# Questions tagged [reference-request]

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

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

13,375
questions

2
votes

1
answer

69
views

Let $A\subseteq [0,1]^d$, $d\geq 2$, a set with Hausdorff dimension $\operatorname{dim}_{\mathcal{H}}A=s$. What is the minimum $s$ (if any) which guarantee that $A$ has non-empty intersections with a ...

13
votes

0
answers

270
views

Let
$\mathsf{AC}_\mathsf{WO}$: Every well-orderable family of non-empty sets has a choice function.
$\mathsf{AC}^\mathsf{WO}$: Every family of non-empty well-orderable sets has a choice function.
My ...

2
votes

0
answers

72
views

Local boundedness of singular parabolic $p$-Laplace equation
$$\partial_t u - \operatorname{div}(|\nabla u|^{p-2}\nabla u)=0,\,1<p<2,$$
requires additional integrability assumption for the ...

5
votes

0
answers

183
views

I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...

1
vote

1
answer

68
views

I'm looking for a reference for the following:
Suppose that $G$ is a finite group, that $M$ is a smooth $G$-manifold, and that $A\subseteq M$ is a closed $G$-invariant subspace of $M$ such that the ...

5
votes

1
answer

138
views

It seems to be considered a classical fact that one cannot have a spherical polyhedral/cone-metric on the 2-sphere with precisely one conical point. However, I've never actually seen it proven ...

-1
votes

1
answer

66
views

With the help of the Schwarz-Christoffel map, for a given polygon (given angle), we can find some points on the boundary of the upper half plane, such that a particular Schwarz-Christoffel map takes ...

3
votes

0
answers

119
views

Is there an account in English of results from "Homologie nicht-additiver Funktoren. Anwendungen" by Dold and Puppe? I am mostly interested in the spectral sequence of cross-effects which ...

2
votes

1
answer

92
views

In Steve Simpson's excellent monograph SOSOA, we find Theorem X.4.4 which contains an equivalence (over RCA$_0^*$) between the following statements:
The induction axiom for $\Sigma_1^0$-formulas (...

1
vote

1
answer

255
views

Let $A$ be an invertible, symmetric and tridiagonal matrix of size $n \times n$. Assume that $A_{i,i}=a \neq 0$ for $i=1\dotsc n$ and all the elements in the sub- and super-diagonal of $A$ are $b \neq ...

4
votes

1
answer

151
views

Consider classical bond percolation on $\mathbb{Z}^d$. Each edge is included with probability $p$ and deleted with probability $1-p$. As is well known, there is a $p_c(d) \in (0,1)$ such that $p>...

12
votes

1
answer

699
views

Many reputable sources (I can give as many as you want) describe Da Vinci as a mathematician, but they never mention a single theorem, result, or lemma that he proved. There's the golden ratio spiral, ...

2
votes

0
answers

18
views

Let $X$ be a standard Wishart matrix, i.e.,
$$
X = \sum_{j=1}^n g_j \otimes g_j \quad \mbox{where} \quad g_j \sim N(0, I_d).
$$
Above, $g_j$ are independent samples from the standard multivariate ...

1
vote

0
answers

74
views

Let $p$ be a prime number and $[p-1]=\{1, 2, \ldots, p-1\}$.
The following proposition is proved: (but I cannot find out where)
Proposition: The non-empty subset sums of $[p-1]$ are equally ...

11
votes

1
answer

435
views

A field $K$ is called pseudo-algebraically closed (PAC) if every absolutely irreducible variety over $K$ has a $K$-point. Let $L$ be the maximal totally real subfield of $\overline{\mathbb Q}$. A few ...

4
votes

3
answers

260
views

Question. Is there any structure theorem for the class of monoids $H$ with the property that $xy = x$ or $xy = y$ for all $x, y \in H$? Or does this look hopeless for some good reasons?
A monoid with ...

0
votes

0
answers

35
views

Let $\Lambda$ be a unimodular lattice in $\mathbb R^d$ (unimodularity is not really necessary here but just for convenience) and let $B$ be a ball centered at the origin that contains $(k+1)$-many $\...

4
votes

1
answer

190
views

$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...

1
vote

0
answers

41
views

Let $\Omega \subset \mathbb R^d$ be an open bounded set with Lipschitz boundary. Let us consider $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ for $\delta >0$. I want to say that the measure of $\...

1
vote

0
answers

56
views

Let $(E,\mathscr{E})$ be a Polish space $E$ together with its Borel $\sigma$-algebra $\mathscr{E}$. Let $(\Omega, \mathscr{F},P)$ be a probability space and let $(X^{(n)})_{n \in \mathbb{N}}$ be a ...

2
votes

1
answer

104
views

There is a result that if $X$ is a locally connected space and $Y$ is a locally compact Hausdorff space, then a function $f \colon X \to Y$ is continuous if and only if $f$ has a closed graph and for ...

0
votes

0
answers

46
views

Let $K>k$ be positive integers. Now assume that $G$ is a $K$-connected graph with $n$ vertices and $m$ edges.
I would like to ask:
QUESTION. Is there an asymptotic for the number of $k$-connected ...

6
votes

0
answers

113
views

The following result is due to R.L. Moore.
Let $K\subseteq\mathbb C$ be compact. Suppose that
$K$ is connected,
and that $\mathbb C\setminus K$ is connected.
Then $\partial K$ is connected.
Does ...

17
votes

2
answers

596
views

$\def\FF{\mathbb{F}}\def\CC{\mathbb{C}}\def\QQ{\mathbb{Q}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}\def\PGL{\text{PGL}}$Let $p$ be an odd prime. The Weil representation is a $p^k$-...

