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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8 votes
1 answer
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Finite domination and Poincaré duality spaces

Here are some definitions: A space is homotopy finite if it is homotopy equivalent to a finite CW complex. A space finitely dominated if it is a retract of a homotopy finite space. A space $X$ is a ...
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2 votes
0 answers
65 views

Sums over lattice points in homogeneously expanding domains

In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $...
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4 votes
1 answer
207 views

What are the jumps in the ramification filtration of the absolute Galois group of a local field?

Let $k$ be a (complete) discretely valued field and $\ell$ a Galois extension of $k$, possibly infinite. The Galois group $\Gamma=\text{Gal}(\ell/k)$ of $\ell$ over $k$ admits a descreasing, $\mathbb ...
14 votes
1 answer
343 views

Fourier's proof of reality of all roots of Bessel function $J_0(x)$

In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real. I want to ask if there is a modern version of this proof exist in literature? If someone ...
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17 votes
3 answers
442 views

Convergence of convex functions

I believe I can prove the following result. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$. Then $...
5 votes
1 answer
284 views

Eikonal equation - Snell's law

I am interested in equations of the form $|\nabla d|= F(x)$, where $F(x)$ is piecewise constant and $d(x) = 0$ on $\Gamma_D$, a subset of the boundary. In particular, like in the figure, one can ...
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4 votes
0 answers
123 views

Inequality for $q$-binomials

Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials) $$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$ Given two ...
0 votes
0 answers
44 views

Sum of products on a directed acyclic graph

Is there a textbook/paper that I can reference for the following problem? I am looking for a concise proof that I can cite. Let $G=(V,E)$ be a weighted directed acyclic graph, and consider $s,t\in V$....
11 votes
1 answer
803 views

Are there textbooks on differential geometry in the language of smooth sets or smooth derived stacks?

In differential geometry it is often natural to speak of infinite-dimensional manifolds (e.g., the manifold of mappings between two smooth manifolds). Different versions of generalized smooth spaces ...
7 votes
0 answers
122 views

Has anyone written about filtered Tannakian categories?

tl;dr Is there any source that discusses the concept of a filtered Tannakian category? I'm writing a paper with this notion and want to know if it's ever been discussed. The original book by Saavedra-...
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0 votes
1 answer
115 views

Class number of imaginary quadratic fields

Let $n$ be a positive squarefree integer, and let $h_n$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$. Then, is it true that $h_n$ is odd if and only if $n$ is a ...
2 votes
1 answer
279 views

Level spacing statistics for primes

In the preprint "Level Spacing Statistics for Primes", we have found some patterns of prime spacings, which may provide new insights on the distribution of primes: We would like to know ...
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3 votes
0 answers
94 views

A special type of differential equations

Working in optimal control of PDEs, I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state. Here is a simplified ...
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1 vote
0 answers
135 views

Special function: Pulse peak modified with a power term

PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...
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2 votes
0 answers
40 views

Topological rings with a final topology

Given a family of ring homomorphisms $ \phi_i : X \rightarrow Y_i $ where each $ Y_i $ is a topological ring and consider the initial topology on $ X $, i.e. the coarest topology such that each map is ...
1 vote
0 answers
39 views

Sherman-Davis type inequalities for non-negative operator in a Hilbert space with trivial kernel

Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of ...
9 votes
1 answer
564 views

The paper behind an olympiad problem

In IMO Shortlist 2013, there is a number theory problem: Determine whether there exists an infinite sequence of nonzero digits $a_1,a_2,a_3,...$ and a positive integer $N$ such that for every integer $...
1 vote
0 answers
26 views

Associating a matroid to a uniform hypergraph

For a fixed ground set $[n]=\{1,\ldots,n\}$, and for any matroid $M$ on $[n]$, specified as a collection of bases $B_M$, the corresponding matroid basis polytope $P_M$ is defined to be the convex hull ...
3 votes
1 answer
211 views

A few reference questions about the Baker–Campbell–Hausdorff formula

I'm looking at the article Baker–Campbell–Hausdorff formula - Wikipedia and I have a few questions. Under the "Special cases" section, there is a notation $\DeclareMathOperator{\ad}{ad}$ $$ \...
3 votes
0 answers
171 views

