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Questions tagged [reference-request]

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

5
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0answers
108 views

Topos with enough projectives

It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...
-4
votes
1answer
180 views

Reference request in optimal stopping [closed]

I am given the following task. Distributed over a trading day, I am supposed to buy a certain quantity of a good. The price of this good changes during the day. The goal is to buy the required ...
0
votes
0answers
52 views

Conjugation of pseudodifferential operators

Let $A$ be a pseudodifferential operator of order $m$ with symbol $a(x,\xi)\in S_{1,0}^m$ where $x,\xi \in \mathbb{R}^n$. We define an operation called conjugation of $A$ given by \begin{equation} A_\...
0
votes
0answers
55 views

Strichartz estimates for fractional Schrodinger equations

A pair $(q,r)$ is $\alpha-$fractional admissible if $q\geq 2, r\geq 2$ and $$\frac{\alpha}{q} = d \left( \frac{1}{2} - \frac{1}{r} \right).$$ We take fractional Schrodinger propagator $U(t)=e^{it (...
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0answers
40 views

Approximation of deterministic problem with stochastic problem

A lot of problems in PDE theory are solved in the following way: The original problem is quite hard and we can't solve it, so we make the approximation problem that we can solve. Than we go back and ...
2
votes
1answer
139 views

Relation between Optimal Transport Cost and Difference between Topological Invariants?

I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of ...
9
votes
1answer
172 views

Lifting of families of curves to characteristic 0

Let $k$ be a finite field, $X_0$ be a smooth affine variety over $k$ and $C\rightarrow X$ a smooth projective family of curves of genus $\geq 2$. By a result of Elkik we can always lift $X_0$ to a ...
4
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0answers
148 views

Mapping class group of $\mathbb{S}^3$

If I recall correctly from a lecture I attended the last year we have that $MCG(\mathbb{S^2})\simeq\frac{\mathbb{Z}}{2\mathbb{Z}}$ by Smale in the 60' and $MCG(\mathbb{S^3})\simeq\frac{\mathbb{Z}}{2\...
7
votes
2answers
156 views

“Overdetermined” Poisson equation

Consider the PDE $-\Delta u = f$ on a bounded domain $\Omega \subset \mathbb{R}^n$, where $f \in C^\infty(\bar{\Omega})$. I wish to consider both the boundary conditions $u = 0$ and $\frac{\partial u}{...
0
votes
0answers
55 views

Complement of a meagre set in a Baire space

I remember reading somewhere that the complement of a meagre set in a Baire space is also a Baire space and this is in fact easy to prove. Looking for this result in the standard collection of ...
15
votes
2answers
421 views

Is Post's tag system solved?

Has the 3-tag system investigated by Emil Post $(0\to00, 1\to1101)$ been solved? Is there a decision algorithm to determine which starting strings terminate, which end up in a cycle, and which (if any)...
2
votes
1answer
130 views

“Compactness in Measure” in Function Spaces

In Chapter 4.9 of the book "Measures of Noncompactness and Condensing Operators" (Vol. 55 of Operator Theory: Advances and Applications), the authors mention the property "compactness in measure". ...
2
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0answers
107 views

English translation of G.Laumon, L.Moret-Bailly book Champs algébriques

Is there an English translation of G.Laumon, L.Moret-Bailly book Champs algébriques. Most questions on this site on stacks received this book as reference in comments/answers. So, I want to ask if ...
-1
votes
1answer
144 views

Riesz representation theorem for Hilbert-to-Hilbert mappings [closed]

Assume $\phi:\mathbb{H}_1\rightarrow \mathbb{H}_2$ is a continuous linear mapping between two real Hilbert spaces $\mathbb{H}_1$ and $\mathbb{H}_2$. If $\mathbb{H}_2=\mathbb{R}$, then the Riesz ...
2
votes
1answer
90 views

Can we extract information from signature (rough path theory) to construct part of signal?

