Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
1,799 questions
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Euler Subgroups and Automorphic L-functions
Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
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Is there research on Machine Learning techniques to discover conjectures (theorems) in a wide range of mathematics beyond mathematical logic?
Although there already exists active research area, so-called, automated theorem proving, mostly work on logic and elementary geometry.
Rather than only logic and elementary geometry, are there ...
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When taking the fixed points commutes with taking the orbits
Let $G$ and $H$ be groups, both acting on a set $X$ on the left, in such a way that the two actions commute. (Equivalently, let $G \times H$ act on $X$.)
The set $\text{Fix}_H(X)$ of $H$-fixed ...
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Aleksandrov's proof of the second order differentiability of convex functions
Aleksandrov [A], proved a remarkable property of convex functions.
Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...
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Average degree of contact graph for balls in a box
Imagine you dump congruent, hard, frictionless balls in a box,
letting gravity compress the balls into a stable configuration
(I believe such configurations are called
jammed.)
Assume the box ...
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Adding inverses to a symmetric monoidal category (Reference?)
As we all know, the forgetful functor $\mathsf{Ab} \to \mathsf{CMon}$ from abelian groups to commutative monoids has a left adjoint, the Grothendieck group. I would like to categorify this ...
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Koebe–Andreev–Thurston theorem - where can I find a proof?
Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two ...
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A conformal map whose Jacobian vanishes at a point is constant?
Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$.
Assume $d \ge 3$ ...
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How to Compute the coordinate ring of flag variety?
Let $G$ be algebraic group over $\mathbb(C)$(semisimple), $B$ be Borel subgroup. consider flag variety $G/B$. It is known that $G/B$ is projective variety. So one consider the homogeneous coordinate ...
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Reference request: Simple facts about vector-valued Sobolev space
Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...
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Toda's book on homotopy groups of spheres
Yesterday one my friend told about recent book of Hiroshi Toda, where the computations of 3-torsion in homotopy groups of spheres are given up to a very high stem (about 75). This book is published ...
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What is a good basic reference on model categories?
I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for ...
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Kan extensions in the $2$-category of monoidal categories
Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of $k$-...
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Books on the analysis of hyperbolic partial differential equations
Most of the present books on pde analysis deal with the elliptic partial differential equations. Is there some book related to rigorous analysis with hyperbolic pdes, and especially hyperbolic systems ...
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On Bailey–Borwein–Plouffe formula for irrational numbers
A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in ...
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On the independence of lower and upper asymptotic and Banach densities
Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \...
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Alternating sum of hook lengths: Part I
Given $\lambda$ an integer partition of $n$, let $h_{ij}(\lambda)$ denote the hook length of cell $(i,j)$ in the Young diagram of $\lambda$.
Is there a closed formula or a generating function for the ...
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Books on reductive groups using scheme theory
Prof. Conrad mentioned in a recent answer that most of the (introductory?) books on reductive groups do not make use of scheme theory. Do any books using scheme theory actually exist? Further, are ...
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Questions about Prikry forcing and Cohen forcing
I have some questions.
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, \mathbb{P}...
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What is a good introduction to cluster algebras from surfaces?
What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in fact,...
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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 ...
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Banach spaces $X$ with $\ell_2(X)$ not isomorphic to $L_2([0,1],X)$
Let $X$ be a Banach space. I think that some time ago I read somewhere that, in general, the space $\ell_2(X)$ of all sequences $(x_n)$ in $X$ with $\sum_{n=1}^\infty \|x_n\|^2<\infty$ is not ...
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Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?
Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion
$$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$
and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke ...
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Reference request: colimits of locally presentable categories
Consider the 2-category of locally presentable categories, cocontinuous functors, and natural transformations. I believe that this 2-category is 2-cocomplete in the sense of containing all small 2-...
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Proofs of some combinatorial identities
Just wondering if anyone knows any references in the literature to bijections corresponding to the following simple generating function identities. Let $B(z)=\dfrac{1}{\sqrt{1-4z}}$ and $C(z)=\dfrac{1-...
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What is the shape of large cardinal tree in implication strength order?
There are two natural orders on large cardinal axioms.
(a) Consistency strength order
$\sigma \leq_C \theta \Longleftrightarrow ZFC\vdash Con(ZFC+\theta)\longrightarrow Con(ZFC+\sigma)$
(b) ...
user45939
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Finite objects for which isomorphism is NP-hard or harder?
