let $V$ be a finite-dimensional $\mathbb{R}$ vector space equipped with a symmetric, bilinear form $b : V \times V \to \mathbb{R}$. My question is if there exists an analog of a Clifford algebra in ...
6
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1answer
151 views

Properness for uncountable models

There are certain generalizations of the notion of a proper forcing to uncountable cardinals in the context of forcing iterations. For example, the ones introduced by Eisworth, and by Roslanowski and ...
33
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2answers
2k views

Can the nth projective space be covered by n charts?

That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$? I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
3
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0answers
48 views

Simple-looking inequality for Fourier series

I am looking for a reference (and possibly some history) for the following inequality for finite sums of complex exponentials. Let $+\infty>b>a>-\infty$. There exists some constant $C>0$ ...
2
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0answers
29 views

Parabolic system with coupling in the diffusion

Let's consider the parabolic system $$ \begin{cases} u_t - \Delta u -a\Delta(uv) = 0 \\ v_t - \Delta v - b\Delta(uv) = 0 \end{cases} $$ with $a,b >0$. What is the name of this system? Are there ...
2
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1answer
210 views

Can $\int \Big{(} \frac{1}{e^{x}-1} - \frac{1}{e^{x}} \Big{)} \Big{(} I_{0}(2 \sqrt{x}) - 1 \Big{)} dx $ be evaluated?

Currently, I'm working on a problem pertaining to certain integrals involving the modified Bessel function of the first kind. On p. 59 of this book by Rosenheinrich, it is stated that $$\int e^{-x} I_{...
3
votes
1answer
167 views

Flatness of finitely presented algebras

Let $R$ be a commutative (noetherian, if needed) ring, let $f_1,\ldots,f_r\in R[x_1,\ldots,x_n]$ and $A=R[x_1,…,x_n]/(f_1,\ldots,f_r)$, when is $A$ flat over $R$? I found a nice answer for the case $n=...
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0answers
135 views

are trivial fibrations of finite CW-complexes soft for normal maps?

Are trivial Hurewicz fibrations of finite CW-complexes soft for normal maps, i.e. is it true that for any trivial Hurewicz fibration $f:Y_1\to Y_2$ and a closed subset $A$ of a hereditary normal space ...
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0answers
56 views

Spectral CLT for random matrices with iid entries

Let $\lambda_1(A_n),...,\lambda_n(A_n)$ be the random eigenvalues of a random $(n \times n)$ matrix $A_n$. We can define the empirical spectral measure $\mu_n^{A_n}$ on $(\mathbb{C},\mathcal{B}(\...
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0answers
77 views

Reference request for convex geometry?

I am looking for a reference for an elementary convex geometry. In Appendix A (page 1810) of this paper https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n3-p08-s.pdf by Green and Tao, ...
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1answer
574 views

Probabilistic proof for derivative of invariant distribution of a Markov chain

Let $P$ be an irreducible Markov matrix, and $\pi$ its stationary distribution. Let $D$ be a perturbation matrix which is zero except for two entries in row $r$: $$D_{rg}=+1 \qquad D_{r\ell}=-1.$$ Let ...
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33 views

A generalized notion of geodesic convexity

I was wondering if there is a notion similar to the one given here in the literature. Let $X$ be a uniquely geodesic space of finite covering dimension. Let $C$ be a subset of $X$ with $\dim C \leq \...
1
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1answer
101 views

Definition of Euler-Lagrange equation and properties, where can I find?

I'm studying a paper and in the introduction appears the following: It is well known that existence of critical points and solvability of Euler-Lagrange equations are related, and there is and ...
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0answers
51 views

On Anderson's treatment of homotopy (co)limits

In "Fibrations and Geometric Realizations"$\,^*$, D.W. Anderson mentions: The results on the existence of homotopy colimits and limits in abstract homotopy theory are also my own, and a ...
2
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0answers
34 views

A generalization of metrics taking values in partial orders

I'm investigating the origin of the following notion: Let $S=(S, +, <, 0)$ be a partially ordered semigroup with minimum $0$ (such that $<$ is invariant by the action of $+$ on both sides). A $S$...
14
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4answers
2k views

Collecting alternative proofs for the oddity of Catalan

Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if and ...
2
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0answers
82 views

Moment of the hitting measure of a subgroup

Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word ...
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0answers
88 views

Are there connected closed 4-manifolds admitting a regular Lagrangian distribution, and which are not Lorentzian?

In the category of real differential manifolds, connected (of $ C ^ {\infty} $ class in the sequel), closed of dimension 4, is there any manifold admitting a regular Lagrangian distribution and which ...
4
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0answers
150 views

As increasingly higher degree terms are added to a "random" polynomial, how fast do the roots approach the unit circle?

As increasingly higher degree terms are added to a "random" polynomial, the roots of a polynomial can be proven to approach the unit circle. For example, see the MathOverflow question Why ...
6
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1answer
144 views

Sharp Craig interpolation theorem for $L_{\omega_1 \omega}$

I’d like to know if a sharp version of Craig’s interpolation theorem for $L_{\omega_1 \omega}$ is already known or exists in the literature. By a “sharp” version of this theorem, I mean something like ...
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0answers
62 views

Objects equinumerous with $3$-ary partitions?

There is a concept of the so-called RP-compositions of an integer discussed by K. Q. Ji and H. S. Wilf in Extreme palindromes. They proved the following result too: Theorem. The number of RP-...
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0answers
132 views

Symplectic structure on moduli space of holomorphic Abelian differentials

I've heard a "symplectic structure" referred to on the moduli space of holomorphic Abelian differentials by numerous people / sources. I do not know how to interpret this - I am looking for ...
29
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1answer
486 views

A strange infinite fraction, and a functional equation

The following curious-looking fraction, with numerical value approximately $1.7302267782385217$, appears in this Reddit question: $$1+\cfrac{2+\cfrac{4+\cfrac{8+\cdots}{9+\cdots}}{5+\cfrac{10+\cdots}{...
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0answers
61 views

Convolution integral and its matrix representation

My background is chemistry and I was exploring some one dimensional deconvolution problems i.e., resolution of two or more overlapping peaks. A lot of excellent work was done in the 1970-80s. However, ...

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