This question is related to rough path theory. Consider we have obtained signature obtained from a set discrete data points postulating linear from one data point to another. Such signature are used ...
3
votes
2answers
138 views

Orthogonal Polynomials and Sturm Liouville operators

Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $...
6
votes
2answers
103 views

Linearization of hamiltonian torus action

Let $(M,\omega)$ be a symplectic manifold with a hamiltonian effective torus action. Suppose it has an isolated fixed point $p$. Is it true that there exists an invariant neighborhood $U$ of $p$ such ...
3
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0answers
32 views

Finding linear order of set maximising number of consecuitive subsets

I have the following combinatorial optimisation problem of which I think someone has probably solved it before. Has someone come across this problem before, maybe in a different setting than in the ...
3
votes
1answer
173 views

derived functor that preserves weak equivalences

Suppose we have a functor $F:A\rightarrow B$ between model categories. 1- Assume that F takes weak equivalences to weak equivalences and cofibrations to cofibrations, can we define the derived ...
5
votes
2answers
221 views

Basic theorem on induction for representations of $p$-adic groups

I know a lot of places where the following is sparsely proved, but I remember there was some paper where I read it in basically the same form I write it, but unfortunately I can't remember where it ...
3
votes
0answers
36 views

Reference on generalization of plane graph duality between bonds and simple cycles

Let $G$ be a plane graph, and $G^*$ its dual. Among the $k$ partitions of the nodes of $G$, I'll call the connected k-partitions those such that each block of nodes of the partition induces a ...
5
votes
1answer
124 views

Anderson localization for fractional Laplacians

There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(\mathbb{Z}^d)$ such as $$ -\Delta+\lambda V $$ where $\Delta$ is the discrete ...
6
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0answers
112 views

Sets $X,Y$ of natural numbers such that any natural $n$ writes uniquely $n=x+y$ [duplicate]

There are many pairs $X, Y$ of infinite subsets of $\mathbb{N}:=\{0,1,2\dots\}$ such that any $n\in\mathbb{N}$ writes uniquely as $n=x+y$, with $x\in X$ and $y\in Y$. An example of such a pair is $(X,...
2
votes
1answer
110 views

Conservativity of language extension

It is folklore that extending a language of classical first-order logic is conservative. That is, given two languages $L \subseteq L'$, a set of $L$-sentences $\Gamma$ and an $L$-sentence $\varphi$, ...
7
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0answers
136 views

Which representations of the Lie algebra of a Lie group come from representations of the group itself?

I think this is very classic mathematics, but I can't find a complete answer in the literature. Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
14
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1answer
652 views

Hadamard theorem about embedding

The following theorem is commonly attributed to Jacques Hadamard. Assume $\Sigma$ is a smooth locally convex immersed surface in the Euclidean space. Then $\Sigma$ is embedded and bounds a convex ...
3
votes
3answers
133 views

Sum of subspaces is closed iff inclination is positive

It is a well-known result in functional analysis that the sum $M+N$ of two subspaces of a Banach space with $M\cap N=0$ is closed if and only if the inclination $$\widehat{(M,N)} := \inf_{x\in M, \|x\|...
1
vote
1answer
52 views

Reference on vector-valued convex conjugate

The following definition of convex conjugate is taken from Wiki: Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$ Denote the dual pairing by $$\langle \cdot ,...
8
votes
2answers
210 views

“Closed bicategories”

I am interested in the following property that a bicategory may or may not have. Let $\mathbf{B}$ be a bicategory. Every one-morphism $f\colon x\rightarrow y$ defines a functor $\mathbf{B}(y,z)\...
2
votes
0answers
34 views

An implementation of Minkowski reconstruction in 3 dimensions

By a theorem of Minkowski from 1903, an $n$-dimensional polytope $P\subset \mathbb R^n$ is determined up to translation by its unit face normal $u_1,\dots,u_k\in S^{n-1}$ and the corresponding $(n-1)$ ...
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0answers
30 views

Has anyone studied Golomb rulers having a spectrum with a minimal $L^2$ norm?

A Golomb ruler can be described as a set of marks on a line having integer positions, such that no two pairs of marks are separated by the same distance. Call the spectrum of a ruler the (multi-)set ...
2
votes
0answers
67 views

Generalization of regularly varying functions

A continuous function $L :\mathbb{R}_+ \to \mathbb{R}_+$ is called regularly varying (at infinity) if for all $a > 0$, $$ \lim_{x\to \infty} \frac{L(ax)}{L(x)}= g(a) $$ for some function $g(a)&...
3
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0answers
68 views

The ring generated by a convex polytope and its faces

Let $V=\Bbb R^n$. Morelli defined the (commutative unital) ring $L(V)$ to be the additive group generated by the indicator functions of convex polytopes in $V$ with multiplication induced by Minkowski ...
6
votes
2answers
387 views

Explicit computation of the Burnside ring

I would like to see explicit computations of the Burnside ring $A(G)$ when $G$ is a small Abelian group, such as $G=\mathbb{Z}/2,\mathbb{Z}/2^n,\mathbb{Z}/p^n$ where $p$ is an odd prime and $n\...
4
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0answers
128 views

Reference request: destroying saturation at an inaccessible?