Explicit computation of hyper Ext in terms of the homologies of the input chain complexes

This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello! Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
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7 votes
1 answer
165 views

Non-recurrent points of $F(a,b)=(b,ba)$ in a compact metric group $G$

Consider a compact metric group $G$ [A compact topological group $G$ where the topology is generated by an invariant metric]. I am particularly interested in the case where $G$ is the $n$-dimensional ...
2 votes
1 answer
82 views

Library/Database of parametric polynomial systems

Could anyone please recommend a known website where I can find a database/library that has systems of polynomial equations with $n$ variables and $m$ parameters? I need some real examples to test my ...
3 votes
0 answers
108 views

Counting monomials modulo prime numbers

The present quest emanates from this study by R. Stanley, including his recent MO question. Define the product (polynomials after full expansion) $$I_n(x)=\prod_{i=1}^n(1+x^{F_{i+1}})$$ based on the ...
6 votes
1 answer
404 views

Center of a monoid ring

According to the Wikipedia page the center of a group ring $R[G]$ is the set: $$ \{ p | \forall g,\, h \in G.\, p(g) = p(hgh^{-1}) \} $$ i.e. class functions which do not distinguish elements of the ...
4 votes
0 answers
111 views

To whom is the classification of atomic, modular finite lattices due?

Here lattice means a poset with meets and joins. A lattice is called atomic if every element is a join of atoms. There are a few different ways to define modular for finite lattices: one is that the ...
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4 votes
0 answers
81 views

Weighted logarithmic Sobolev inequality

$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that $$ \Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu $$ where the entropy $$ \Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
8 votes
1 answer
219 views

Why operator systems?

A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...
1 vote
0 answers
81 views

Intersection multiplicity on a surface with only normal-crossing singularities

Let me begin by a definition. An algebraic surface $X$ over $\mathbb C$ is called a surface with normal crossing singularities if at every closed point $x\in X$, the analytic local ring $\widehat{\...
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2 votes
1 answer
329 views

First use of corner quotes for Gödel numbers

Who first used the corner quotes, $\ulcorner$ and $\urcorner$, for the notion of Gödel number? They can also be written as\Godelnum with Sam Buss's macro. They were ...
5 votes
1 answer
129 views

How strong is separation + reflection of unbounded quantifiers?

Consider a set theory with the following axioms: separation: $\exists y \forall x (x \in y \leftrightarrow \phi \land x \in a)$ where $y$ is not free in $\phi$ reflection: $\phi \to \exists u \phi^u$ ...
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3 votes
0 answers
81 views

L¹ norm of Riesz potentials on flat tori

Let $g$ be the distribution whose Fourier coefficients are given by $$\hat{g}(k) = \begin{cases} 0, & {k=0} \\ |k|^{s-d}, & {k\in \mathbb{Z}^d\setminus\{0\}},\end{cases} \qquad 0\leq s<d,$$ ...
12 votes
4 answers
2k views

What can be preserved in mathematics if all constructions are carried out in ZF?

This is inspired by this discussion. I see that the debates about the necessity of the axiom of choice in this or that statement are still ongoing. In this regard, I became interested in whether there ...
6 votes
1 answer
337 views

Formalisation of the Riemann Hypothesis

Might there be a research team that has formalised the Riemann Hypothesis? So far I have encountered two related questions: Is there a formulation of the Riemann Hypothesis in first-order arithmetic? ...
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6 votes
2 answers
189 views

How does one prove the consistency of $\mathrm{PA}$ in $\mathrm{Z_2}$?

It is "well-known" (e.g. stated here without proof and sketched here) that $\mathrm{Z_2}$ proves $\mathrm{Con(PA)}$ using the "usual" model-theoretic proof, that is one can build a ...
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2 votes
0 answers
52 views

Mean width of a simplex as one edge becomes longer

Consider an edge $e$ of a simplex $C$. If we increase the length of $e$ while keeping the length of other edges constant, the width of $C$ increases in some directions and decreases in other ...
9 votes
1 answer
308 views

To whom is Bézout's theorem for varieties due?

