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

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3
votes
1answer
172 views

Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...
3
votes
1answer
117 views

How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?

Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation) $$ ...
4
votes
1answer
127 views

how to define the injectivity radius of manifolds with boundary?

For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...
1
vote
1answer
66 views

Asymptotic for zeroes of $L(s,\chi)$ in a disk $|s|<R$

In 'Remarks on Weil's quadratic functional..' p.191 Bombieri claims any given $L$-function $L(s,\chi)$ has at least $$\big(\frac{1}{\pi}+o(1)\big)R\log R$$ zeroes in a disk $|s|<R$. Is there a ...
3
votes
1answer
137 views

Schubert varieties and Young diagrams

In his book Young Tableaux, Fulton asserts, in Exercise 9.4.18 on p. 152, that the Schubert variety $\Omega_{\lambda}$ is defined by the conditions $\text{dim}(V \cap F_{n+i- \lambda_{i}}) \geq i$ for ...
6
votes
1answer
196 views

Theorem of Bukovsky characterizing ground models

It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$: (1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in ...
3
votes
0answers
50 views

Transverse intersection in the $G$-orbit of paths

I know how to prove the following lemma but I assume that it is well-known. Can someone provide a reference for it? Let $d>1$ and let $M$ be a $d$-dimensional connected smooth manifold with a ...
4
votes
1answer
96 views

The volume around a singular isolated root when equalities are loosened

Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a ...
4
votes
1answer
86 views

Symmetric matrix formula for Gaus-Legendre quadrature

While searching the web, I came across the following algorithm for the Gaus-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...
10
votes
1answer
594 views

Foundations of topology

I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here. Also some time ago I read ...
5
votes
0answers
77 views

Reference request: "effective'' semistable reduction

I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...
13
votes
0answers
378 views

Chasing a 1950s thesis from the University of Dhaka on block designs

On behalf of a friend I am searching for a thesis on block designs from the 1950s. The details are below. Author: Qazi Motahar Husein (Sometimes Husain or Hussein). Title of the Thesis: Symmetrical ...
2
votes
0answers
45 views

Literature on transformed Gaussian matrices

I am considering real $n$-by-$m$ matrices of the following type: $$ M=SM^\prime,\\ M^\prime_{ij}\sim^{iid}N(0,1). $$ Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...
5
votes
1answer
133 views

Reference for affine Grassmanian

Could someone please provide a precise reference in the literature where the following well-known fact is proved. Also if someone could write out the proof that would be great. ...
3
votes
2answers
118 views

An English version Borok's work on finite-infinite systems of ordinary differential equations

I am looking for the English translation of the paper by V. M. Borok (originally in Russian) The Cauchy problem for finite-infinite systems of linear differential equations. This work is about the ...
7
votes
2answers
409 views

Reference for Manin's idea on algebraic geometry over the symmetric monoidal model category of Motives

Reference for Y. Manin's idea of "algebraic geometry over the symmetric monoidal model category of motives." Has been sugested to me that this was made in a Manin's letter. There is an escaned copy? ...
4
votes
2answers
129 views

Classification of symmetries of tilings in surfaces?

Is there a general study of the symmetries of tilings on surfaces? Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. ...
2
votes
1answer
183 views

An electronic copy of Vishik's work on $p$-adic $L$-functions for modular forms

This question is very simple. Would someone be so nice as to send me an electronic copy of M. M. Vishik, Non-Archimedean measures connected with Dirichlet series, Mat. Sb. (N.S.), 1976, Volume ...
0
votes
0answers
27 views

Fubini's Theorem for Lévy bases

Let $M$ be an infinitely divisible independently scattered and homogeneous random measure on $\mathbb R^d$ (ie a homogeneous Lévy basis). Let $\nu$ be a sigma finite measure on $\mathbb R^k$. Let ...
11
votes
2answers
225 views

Homotopy property of constructible sheaves on stratified spaces

Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...
4
votes
0answers
45 views

Looking for a Collection of Examples and Counter Examples for Assumptions about the Properties of Planar Euclidean TSP Instances?

Where can I find example and counter examples to seemingly plausible assumption about the properties of optimal solutions of planar euclidean TSP instances? The reason for asking is that the ...
11
votes
2answers
279 views

Tableaux with limited rows and complementary skew shapes

Given a partition $\mu=(\mu_1,\mu_2...,\mu_d)$, define $\bar\mu=(\mu_1-\mu_d,\mu_1-\mu_{d-1},...,\mu_1-\mu_2,0)$, the complementary shape in the $d\times \mu_1$ rectangle. Then the number of skew ...
1
vote
0answers
45 views

Multilinear Interpolation

Suppose I have a multilinear map $\Gamma(u,v)$ satisfying \begin{align} \big\| \Gamma(u,v)\big\|_{L^2} &\leq \big\| u\big\|_{L^2} \big\| v\big\|_{L^2} \\ \big\| \Gamma(u,v)\big\|_{L^\infty} ...
5
votes
2answers
148 views

Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?

