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
votes
0answers
73 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 ...
6
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1answer
185 views

Lipschitz-continuity of convex polytopes under the Hausdorff metric

Recently, I proved the following Lipschitz-continuity like result for convex polytopes: Let $A\in\mathbb R^{m\times n}$ and $b,b'\in\mathbb R^m$ be given such that $\{x\,:\,Ax\leq 0\}=\{0\}$ (which ...
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0answers
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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0answers
22 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 ...
6
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2answers
197 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 ...
2
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0answers
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 ...
6
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0answers
190 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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0answers
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 ...
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 ...
2
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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 ...
7
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4answers
180 views

Best source for classification of right-angled hyperbolic hexagons

A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose ...
8
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1answer
257 views

Reference for "Every compact quasinilpotent operator is the limit of nilpotent ones"

It was mentioned on Page 916 Problem 7 of Halmos's "Ten Problems in Hilbert space" that there is a proof for "Every compact quasinilpotent operator is the limit of nilpotent ones" ...
4
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0answers
132 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 ...
6
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1answer
188 views

Is the projectivization of a topological vector space Tychonoff?

Let $E$ be a locally convex topological vector space over $\mathbb{R}$. The projectivization $PE$ is the quotient of $E\backslash\{0_{E}\}$ with respect to the equivalence relation $e\sim f$ if $e=\...
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 ?
7
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2answers
160 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 ...
7
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1answer
489 views

Wu relation for Stiefel-Whitney classes and Steenrod square

Wu relation for Stiefel-Whitney classes and Steenrod square is given by the following cochain equation $$ Sq^n(x_{d-n})=u_n \smile x_{d-n} + \text{d} y_{d-1} \ \text{ mod } 2 $$ on a $d$-dimensional ...
18
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2answers
966 views

What sort of cardinal number is the Löwenheim–Skolem number for second-order logic?

In their paper “On Löwenheim–Skolem–Tarski numbers for extensions of first order logic”, Magidor and Väänänen make the following statement: “For second order logic, $\mathrm{LS}(L^{2})$ [the Löwenheim–...
5
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1answer
144 views

Permutations of a group that are eventually left translations

$\DeclareMathOperator\FSym{FSym}\DeclareMathOperator\Sym{Sym}$Notation: for $X$ a set, $\Sym(X)$ the group of permutations of $X$, and let $\FSym(X)$ be the subgroup of finitely supported permutations ...
2
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1answer
83 views

Change of variable and boundary data for Laplace equation

Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem $$ \begin{cases} -\Delta u = 0 & x \in \Omega \\ u = 1 & x \in \partial \Omega \end{cases} $$ Does it make ...
7
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0answers
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'...
4
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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 ...
2
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1answer
126 views

Wavelet transform stability to deformations

I've come across the following claim in a paper of Mallat: "High frequency instabilities [of a signal representation] to deformations can be avoided by grouping frequencies into dyadic packets in ...
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 ...
4
votes
1answer
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 ...
8
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0answers
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 ...
2
votes
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(...
7
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2answers
2k views

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 ...
4
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0answers
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 ...
5
votes
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 ...
25
votes
1answer
699 views

Looking for source: "How not to be a graduate student"

I remember having read, about 15 years ago, a transcript of a lecture given by Richard K. Guy, titled "How not to be a graduate student". He gave lots of advice, mostly humorous, concealing ...
2
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0answers
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 ...
11
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3answers
405 views

An open triangle problem in plane geometry

Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following: Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is ...
5
votes
1answer
291 views

Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract of $\mathcal{H}$

I am interested in sufficient criteria which ensure that a subset $X$ of a Hilbert space $\mathcal{H}$ is a Lipschitz (or at least uniformly continuous) retract of $\mathcal{H}$. Under which ...
2
votes
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 ...
4
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0answers
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 ...
3
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4answers
2k views

A chess question of W.T. Tutte [closed]

In "Graph theory as I have known it", p.12, Knights Errant, the late Tutte mentions as an aside the chess question "Does either Black or White have a certain win from the initial ...
9
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2answers
1k views

Grauert's criteria for ample line bundles

In their book "Compact complex surfaces", W.P. Barth, K. Hulek, C.A.M. Peters and A. Van de Ven refer to the following theorem: Let $X$ be a compact complex space and $L$ a holomorphic line ...
11
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1answer
964 views

Cohomology ring of classifying space of spin group $\mathrm{BSpin}(n)$

$\DeclareMathOperator\BSpin{BSpin}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\BSO{BSO}\DeclareMathOperator\BO{BO}\DeclareMathOperator\BPin{BPin}$In the answer for question: Homology of ...
0
votes
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 ...
2
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0answers
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 ...
7
votes
3answers
1k views

Is assigning the endomorphism object in some sense functorial?

Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the $\mathcal V$-valued hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's ...
33
votes
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 ...
0
votes
0answers
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, ...
3
votes
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 ...
6
votes
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 ...
370
votes
79answers
170k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...
2
votes
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 ...
23
votes
1answer
1k views

Definition of an n-category

What's the standard definition, if any, of an $n$-category as of 2020? The literature I can tap into is quite limited, but I'll try my best to list what I had so far. In [Lei2001], Leinster ...
28
votes
19answers
35k views

Good books on problem solving / math olympiad [closed]

I would want all book tips you could think of regarding problem solving and books in general, in elementary mathematics, with a certain flavour for "advanced problem solving". An example ...

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