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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12 votes
2 answers
364 views

Non-integrability of Abel's equation

I frequently encounter in the literature the statement that Abel's equation $$\frac{dy}{dx}=x+y^3$$ is not integrable. This is always stated without reference. My questions are a) What is the precise ...
10 votes
1 answer
269 views

Bibliography request: Entropy for continued fractions

Given a strictly positive real number $x$ we set $e(x)=\log(1+x)$ if $x$ is an integer and $$e(x)=\log(1+x)+\frac{1+\lbrace x\rbrace}{1+x}\left(e(1/\lbrace x\rbrace)-\log(1+\lbrace x\rbrace)\right)$$ ...
5 votes
2 answers
467 views

On the derivative of the Bernstein polynomial

$\newcommand\Z{\Bbb Z}\newcommand\De{\Delta}$For a natural $n$ and a function $g\colon\Z\to\Bbb R$, let $B_n g$ be the corresponding Bernstein polynomial, so that $$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...
0 votes
1 answer
217 views

Unpacking the plethystic substitution $h_n[n\mathbf{z}]$ in a paper by Aval, Bergeron and Garsia

I'm not familiar with the formalism of plethysm, so I need some help in unpacking the plethystic substitution $h_n[n\mathbf{z}]$ found in eqns. 5.6 and 5.9 of "Combinatorics of labelled ...
0 votes
0 answers
54 views

Reference request for equivalent Lipschitz smoothness conditions

For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
4 votes
1 answer
145 views

Reference request: Fréchet embedding

Given a separable metric space $(X,d)$, we have an isometric embedding $\iota:X\to\ell^\infty$ given by taking $(x_n) _{n \ge 0}$ to be the countable dense subset and sending $\iota(x)_n=(d(x,x_n) - d(...
2 votes
0 answers
208 views

What are the Hodge and log Hodge groups of $M_{g,n}$?

I would like to know, ideally with a reference, what the Hodge and log Hodge numbers of the moduli space of stable curves $\bar M_{g, n}$ are. At the very least I'd like to know the genus zero case $g ...
7 votes
0 answers
115 views

Stochastic analysis on nuclear Fréchet spaces

This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise. A lot of the time in infinite-...
0 votes
0 answers
25 views

reference request: mercer expansion and kernel underlying Sobolev spaces?

Let us define the periodic Sobolev spaces, for $s > n/2$ by $$ H_{s}([0, 1]^n) = \{f : [0, 1]^n \to \mathbb{R} :\mbox{for}~j\leq s, f^{(j)} |_{\partial[0, 1]^n} \equiv 0, ~~ \int_{[0, 1]^d} (f^{(s)...
32 votes
3 answers
2k views

Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?

Updated on Feb.16.2024 Fortunately for three of these series, Eqs. (1), (3) and (4), I have found a proof using classical methods which I placed in the Answers section below. (I doubt that there is a ...
24 votes
3 answers
1k views

References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...
9 votes
5 answers
945 views

English Reference for the Bénabou-Roubaud theorem

The Bénabou-Roubaud theorem links fibrational descent theory with monadicity. Particularly, it says that given a bifibration satisfying the Beck-Chevalley condition w.r.t some arrow $p$ in the base ...
0 votes
0 answers
137 views

Connectedness of deleted symmetric product

Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
4 votes
1 answer
414 views

Counting number of points on a lattice in a hypercube

Suppose I have a lattice $\Lambda \in \mathbb{R}^n$. Let $X_i >0$ for $i=1,..,n$. I am interested in some references regarding counting number of points of $\Lambda$ inside $[-X_1, X_1] \times \...
2 votes
2 answers
173 views

A Inequality in the paper by Kenig, Ponce and Vega

I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle", ...
1 vote
0 answers
61 views

Finitely presentable groups are residually finite if and only if they are universally pseudofinite

Suppose $G$ is finitely presentable. Then residual finiteness of $G$ is equivalent to $G$ satisfying the universal theory of finite groups (equivalently, to every existential statement true in $G$ ...
11 votes
3 answers
541 views

Was the small Desargues Theorem known to ancient Greeks?

