Questions tagged [reference-request]

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

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3 votes
1 answer
138 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^\...

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