Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,545
questions
3
votes
1
answer
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views
Inequality for generalized Laguerre polynomials
Please. Does anybody know a proof of this inequality
$$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and
$L^{\alpha}_n$ ...
12
votes
4
answers
1k
views
How dense is the set of asymmetric graphs?
On $n$ nodes, we have $2^{n(n-1)/2}$ graphs. Asymmetric graph is a graph that has only trivial automorphism. We known that asymptotically almost all finite graphs are asymmetric. Therefore, in the ...
2
votes
0
answers
121
views
Homotopy invariant $K$-theory spectrum version vs space version
Let $K^H$ be the homotopy $K$-theory spectrum. It is defined as a colimit of the form $|\mathbb{K}(X\times \Delta^{\bullet})|$. Here $\Delta^{\bullet}$ is the co-simplicial scheme defined by the ...
1
vote
0
answers
96
views
Generalizion of Euler identity with infinite sum of inverse squares
For $x,y\in \mathbb{R}\backslash \mathbb{Z}$ let
$$
f(x,y)=\sum_{n\in\mathbb{Z}} \frac{1}{(n-x)(n-y)}
$$
Is there a closed formula for $f(x,y)$?
What is known:
We have
$$
f(x,x)=\left(\frac{\pi}{\sin(\...
4
votes
1
answer
251
views
Is there a C*-algebra whose Pedersen ideal is not proper?
In [1, 5.6.3] Pedersen states without proof or reference that there are non-unital C*-algebras whose Pedersen
ideal is the whole algebra.
Does anyone know where can I find such an example?
Is it ...
3
votes
1
answer
421
views
Borel–Weil–Bott for partial flag varieties
Is there a generalization of Borel-Weil-Bott for partial flag varieties, i.e. homogeneous spaces of the form $G/P$ with $P$ parabolic and $G$ semisimple? If so, I would like a reference.
10
votes
1
answer
273
views
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.
Let $N^+$ denote the space of uni-upper-triangular ...
11
votes
6
answers
2k
views
The Wiener-Ikehara approach to the PNT
Was providing an alternative proof of the PNT one of the main impulses that led to the discovery of the Tauberian theorem of Wiener and Ikehara or the other way around?
In any case, do you know who ...
19
votes
0
answers
519
views
univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
9
votes
5
answers
2k
views
A survey on positive mass theorem?
Could you suggest a good survey paper on positive mass theorem?
16
votes
2
answers
1k
views
Good overviews on $\phi^{4}$-field theory?
I'm looking for nice overviews on $\phi^{4}$-field theory from the mathematical-physics point of view. To be a little more specific, here are some topics I'd like to read about:
(1) What are the ...
2
votes
1
answer
120
views
Fiberwise skeleton vs. category of isomorphism types
What is the relationship (if any) between the process of taking a skeleton of a category, taking a fibered skeleton of a fibered category, and taking isomorphism classes of a category as objects of a ...
0
votes
0
answers
84
views
Reference request for additive persistence of a number
It is well known fact that each natural number can be represented uniquely in any base. So we can define digit sum function whose value is sum of digits of the natural number in given base.
Let $f(n,b)...
1
vote
1
answer
140
views
Complexity of solving $\sum_i A_i X B_i = C$
Is anything known about computational complexity of finding $X$ which satisfies the following matrix equation?
$$\sum_i^n A_i X B_i = C$$
With $A_i,B_i,C$ dense $d\times d$ matrices. Any literature ...
18
votes
5
answers
4k
views
Are there textbooks on logic where the references to set theory appear only after the construction of set theory?
This is cross posted from MathStackExchange. Since this is a reference request, I believe there will not be duplications of efforts in answers. This is also related to the question here.
In textbooks ...
4
votes
0
answers
204
views
Fréchet subdifferentiation on riemannian manifolds
Context. I'm looking for a "natural" definition of subdifferentials on riemannian manifolds.
Given a function $F:\mathbb R^m \to \mathbb R$, its Fréchet-subdifferential at a point $w \in \...
1
vote
0
answers
409
views
Examples of almost Dedekind domains that are not Dedekind
All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...
0
votes
1
answer
177
views
English translation of Hasse's "Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage"
I want to read through Hasse's paper about cubic number fields: Arithmetische Theorie der kubischen Zahlkorper auf klassenkorpertheoretischer Grundlage, Mathematische Zeitschrift 31 (1930) pages 565-...
