Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,575
questions
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Why is the inverse of a bijective rational map rational?
Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $...
6
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2
answers
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Euler characteristics with and without compact support of algebraic varieties
Let $X$ be a complex algebraic variety, possibly singular and/or non-compact. It is well known that if $X$ is smooth then its Euler characteristic is equal to its Euler characteristic with compact ...
- 21.1k
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2
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488
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Forcing for Arbitrary First Order Theories
Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of ...
user45939
6
votes
3
answers
759
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Poincare recurrence theorem and convergence on compact metric spaces
I am looking for a proof (or a reference to a proof) of the following theorem:
Let $X$ be a compact metric space with metric $d$, endow $X$ with the Borel $\sigma$-algebra and a probability measure $\...
- 245
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votes
3
answers
980
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PDEs involving measures; where to begin?
If I want to learn about existence of weak solutions to PDEs of the form
$$u_t + Au = f$$
or
$$Au = f$$
where $A$ is elliptic and $f$ is a measure, where do I start? I know the Galerkin method for ...
- 375
6
votes
1
answer
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Caccioppoli-Leray Inequality for De Giorgi's theorem proof
I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients.
This is the translation of the original paper
De Giorgi paper
At page ...
- 255
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1
answer
526
views
What is known about a 3-manifold $M$ when its fundamental group is linear?
Suppose we have a 3-manifold $M$ and its respective fundamental group $\pi_1(M)$. An important question about its fundamental group is to ask if it is linear, i.e. they are isomorphic to a subgroup of ...
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2
answers
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Continuous choice of Hahn-Banach extensions
QUESTION RETRACTED - My original argument was fundamentally mistaken (mixing up lower and upper semi-continuity). Sorry (and thanks for the useful comments)
I need, and (unless I am seriously ...
- 587
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2
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663
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Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"
Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?
Example: Given a dg ...
- 53.1k
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Vladimir Voevodskys 2002 ICM Lecture.
Is Vladimir Voevodskys ICM lecture available in videotaped format somewhere?
Strangely it is not at the IMU homepage (but Lafforgues is) http://www.mathunion.org/Videos/ICM2002/
Was it not taped (Why ...
- 235
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3
answers
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Surveys on Navier Stokes Equations and its physical implications
Hi,
I'm a beginning graduate student, and I'm interested in learning more about Fluid Mechanics and, in particular, the Navier stokes Equations. I would like to know: are there are some sort of free ...
6
votes
5
answers
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Reference for quantum Schur-Weyl duality
I am trying to prove a version of quantum Schur-Weyl duality. I hope to be able to generalize the proof of the Schur-Weyl duality between $U_q(\mathfrak{gl}_n)$ and the Hecke algebra $H_r$. So I am ...
- 1,001
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1
answer
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views
How much choice is needed to prove the completeness of equational logic?
All the proofs of the completeness of (Birkhoff's) equational logic I have read seem to pick representatives for equivalence classes of terms and hence require the axiom of choice. Is AC (or a weak ...
- 83
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answers
360
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Twisted forms with real points of a real Grassmannian
Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$.
We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{\...
- 13.6k
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Need a reference for a trigonometric inequality
In my old high school notebook (20 years ago), the following inequality appears with its proof:
$$1+\cos x + \frac{1}{2}\cos 2x + \cdots + \frac{1}{n}\cos nx \geq 0$$
for any real $x$ and positive ...
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Who was Bickart?
The term "Bickart points" is often used for the foci of the Steiner circumellipse of a triangle. Who was Bickart, and what was the first publication to use the term?
- 3,443
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Algebraic geometry additionally equipped with field automorphism operation
I am looking for some facts on theory, which is essentially algebraic geometry but with field automorphisms added as 'basic' operations. (Precisely, I mean universal algebraic geometry for (universal) ...
- 63
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votes
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answer
394
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High sum of fractional parts
Let $n\geq 2$ and $x_1,\ldots,x_n > 0$ be such that $x_1+\cdots+x_n =1$. Is it true that there must exist a positive integer $k$ such that $$\{x_1k\}+\cdots+\{x_nk\} = n-1?$$
This looks closely ...