The following is a modern, fairly general form of Bézout's theorem. (There are forms that are more general and/or more precise; bear with me.) Define the degree of a reducible variety to be the sum of ...
1 vote
0 answers
89 views

Reference request: Time and proofs of shared pasts

Is there research about structures for notions of time with distributed systems of information, as with blockchains? I am thinking of tuples $(I, T, P, A, \prec, s, \eta, u)$ where $I$, $T$ and $P$ ...
1 vote
0 answers
78 views

Approximation in Bochner spaces

Is there any result like the Bramble-Hilbert lemma for Bochner spaces? More specifically: let $H$ a (e.g.) Hilbert space and $I$ a compact interval, $L \in L(H^k(I,H), Y)$ for $Y$ a normed space. If $...
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7 votes
1 answer
254 views

"Novelty" maximal subgroups in $S_n$

What are the maximal subgroups $M < S_n$ such that $M \cap A_n$ is not maximal in $A_n$? Maximal subgroups of $S_n$ are described by the O'Nan-Scott theorem and very extensively studied in many ...
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7 votes
1 answer
155 views

Origin of the abacus bijection

What is the origin of the abacus bijection (aka the rim hook bijection, aka the Stanton-White bijection, aka James's bijection)? Igor Pak, in his 2000 article "Ribbon tile invariants" (...
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2 votes
0 answers
74 views

Inequality on polynomials

Recall $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ and the Gaussian polynomial $\binom{a}{b}_q=\frac{[a]!_q}{[b]!_q[a-b]!_q}$ with $[0]!_q:=1$. Given two polynomials $U(q)=\sum_k\alpha_kq^...
1 vote
1 answer
168 views

Name of a space with both a topology and a metric that are not compatible?

Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$. Is there a canonical name for such a structure (maybe ...
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1 vote
0 answers
85 views

N-wave solution of conservation law $u_t + (u - u^2)_x = 0$

How can we compute the "N-wave" source-solution of the conservation law $$u_t + (u - u^2)_x = 0, $$ that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
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2 votes
1 answer
509 views

Most efficient way of getting a brief overview of the current active research areas in Algebraic Topology

I'd be applying for a Ph.D. at various grad schools in the U.S. in the coming months and while I know I'd like to pursue research in the field of Algebraic Topology, I am not knowledgeable enough yet ...
7 votes
2 answers
196 views

Congruences of binomial sums

Let $a_n$ is a binomial sum, for example $$ a_n := \sum_{k} \binom{n-k}{k} \quad \text{or} \quad \sum_{k=0}^n\binom{n+k}{n-k}\binom{2k}{k} \quad \text{or} \quad \sum_{k=0}^n\sum_{\ell=0}^k\binom{n}{k}\...
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3 votes
1 answer
136 views

Correlated Brownian motions across different times and representation with independent processes

This is a more wide-net question of Two increasingly correlated Brownian motions and Williams decomposition. In our problem we have two correlated Brownian motions $B^{1},B^{2}$ (starting at time $t=0$...
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16 votes
0 answers
300 views

Joyal's topos in which $[0,1]$ fails to be compact

Some time around 1977, André Joyal constructed a topos (actually a locale, i.e., a localic topos, necessarily non-spatial) in which the closed unit interval $[0,1]$ fails to be compact. There are ...
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4 votes
0 answers
96 views

A particular family of symmetric functions (sums of powers of sums of subsets)

For any $m,k$ define $$ f_{m,k}(x_1,\ldots,x_n) = \sum_{1\le i_1<i_2<\cdots<i_m\le n} (x_{i_1}+\cdots+x_{i_m})^k. $$ Do these symmetric polynomials have a name and any theory?
9 votes
1 answer
300 views

Propagators and PDEs

I have already asked this at MSE but did not get an answer. In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. ...
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1 vote
1 answer
84 views

Are simplicial polytopes a dense set?

Consider the space of non-empty, compact, and convex subsets of $\mathbb{R}^d$ equipped with the Hausdorff metric. Are simplicial polytopes a dense subset of that space? Probably this is just a ...
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