My question concerns the classical Desargues Theorem and its simplest version The small Desargues Theorem: Let $A$, $B$, $C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...
1 vote
0 answers
106 views

Multiplicities of Galois representations in the semisimplification of the reduction of a Tate module

Let $C$ be a smooth proper curve, of genus $g$ over a number field $K$. Let $v$ be a prime of good reduction for $C$ above $p>2$, and let $T_pJ$ denote the $p$-adic Tate module of $J$, the Jacobian ...
0 votes
0 answers
132 views

A question and reference about Bombieri's article continued fraction of algebraic numbers

Above the Comments in the article continued fraction of algebraic numbers, there are some words on the unboundedness/cycle of coefficients of continued fraction of algebraic numbers "Thus, ...
0 votes
0 answers
17 views

reference request: product measures defined by a subsequence of measures

Suppose $\{\mu_n\}_{n\in\mathbb{N}}$ is a sequence of pairwise equivalent probability measures, each of which is defined on $\mathbb{R}$. Let $\bigotimes_n\mu_n$ be the product measure defined on $\...
2 votes
2 answers
73 views

Reference request for a subfamily of regular graphs

[Repost of same question math stack exchange which got no answers] I'm looking for literature on the following family of graphs: Call a regular graph $G=(V,E)$ (of regularity degree $d$) nice if there ...
2 votes
2 answers
193 views

Name for a certain type of cardinal

I'm not a set-theorist, but I hope this question is appropriate. This is just a question about names: Fix a cardinal $\lambda$. I'd like to know if there is a name for regular cardinals $\kappa$ such ...
5 votes
1 answer
778 views

Mathematics research relating to machine learning

What branch/branches of math are most relevant in enhancing machine learning (mostly in terms of practical use as opposed to theoretical/possible use)? Specifically, I want to know about math research ...
5 votes
0 answers
116 views

If chromatic polynomials for two graphs agree, can I always find an edge such that the two deletion-contraction minors have same chromatic polynomial?

Suppose I have non-isomorphic graphs $G$ and $H$ (which have at least one edge), but such that their chromatic polynomials are the same. Can I then always find an edge $e$ in $G$ and $f$ in $H$ such ...
2 votes
1 answer
65 views

Seeking Article "Generating random lattices according to the invariant distribution" by M. Ajtai

I am searching for a specific article titled "Generating random lattices according to the invariant distribution" authored by Ajtai. Despite being widely cited in various papers, I have been ...
18 votes
2 answers
8k views

The canonical line bundle of a normal variety

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...
5 votes
0 answers
181 views

Friedman's proof of covering lemma for $L$

There is a two-page proof of the covering lemma for $L$ using $\Sigma_n$ substructures (Theorem 3.10) in Sy Friedman's Fine Structure and Class Forcing, compared to the proof that spans about twenty ...
7 votes
1 answer
621 views

Fermat-quotient of "order" 3: I found $68^{112} \equiv 1 \pmod {113^3}$ - are there bigger examples known?

(I've taken this from MSE, it seems to be more appropriate here) I'm rereading an older text on fermat-quotients (see wikipedia) from which I have now the Question for $$ b^{p-1} \equiv 1 \pmod{ ...
4 votes
2 answers
310 views

Reference on Operads

I was reading 'Category for Scientists' by David Spivak and I'd like some references on operads with that kind of approach, using only "basic category theory", nothing too advanced. I ...
1 vote
1 answer
142 views

Name for an easy combinatorial game

What is the name of the following combinatorial game: Two players, moving in turn. Positions: $0,1,2,\ldots$. Moves: $n\longmapsto n-1$ or $n\longmapsto \lfloor n/2\rfloor$ if $n>0$. No move for $0$...
7 votes
1 answer
136 views

How to prove a 1-localization of a 1-category is already an $(\infty,1)$-localization?