46
votes
11
answers
5k
views
Reference request: Examples of research on a set with interesting properties which turned out to be the empty set
I've seen internet jokes (at least more than 1) between mathematicians like this one here about someone studying a set with interesting properties. And then, after a lot of research (presumably after ...
9
votes
1
answer
436
views
$M = AA^t$ where $A$ has unit norm columns
Let $M \in \mathbb{R}^{k\times k}$ positive definite with $\operatorname{tr} M = m$, where $m$ is an integer such that $m \geq k$. I have found a way (using this answer) to decompose $M = AA^t$ with $...
0
votes
1
answer
322
views
Hitting probability for mean-reverting stochastic process
I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...
0
votes
1
answer
353
views
Where can I find the problem by Lagarias?
Jeffrey Lagarias proved, unconditionally, that:
$$
\sigma(n)<H_n+2\exp(H_n)\log(H_n)\qquad n>1
$$
This was posed as a problem in:
J. C. Lagarias, Problem 10949: A generous bound for divisor ...
15
votes
5
answers
2k
views
Reference request: Recovering a Riemannian metric from the distance function
Let $M = (M, g)$ be a Riemannian manifold, and let $p \in M$.
Writing $d$ for the geodesic distance in $M$, there is a function
$$
d(-, p)^2 : M \to \mathbb{R}.
$$
This function is smooth near $p$. ...
3
votes
1
answer
122
views
Reference Request: Comprehension for multicategories
I recently came across the notion of comprehension for a fibration of categories $p:E\to B$ that subsumes the axiom of separation and the grothendieck construction for fibred categories. A nice ...
3
votes
1
answer
225
views
Metric "in the limit"?
Let's say that a function $d:S\times S\to [0,\infty)$ for a countable set $S$ is a metric in the limit if
$$d(x,y)\le \liminf_{n\to\infty} d(x,z_n)+d(z_n,y),$$
$$\lim_{n\to\infty} d(z_n,z_n)=0, \quad\...
1
vote
0
answers
284
views
Codifferential of wedge of two 1-forms
Let $\omega,\eta$ be two 1-forms on a manifold $M$. I'm interested in an expression for
$$
\delta(\omega\wedge\eta)
$$
where $\delta$ is the co-differential operator $\Lambda^2(M)\to\Lambda^1(M)$. ...
2
votes
0
answers
149
views
$k$-invariants of $KO$ and $ko$ and differentials in the AHSS spectral sequence
Let $KO$ and $ko$ denote real $K$-theory and connective real $K$-theory. It appears to be a well done result that the $k$-invariants can be used to determine the early differentials in the Atiyah-...
6
votes
1
answer
564
views
Does the purported proof of Rota's conjecture provide an algorithm for calculating the forbidden minors of matroids over arbitrary finite fields?
About six years ago there was a proof announced and later outlined in a notice from AMS. However right now I can only seem to find forbidden minor characterizations for matroids linearly ...
2
votes
2
answers
183
views
What is the minimum number of triangle centers sufficient to unambiguously describe a triangle?
I am looking for a minimal number of properties describing a triangle so that these properties are invariant to the choice of a Cartesian coordinate system as well as to the order in which the ...
2
votes
0
answers
179
views
Dyadic models in number theory and "spillover"
In a classic blog post, Tao discusses the appearance of "dyadic models" in various guises in various areas of math. The number-theoretic version of the idea is to study polynomials over a ...
2
votes
0
answers
57
views
Limit set for IFS has either empty interior or dense interior
Let $f_1,\ldots,f_k:\mathbb R^n\to\mathbb R^n$ be contracting affine maps. By the theory of iterated function systems, there is a unique minimal compact $K\subseteq\mathbb R^n$ such that $K=f_1(K)\cup\...
9
votes
2
answers
607
views
Definition of subcoalgebra over a commutative ring
Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$.
Notes I'm reading give the following definition:
$D$ is called subcoalgebra of $C$ if the ...
14
votes
1
answer
841
views
Why there is no 3-category or tricategory of bicategories?
I recently asked this question on MSE.
So I want to move it here in hope to gain a more wordy answer.
I have read around about bicategories, lax functor, lax natural transformation and modifications. ...