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Automorphisms of a modular tensor category
I would like to ask for references on automorphisms of a modular tensor category, that do not change the objects. Some special cases, such as automorphisms of a quantum double, are also helpful.
- 4,716
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Serre duality in families
In Ravi Vakil's lecture notes ("Foundations of Algebraic Geometry", Classes 53 and 54) one can find a relative version of Serre duality (Exercise 6.1), namely:
"Suppose $\pi: X\rightarrow Y$ is a ...
user94539
6
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1
answer
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rationality of weighted projective space
A complex weighted projective is $\mathbb{P}(k_1, \cdots, k_{n+1})=Proj(\mathbb{C}[x_1, \cdots, x_{n+1}])$ with $x_i$ of degree $k_i$ (sometimes people ask for each $n$ of the weights being coprime). ...
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Texts on moduli of elliptic curves
I want to study FLT (Fermat's Last Theorem), and now I'm studying moduli of elliptic curves.
I've heard that Deligne-Rapoport, Katz-Mazur, Mazur's "Modular curves...", and Katz's "p-adic..." are very ...
- 1,352
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2
answers
359
views
Symmetric monoidal category with trivial switch morphisms
Is there a specific terminology for a symmetric monoidal category in which for any object $x$ the switch map $x\otimes x\to x\otimes x$ is the identity ? (Or alternatively the action of the symmetric ...
- 262
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1
answer
478
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A question about the (motivic) integral cohomology of the Eilenberg-MacLane spectrum
Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum. Let $n\geq 0$ be any integer.
Is it known the structure of the group $[H\mathbb{Z},\Sigma^{n}H\mathbb{Z}]$?
Is there any reference in this ...
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702
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Books on the History of math research at European universities
Are there good books that cover the history of math and mathematical science (ex. physics, chemistry, computer science) PhD programs in the Occident? My primary motivation is to figure out how the PhD ...
- 3,827
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1
answer
628
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Best estimate of the Mertens function without assuming the Riemann Hypothesis
I'm searching the best known upper bound for the Mertens function, but without assuming the Riemann hypothesis.
Landau, in 1901, have proved that $M(x)= O(x \exp(-c\sqrt{\ln x})$, but I am unable to ...
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542
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Commutative group algebraic spaces
Is the category of commutative group algebraic spaces (commutative group objects in algebraic spaces) locally of finite type over a field, an abelian category?
I would benefit from a reference
user119470
6
votes
2
answers
683
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Reference for LIL for fractional Brownian motion
(Cross-posted to https://math.stackexchange.com/questions/2377810/law-of-iterated-logarithm-for-fractional-brownian-motion.)
It seems strange but, even after consulting several books, and hours ...
- 759
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1
answer
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Yoneda extensions in exact categories and their derived categories
If $\mathcal{A}$ is any abelian category, then for all objects $X,Y$ in $\mathcal{A}$ and for all integers $i \geq 0$, there is an natural isomorphism
$$\mathrm{Ext}_\mathcal{A}^i(X,Y) \simeq \mathrm{...
- 941
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Existence of well-ordering of epsilon_0 in weak theories
In a discussion on a youtube video on the hydra game I jokingly mentioned how everyone was assuming that $\varepsilon_0$ was well-ordered. This lead to a bit of disagreement (in a nice way!) about the ...
- 33.9k
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Numbers divisible only by primes of the form 4k+1
Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?
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How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?
I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...
- 6,186
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Heyting algebras originating from directed graphs
The category RefGph of reflexive directed graphs is the functor
category $\hat{∆}_1=\mbox{Fun}(∆^◦_1,$Set), where $∆_1$ is
the simplex category truncated at level 1.
Hence the poset Sub(X) of ...
- 457
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votes
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answer
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Embedding of real trees into $\ell_1(\Gamma)$
It seems plausible that any real tree or ${\mathbb{R}}$-tree in the sense of the definition in https://en.wikipedia.org/wiki/Real_tree admits an isometric embedding into the Banach space $\ell_1(\...
- 4,868
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votes
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802
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Fundamental solution of Discrete Laplace in the plane
We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator
Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)...