An ideal $I$ on $P(\kappa)$ is said to be $\kappa^+$-saturated if there is no sequence $\langle S_\alpha \mid \alpha<\kappa^+\rangle$ of $I^+$ sets such that $\alpha<\beta<\kappa^+\implies S_\...
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0answers
100 views

Reference request for bounds of $n$-th composite

Motivation I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions. Recently during trying to understand and prove the ...
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0answers
143 views

Quantitatively characterizing the failure of the converse of Dirac's theorem

First, I am an undergraduate so I apologize if this is trivial and certainly understand if it is closed immediately. I am currently in a combinatorics and graph theory class and recently we have ...
2
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0answers
61 views

Reference Request: A “Chevalley-Eilenberg”-style formulation of the $L_\infty$ algebra minimal model theorem?

The nicest definition of $L_\infty$-algebras ---which I will call a "Chevalley-Eilenberg" style definition after the obvious analogy with the Chevalley-Eilenberg differential of Lie algebras--- is the ...
5
votes
1answer
168 views

Relationship between Hilbert-Samuel multiplicity and polar multiplicity

Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows: $$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \...
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0answers
78 views

unknown sequences of rational numbers with sum of a transcendental number [closed]

Starting with a transcendental number like $e$, it is known that we can write it as a sum of infinitely many rational numbers of the form : $e= \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}...
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0answers
63 views

Meyer's class number formulas

In my previous question I asked for a reference for $L$-series of quadratic orders in connection with a certain class number formula. It seems that this had been investigated by Curt Meyer. Is there ...
7
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0answers
121 views

Determinants associated to orthogonal polynomials

Let $${p_n}(x) = \sum\limits_{j = 0}^{n } {{{( - 1)}^{n - j}}p(n,j){x^j}} $$ be orthogonal polynomials satisfying $${p_n}(x) = (x - {s_{n - 1}}){p_{n - 1}}(x) - {t_{n - 2}}{p_{n - 2}}(x)$$ with ...
3
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0answers
65 views

Powers of ergodic transformations

Here is a lemma that I know to be true, and can prove in half a page or so, but I'm wondering: can anyone supply a reference so that it can simply be quoted in a paper? Lemma Let $T$ be an ergodic ...
0
votes
1answer
85 views

Reference from the article “Random Ordinary Differential Equations”, by J.L. Strand

In the article Random Ordinary Differential Equations, Journal of differential equations 7, 538-553 (1970), by J.L. Strand, reference number 6 refers to his PhD thesis: Stochastic Ordinary ...
6
votes
1answer
237 views

$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer

Does anyone know if the following problem has ever been studied? Let $a$ and $b$ be two real numbers and consider the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$$ where $n$ is a ...
5
votes
0answers
278 views

Elementary questions about Morse-Bott functions

Let $M$ be a manifold, $F$ be a Morse-Bott function, $c$ be a critical level, and $M_c$ be the corresponding critical submanifold. Let us assume that $M_c$ is connected, and the index of $M_c$ is ...
2
votes
0answers
120 views

Infintely iterated and functional integration in constructive math

Looking for references on constructive derivations of (elements of) functional integration -- in particular, those used in the classical construction of the Wiener measure. It seems such ...
5
votes
0answers
230 views

Applications of E8 manifold

The $E_8$ Cartan matrix is given by, $$ K_{E_8}=\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ ...
10
votes
1answer
234 views

Positive Ricci curvature on fiber bundles

My advisor and I are working on Ricci curvature and an anonymous referee pointed out the following conjecture: Let $F\hookrightarrow M\stackrel{\pi}{\to}B$ be a fiber bundle from a compact manifold ...
3
votes
1answer
116 views

Liouville theorem for fractional Laplacian

Is there any Liouville type theorem for the half space problem \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s v &= 0 &&\text{in } \mathbb R^N_+\\ v & =0 &&\text{...