I don't even know this fits in here or in Mathematics Stack Exchange, but let me ask. I'm new to simplicial stuff, so a good reference would be quite helpful. Let's say $C$ is a certain category, and ...
24 votes
1 answer
888 views

Reference request for a proof of the two-square Theorem

One can show (see below for a sketch of a proof) that every odd prime number $p$ can be written in exactly $(p+1)/2$ different ways as $$p=a\cdot b+c\cdot d$$ with $a,b,c,d\in\mathbb N$ satisfying $\...
4 votes
1 answer
229 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 ...
2 votes
0 answers
101 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
1 vote
0 answers
53 views

A question on generalized bases

I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
7 votes
1 answer
312 views

Literature about formalization of "natural reasoning" in mathematical logic

In "Logic of sheaves of structures", X. Caicedo justifies the logic he introduces stating (more or less) that assertions about a point should really be understood as assertions about a ...
8 votes
0 answers
175 views

Elkies' family of elliptic curves of rank 19

There is a widely cited fact that Elkies had found that infinitely many curves of rank 19 in 2006, in "Z^28 in E(Q), etc. Email to the number theory mailing list at [email protected]&...
17 votes
1 answer
834 views

Where did the military money go?

In older papers, one sometimes finds references to sources of funding directly linked to or overseen by military agencies. For example, I have memories of seeing acknowledgments to DARPA funding in a ...
8 votes
1 answer
186 views

A reference for forcing projections

The idea of a projection $\pi\colon\mathbb{Q}\to\mathbb{P}$ of forcing notions is something like a combinatorial stand-in for the fact that forcing with $\mathbb{Q}$ produces a generic for $\mathbb{P}$...
0 votes
0 answers
43 views

Two-parameter “$\varepsilon$-$\delta$ filtration” given a function between metric spaces

Let $X,Y$ be metric space and $f : X \to Y$ a (not necessarily continuous) function. I'm interested in the two-parameter filtration $(X_{\varepsilon, \delta})_{{\varepsilon, \delta} > 0}$ where $X_{...
3 votes
1 answer
286 views

Counting points on elliptic curves

Consider the Legendre family of elliptic curves $$E_a: y^2=x(x-1)(x-a).$$ Let $p$ be an odd prime. QUESTION. Is the following true? If $p\equiv 3\pmod4$ then number of solutions to $E_2$ over the ...
10 votes
0 answers
339 views

Examples of games developed purposely to analyze players' strategies for mathematics research

Background This question is about games that were created, developed, deployed and popularized1 by researchers because they wanted to learn more about some mathematical structure, and did so by ...
6 votes
0 answers
80 views

The meet of two dominant permutations in weak order of $S_n$

A permutation is called dominant if its Lehmer code is a partition, or equivalently if it avoids the pattern $132$. I can prove that given a permutation $v\in S_n$, there is a unique dominant ...
15 votes
4 answers
1k views

Geodesics in $\mathbb{R}^2 \times \mathbb{S}^1$ under "segment" metric

Represent the position of a unit-length, oriented segment $s$ in the plane by the location $a$ of its basepoint and an orientation $\theta$: $s = (a,\theta)$. So $s$ can be viewed as a point in $\...
23 votes
3 answers
2k views

Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$. Say that $S$ is an integer-distance set if every pair of points in $S$ is separated by an integer Euclidean distance. ...
2 votes
0 answers
47 views

Does this filtration have a name?

In the context of Ethier&Kurtz Markov Processes: Characterization and Convergence (Chapter 4, equation (3.2)) as well as the two papers Martingale problems for conditional distributions of Markov ...
7 votes
1 answer
151 views

Reference request for equivalences between different models of lax limits

There are several models for lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...
1 vote
0 answers
102 views

PDE coupled with the pronic numbers (related to triangular numbers)

I am studying the linear PDE: $$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
9 votes
1 answer
684 views

Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection

I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
1 vote
0 answers
108 views

What is the "weight" of an automorphic form for $\mathrm{PGL}_2$?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in ...