(this question is about a particular aspect of a previous question, which was not duly stressed) Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let $$ \widetilde{M}:=\mathbb{P}T^*M $$ be the ...
1
vote
0answers
16 views

Approximation error estimate

I would like to find a good reference for the following or a similar, probably well-known, approximation error result: Let $\Omega\subset \mathbb{R}^d$ be bounded, $p\in [1,\infty]$, $l, m\in ...
13
votes
1answer
552 views

Resolution of singularities in étale cohomology

The lack of a suitable resolution of singularities comes up often in work on étale cohomology from the 1960s and 70s, And I think even the latest version of Milne's lecture notes says "It is likely ...
2
votes
0answers
63 views

Action of orthogonal group on the free Lie algebra

This question is somewhat related and inspired by this post of professor Montgomery. The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of ...
2
votes
1answer
152 views

Applications of Level Lowering

What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
4
votes
2answers
228 views

Moment problem on [-1,1]: necessary and sufficient conditions

Consider a sequence of real numbers $s=(s_0,s_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that $$ s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;? $$ ...
7
votes
2answers
381 views

Famous results about the value of a given limit assuming it exists

Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
13
votes
1answer
323 views

Homotopy fiber of a map between classifying spaces

I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong): Let $G$ and $H$ be topological groups and $f : G \to H$ be ...
1
vote
0answers
74 views

Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$

I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when: (1.) $q=p$ and/or (2.) $E$ has multiplicative reduction at $q$. Here, $E$ is an ...
6
votes
0answers
112 views

Reference for supergroupoids in supersymmetry?

I would like to know some references on supergroupoids in supersymmetry. A supersymmetry is invariance under a supergroup action (nLab, Supersymmetry). It is know that groupoids provide a local ...
0
votes
1answer
65 views

Can we always extend a vector bundle on an open subset of a ringed space with soft structure sheaf?

Let $(X,\mathcal{O}_X)$ be a ringed space with soft structure sheaf. Moreover let $X$ be paracompact. Let $U$ be an open subset on $X$ and let $E$ be a finite dimensional vector bundle on $U$, i.e. ...
2
votes
1answer
82 views

Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate. Longer: If $p_n$ is the $n$-th Legendre polynomial, ...
1
vote
1answer
53 views

Extremal of an L^1 continuous functional on a compact bounded set

Please, I need a small help with a reference. Lets say we do have a continuous functional $f$ on $L^1$ space and we want to prove the existence of extremals $f(\Omega)$, where $\Omega$ is compact and ...
2
votes
0answers
167 views

Equivariant Riemann-Roch on DM stacks?

Does an equivariant version of (Toen)-Riemann-Roch theorem hold say over a smooth Deligne-Mumford stack $X$ over the complex numbers? Any references that state this explicitely? Are there formulas ...
2
votes
1answer
108 views

Universal space for the family of subgroups of a finite cyclic group

Let $G$ be a compact Lie group and let $\mathcal{P}_G$ denote the family of proper subgroups of $G$. The universal space for the family $\mathcal{P}_G$ is a cofibrant $G$-space which does not have ...
14
votes
2answers
417 views

$RO(G)$-graded homotopy groups vs. Mackey functors

Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work $\infty$-categorically. Also, I've looked through other similar MO questions, but I didn't find ...
4
votes
0answers
74 views

References for “quadratic” Dirichlet series

(Please pardon the use of nonstandard terminology, as I know not the accepted names for my entities of interest.) Some personal research I have been doing led me to consider series of the form ...
4
votes
0answers
209 views

Is the two variable fragment of arithmetic, i.e., theory of ($\mathbb{N}, + ,\times$), decidable?

Any references would be appreciated. Most places only address different vocabularies (e.g. a survey of arithmetical definability by Bes).
1
vote
0answers
37 views

Alternative to generic chaining bounds for a particular family of stochastic processes

Generic chaining provides a general but rather abstract framework to bound suprema of stochastic processes. In many applications, however, we know more about the expression of the stochastic process. ...
7
votes
1answer
187 views

Source request for $H^*(B\mathrm{TOP},\mathbb{Q}) \cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the ...
0
votes
0answers
31 views

Interpolation functional for BV spaces?

Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...
12
votes
2answers
873 views

Do non-stable Banach spaces exist?

Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties: Is every infinite ...
5
votes
1answer
243 views

References for higher descriptive set theory surveys

A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and ...
2
votes
0answers
81 views

Balanced partitions of vector sets

We are interested in the following Lemma. Let $V\subset [0,1]^n\subset\mathbb R^n$ be a set of $n$-dimensional vectors. Then for each $r\le |V|$ there exists a partition $$V=V_1\cup V_2\cup\dots \cup ...
1
vote
0answers
33 views

Functional Derivative estimate

Recently I've conisdered a functional derivative estimate on the first spatial derivative of bounded classical solutions $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ to $$ u_t - u_{xx} - f(u) = 0 \ \ \ ...
6
votes
1answer
114 views

Convex Hull of univalent functions and Bieberbach Conjecture

Consider the class $S$ of univalent functions on the unit disk $D$ normalized so that $f(0)=0$ and $f'(0)=1$. Each function in $S$ satisfy the Bieberbach conjecture, that is the $n$-th coefficient in ...
2
votes
1answer
71 views

Dimension of preprojective algebra of Dynkin type

Fix a field $\Bbbk$. Let $Q$ be a Dynkin quiver and let $\Pi(Q)$ be its preprojective algebra. It is well-known that in this case $\Pi(Q)$ is finite-dimensional, but I've been unable to find a ...