0
votes
1
answer
334
views
How do I show that any finite-dimensional (absolute) CW-complex $X$ is locally contractible?
I know that it holds even if $X$ has infinite dimension, but I am looking for a specific argument in the finite-dimensional case.
1
vote
0
answers
133
views
Reference request for a paper of Berard-Bergery
I was wondering if anyone could point me to a pdf copy of the following paper by Lionel Berard-Bergery:
"Scalar curvature and isometry group", in Spectra of Riemannian Manifolds, Kaigai ...
1
vote
0
answers
87
views
Proving that a model exhibits either a first or second order phase transition
Motivating example:
Take the (wired) random cluster model $\phi^1_{p,q}$ with parameter $q$ (see http://arxiv.org/abs/1707.00520 for an introduction).
It is now known on $\mathbb{Z}^2$ that it has a ...
4
votes
0
answers
200
views
How does a Lyapunov vector evolve along a trajectory?
First I introduce the Lyapunov vectors. Here I follow the notations of a previous answer I got on MO.
We have a dynamic system with discrete time $t$ (integer values). The
time evolution is defined by ...
9
votes
1
answer
1k
views
Stacks for a string theory student
First, I'm a string theory student hoping to grasp some math involved in some physics developments.
I'm hoping to read the famous Kapustin-Witten Paper "Electric-magnetic duality and the ...
8
votes
2
answers
913
views
Lower bound on exponential sums
Let $k\geq 2$. Consider the following norm of exponenetial sum:
$$
I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy.
$$
Bourgain mentioned on Page 118 of
https://...
16
votes
2
answers
687
views
Is this kind of "Gerrymandering" NP-complete?
[I posted this on Math Stack Exchange about two weeks ago, but didn't get any reply, so I'm trying it here.]
Consider the following simplified form of "Gerrymandering": You have $n^2$ ...
8
votes
1
answer
350
views
Reference request: Origins of differential homological algebra
Differential homological algebra in its initial formulation is due to Eilenberg and Moore, who published the homological version of the Eilenberg–Moore spectral sequence in 1965 (and the cohomological ...
14
votes
5
answers
2k
views
Largest Hausdorff quotient
The inclusion of the full subcategory of Hausdorff topological spaces into the category of topological spaces has a left adjoint, which can be proven easily by the Adjoint Functor Theorem (see for ...
0
votes
0
answers
43
views
Solving nonlinear equations involving expectations
Let $X$ be a random variable and $g(x,y)$ be a function of two variables. Consider the equation
$$
\mathbb{E}_Xg(X,y) = 0
$$
Are there any specialized techniques for solving such equations (...
10
votes
3
answers
844
views
Explicit free subgroup in Thompson's group $V$
R. Thompson introduced three groups $F\subset T\subset V$. The question concerning amenability of $F$ is still unanswered and has attracted much attention. I have read that Thompson group $V$ contains ...
9
votes
1
answer
487
views
Proof of a 'well-known' result of Shimura on periods of modular forms
It is often noted in the literature that there are certain complex periods that allow one to normalize the modular symbol associated to a modular form in such a way that its coefficients are algebraic....
6
votes
1
answer
167
views
Finitely presented modules admitting projective covers
A ring $R$ is called semi-perfect if every finitely generated $R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely ...
3
votes
0
answers
202
views
Quick proof of the first part of Kaplansky's Theorem on characterization of Noetherian domains with all maximal ideals principal
I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-...
2
votes
1
answer
800
views
On matrices that almost have the same eigenvalues
Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if
$$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and ...
4
votes
1
answer
206
views
Tachikawa conjecture for finite dimensional commutative monomial algebras
Let $A=K[x_1,...,x_n]/I$ be a finite dimensional local algebra with a monomial ideal $I$.
The Tachikawa conjectures are conjectures for finite dimensional algebras. Im interested in them here for such ...
7
votes
2
answers
407
views
References about "monoidal fibrations" in $\infty$-category theory
$\newcommand{\cat}{\mathsf} \newcommand{\fun}{\mathrm{Fun}} \newcommand{\calg}{\mathrm{CAlg}}$
Let $\cat C^\otimes,\cat D^\otimes, \cat E^\otimes$ be symmetric monoidal $\infty$-categories, and $p^\...