Are there finite objects for which deciding isomorphism
is NP-hard or harder?
Graphs and groups are not solutions.
Searching the web didn't return answer for me.
Partial result based on Chow's ...
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The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$
The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c}
a\...
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Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces
This question was posed on MathStackExchange but did not get an answer (even with a bounty).
In all books that I have checked the spectral theorem (every self-adjoint unbounded operator on a Hilbert ...
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Retraction of a Riemannian manifold with boundary to its cut locus
This question is edited following the comment of Joseph. He pointed out that the main object of the first version of this question is the cut locus.
Recall that the cut locus of a set $S$ in a ...
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Upper bound on order of finite subgroups of GL_n(Z_p)?
Fix a prime $p$ and integer $n>1$, along with the ring $R$ of integers in a finite extension of the field $\mathbb{Q}_p$ (for example $R = \mathbb{Z}_p$).
Is there an upper bound $C(n,p)$ on ...
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Example for non equivalent rational full CFTs with same modular invariant (partition function)
I am looking for a counter example which shows, that a full rational 2D CFT (with respect to a given chiral subtheory) is not characterized by its modular invariant partition function. People tell me ...
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Reference for Kronecker-Weyl theorem in full generality
The Kronecker-Weyl theorem asserts the following: fix real numbers $\theta_1,\dots,\theta_d$, and consider the infinite ray $t(\theta_1,\dots,\theta_d)$ $(t\in\Bbb R)$ inside the $d$-dimensional torus ...
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Detailed modern references for basic properties of Pfaffians over commutative rings
Pfaffians are important to algebraic combinatorics, at least.
This is to propose the making of a 'wiki' list, more modern, precise and compressed than e.g. the relevant Wikipedia page (nothing against ...
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Which mathematician sampled published proofs and found one third of them to have errors?
A recent question about whether/how we can trust mathematics in the face of human fallibility reminded me of a paper or article I read probably more than twenty years ago about a mathematician who was ...
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What is the precise relationship between forcing on a poset and the topos of double-negation sheaves on this poset?
I've seen various statements that the Boolean-valued models of ZFC occurring in model-theoretic forcing are "really" the topos of sheaves on an appropriate site, but never a fully precise statement. ...
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Representations of the automorphism group of graphs via spectral graphs theory
Given a (simple) graph $G=(V,E)$ with $V=\{1,...,n\}$ and let $A$ be its adjacency matrix.
I am interested in the representation theory (over $\Bbb R$) of the automorphism group $\def\Aut{\mathrm{Aut}...
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Classification of (not necessarily connected) compact Lie groups
I am looking for a classification of compact (not necessarily connected) Lie groups. Clearly, all such groups are extensions of a finite "component group" $\pi_0(G)$ by a compact connected ...
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Where can you find Grothendieck's "Récoltes et Semailles"?
Where can you find Grothendieck's "Récoltes et Semailles"?
Is it available anywhere?
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When do tensor products of C*-algebras commute with colimits?
Let $I$ be a filtered poset, which you should think of as being huge. Let $A_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume ...
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Functions that are approximately differentiable a.e
The classical definition of an approximately differentiable function is as follows:
Definition.
Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. ...
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3-manifolds with isomorphic fundamental groups
There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces ...
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Origin of term Ahlfors-David regular
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...
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Tensor products of $\mathbb{E}_\infty$-spaces
In the $\infty$-world, connective spectra play the role of abelian groups, while $\mathbb{E}_\infty$-spaces play that of commutative monoids. This may be rephrased by saying that we may identify the $\...
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residually finite-by-$\mathbb{Z}$ groups are residually finite
I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite.
However, ...
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Complexes of representations with complementary central charges
This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary ...
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Consequences of Legendre's conjecture
I am looking for a list/reference which explores the consequences of Legendre's conjecture, which states that one can always find a prime number between $n^2$ and $(n+1)^2$.
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Reference request : Grothendieck's topological space valued integral
As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...
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A result attributed to Whitney
One of the basic results of real analysis says that any closed subset of a smooth ($C^\infty$) manifold $M$ is the set of zeros of some map $\lambda\in C^\infty(M;[0,1])$. This result (or some ...
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Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?
It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: ...
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