15
votes

1
answer

991
views

I am reading the three texts on condensed mathematics by Scholze and Clausen. I am also interested in paper "A $p$-adic 6-functor formalism in rigid-analytic geometry" by Lucas Mann.
To ...

2
votes

1
answer

121
views

I want to study lattices as a structure related to ring theory. I am familiar with lattices as a beginner but I want to go further and know their connections to ring theory. Do you know a book which ...

15
votes

2
answers

371
views

If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...

0
votes

0
answers

39
views

Let $f$ be a non-constant meromorphic function of finite order in $\mathbb{C}$ having zeros of multiplicity at least $k+1,~k\geq 1,$ and define $$M[f]:=g\cdot\prod\limits_{j=0}^{k}\left(f^{(j)}\right)^...

4
votes

0
answers

32
views

You can interpret a featureless graph as product of featureless abstract tensors; the tensors are then automatically totally symmetric as "leg crossing" in the graph interpretation is the ...

6
votes

1
answer

130
views

Let $c>1$, $c\not\in\mathbb{Z}$ and consider the sum
$$
\sum_{n\leq x} \tau(\lfloor n^c \rfloor),
$$
where $\tau(n)$ is the number of divisors of $n$. I'm almost certain I've seen an evaluation of ...

2
votes

1
answer

87
views

Question. What is a good reference (textbook, article etc.) to learn more about harmonic functions on finite (and infinite) cylinders?
I am trying to gain a better understanding of the behavior of ...

3
votes

2
answers

214
views

Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials)
$$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$
Given two ...

2
votes

0
answers

99
views

Let $\Bbb Z[\zeta_p]$ denote the cyclotomic integers where $p$ is a prime and let $h_1 = h_1(p)$ denote its relative class number.
Question: Is it known whether there are infinitely many primes $p$ ...

0
votes

0
answers

33
views

What is the state-of-the-art algorithm for approximating the Cheeger constant, given a regular graph? Ideally, such an algorithm would run in polynomial time (in the size of the graph, and could be ...

0
votes

1
answer

73
views

The following interesting problem was asked at Aops and I wonder if it was based on some research paper:
Let $K$ be a convex body in $\mathbb R^2$, such that the diameter of $K$ is less than $\sqrt2$....

2
votes

1
answer

117
views

Question: If $M$ is a compact smooth finite-dimensional manifold with boundary, is the inclusion of a closed subspace $A \subseteq M$ a cofibration? (I'm specifically interested in the case when $A$ ...

6
votes

1
answer

227
views

Let $u_n : \mathbb{R}^n \to \mathbb{R}$ be a sequence of harmonic functions which converge uniformly on compact subsets. The limit function $u$ (which we assume to be not identically $0$) is clearly ...

3
votes

0
answers

82
views

Set-up. Consider a random walk $S_n=\sum_{i=1}^n X_i$, where $\{ X_i, 1\leq i < \infty \} $
is a sequence of i.i.d. random variables with distribution $\mu$, $\mathbb{E}X_1 = 0$. Let $a > 0$.
...

4
votes

1
answer

174
views

Context
Given a finitary monad $T:\operatorname{gSet}_n\to\operatorname{gSet}_n$ we can define categories $\operatorname{Comp}_k^T$ of $k$-computads for $T$, for any $k=0,\cdots,n+1$. This is nicely ...

3
votes

2
answers

168
views

Let $\Omega$ be a bounded smooth domain,
$Lu = D_i \left( a^{ij} (x) D_ju \right)$, and two constants
$\lambda, \Lambda > 0$. Suppose the coefficient $a$ is measurable,
symmetric, and satisfies
$$
...

1
vote

0
answers

58
views

Let us consider the Burgers equation
$$u_t + (u^2)_x = 0$$
In
Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...

2
votes

1
answer

185
views

Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law
$$u_t + f(u)_x = 0,$$ satisfying the entropy condition
$$\eta(u)_t + q(u)_x \le 0$$ in the ...

4
votes

1
answer

145
views

Lurie's $\infty$-categorical Dold-Kan Correspondence relates simplicial objects and sequential diagrams in a stable $\infty$-category. Is there any reference for an equivalence to a category of ...

1
vote

1
answer

119
views

This is a 2 part question:
1). I am looking for a (hopefully accessible to beginning grad student who knows matrix perturbation theory) reference for doing concrete calculations of perturbed discrete ...

4
votes

0
answers

78
views

Median orders are great tools for dealing with a-priori unknown orientations of edges in tournaments, because they provide us with local properties on oriented edge density.
I've been wondering if ...

5
votes

0
answers

50
views

In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...

0
votes

0
answers

61
views

This seems like something that would have a well-known treatment somewhere, but I'm not sure where to look. If we have a vector space $V$ (or maybe even a module), we can consider an abstract ...

0
votes

0
answers

65
views

Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials)
$$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$
Given two ...

2
votes

0
answers

366
views

Let $m>1$ and consider locally bounded weak solutions $u(x, t)$ of the parabolic porous medium equation, meaning that $$u\in C_{loc}\left([0, T); L_{loc}^{2}(U)\right)\cap \left\{u^{\frac{m+1}{2}}\...

2
votes

0
answers

156
views

Let $V\to X$ be a vector bundle (over say a scheme).
Then the cohomology of its projectivisation is
$$\text{H}^*(\mathbf{P}V)\ =\ \text{H}^*(X)[t]/(t^{n+1}+c_1(V)t^n+\cdots+c_n(V))$$
as an algebra, ...