- 4,761
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votes
1
answer
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If a topological space has vanishing $n$th homology for every possible homology theory, does it have vanishing $n$th homotopy?
I don't have any strong preference as to whether or not the homology theories are required to be ordinary.
Also, if this does not hold in general, does it hold for some nice category of spaces, like ...
- 363
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votes
1
answer
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Boardman-Vogt tensor product
Let $\mathbf{sSet}$ be the model category of simplicial sets and $\mathbf{Op}$ the model category of symmetric operads. Equipped with Boardman-Vogt tensor product $ \otimes_{BV}$, the category $\...
- 1,607
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votes
1
answer
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Is $G_{\operatorname{red}}$ normal in $G$?
Let $G$ be an affine group scheme of finite type over a field $k$. It is well known that the associated reduced subscheme $G_{\operatorname{red}}$ of $G$ is a subgroup if $k$ is perfect. So let us ...
- 352
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votes
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What did Shimura say about $y^2 + y = x^3 - x$?
From the introduction of Ribet-Stein:
Shimura showed that if we start with the elliptic curve $E$ defined by the equation $y^2 +y = x^3 −x^2$ then for “most” $n$ the image of $\rho$ is all of $\...
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Reference for tetrahedral Coxeter group
Let G be the group with 4 generators, each of order 2, such that the product of any 2, say ab, has order 3 (i.e., ababab=e). That is, this is an infinite reflection group with Coxeter diagram a ...
- 63
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The 2-group of extensions
Let $A,B$ objects of an abelian category. Then we can define the abelian group $\mathrm{Ext}^1(A,B)$ as the set of isomorphism classes of extensions $0 \to B \to E \to A \to 0$, endowed with the Baer ...
- 61.5k
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2
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329
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Continuity of the spectrum with respect to the metric
The following question is quite natural, but I am not aware of a reference dealing with it: let $M$ be a compact (smooth) manifold (posssibly with boundary) and $E$ a vector bundle on $M$ with an ...
- 3,349
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votes
1
answer
469
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Sets of reals and absoluteness
Schoenfield's absoluteness states that if $\phi$ is $\Sigma^1_2$ then $V\models \phi$ iff $L\models \phi$. The set of reals in $L$ is $\Sigma^1_2$ and it is the largest countable $\Sigma^1_2$ set of ...
- 3,766
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2
answers
423
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Combinatorics of folding digit strings
Say that a string of $n$ digits, each from $\lbrace 0,1,2,\ldots,b-1 \rbrace$,
is foldable if, were each
digit on its own stamp in a sequence of connected stamps,
one could fold the stamps so that ...
- 149k
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votes
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answer
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Extreme points of a compact convex set are a $G_\delta$?
Dear All,
I'm reading a paper (Residuality of Dynamical Morphisms by Burton, Keane and Serafin) that makes a claim that I've been unable to verify or find a reference for. The claim is made that the ...
- 22.5k
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votes
1
answer
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Löwner-John Ellipsoid: incribed and circumscribed
I have two questions about the
Löwner-John ellipsoid, one just terminology, the other
more substantive.
Let $K$ be a convex body in $\mathbb{R}^d$.
Q1.
Is "the
Löwner-John ellipsoid"
the ...
- 149k
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votes
2
answers
949
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Literature on behaviour of eigenfunctions under multiplication?
Dear community,
I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable ...
- 199
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votes
2
answers
799
views
A dual theory to the theory of currents?
The k-currents are defined as dual space to the spaces of all smooth k-forms.
(These monsters are used to work with the minimal k-surfaces.)
Assume I want to look at the generalized k-forms;
they can ...
- 1,785
6
votes
2
answers
690
views
4-coloring maps of pentagons
Is there a simple proof for the 4-color theorem when restricted to (finite) maps all of whose
(internal) regions are pentagons? I am in fact most interested in convex pentagons, if that additional
...
- 149k
6
votes
1
answer
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Spencer-Brown's claimed proof of the four color theorem [closed]
I read on Wikipedia that G. Spencer-Brown gave a non-computer based proof of the four color theorem. As I'm not an expert in the subject I'm unable to verify that claim. Does any one have an idea ...
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