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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2 votes
1 answer
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Hausdorff dimension and non-empty intersections with lines

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

$\mathsf{AC}_\mathsf{WO}+\mathsf{AC}^\mathsf{WO}\Rightarrow \mathsf{AC}$?

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 ...
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2 votes
0 answers
72 views

Boundedness for singular parabolic p-Laplace equation

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

Have we discovered constructions for natural fractional dimensional spheres?

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

Relative equivariant Thom transversality

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 ...
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5 votes
1 answer
138 views

Nonexistence of sphere with one conical point [reference request]

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

Related to the Schwarz Christoffel map

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 ...
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3 votes
0 answers
119 views

An account of "Homologie nicht-additiver Funktoren. Anwendungen"'s results

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

Equivalences between statements of (seemingly) different order

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 (...
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1 vote
1 answer
255 views

Kronecker product: Is it possible to simplify this product $e^{-A} \otimes e^{A}$ where $A$ is an invertible and symmetric matrix [closed]

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 ...
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4 votes
1 answer
151 views

Percolation: at what length scale do we see it?

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

Reference request: Leonardo Da Vinci's supposed math results

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

Expected value of $\mathrm{tr}((X + D)^{-1})$ where $X$ is Wishart and $D$ is diagonal?

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

Reference request for a result in additive combinatorics

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

PAC and totally real fields

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

Structure theorem for a class of idempotent monoids (where $xy = x$ or $xy = y$ for all $x, y$)

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

Extension of primitive set of vectors and reduction theory

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 $\...
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4 votes
1 answer
190 views

Wedderburn decomposition of special linear groups

$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$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

Uniform bound on the measure of $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ if $\Omega$ is an open bounded set with Lipschitz boundary

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 $\...
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1 vote
0 answers
56 views

Reference request: rates of weak convergence of Polish space-valued random variables

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 ...
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2 votes
1 answer
104 views

A characterization of continuity in terms of preservation of connected sets. Where to find the result?

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

Asymptotics for enumerating graphs

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

A theorem by R.L. Moore

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 ...
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17 votes
2 answers
596 views

Has anyone seen this construction of the Weil representation of $\mathrm{Sp}_{2k}(\mathbb{F}_p)$?

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

Derived categories and $\infty$-categories necessary for condensed mathematics

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

Reference for lattices as algebraic structures

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 ...
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15 votes
2 answers
371 views

Generalizations of summation methods of divergence series

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

Exceptional values of differential monomials of meromorphic functions with multiple zeros

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)^...
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4 votes
0 answers
32 views

On the connection graphs-knots-tensors

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

Mean value of the divisor function over Piatetski-Shapiro sequences

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

Reference for harmonic functions in cylinders

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 ...
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3 votes
2 answers
214 views

Inequality for Gaussian polynomials III

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

On the relative class number of a cyclotomic extension

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 $p$ ...
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0 votes
0 answers
33 views

State-of-the-art for approximating the Cheeger constant (for graphs)

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 ...
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0 votes
1 answer
73 views

Lattice-point-free body diameter

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 $\sqrt2$....
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2 votes
1 answer
117 views

If $M$ is a compact smooth finite-dimensional manifold with boundary, is the inclusion of a closed subspace $A \subseteq M$ a cofibration?

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

Limit of zero sets of harmonic functions

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

Conditional distribution of steps of random walk given the sum

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 $a > 0$. ...
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4 votes
1 answer
174 views

Is the category of computads for a finitary monad on $n$-globular sets cocomplete?

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

Regularity of eigenfunctions of a self-adjoint differential operator in Gilbarg-Trudinger

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 $$ ...
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1 vote
0 answers
58 views

"N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$

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). ...
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2 votes
1 answer
185 views

Strategy of the proof of the "minimal entropy condition" for scalar conservation laws

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 ...
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4 votes
1 answer
145 views

Reference for the equivalence between chain complexes and sequential diagrams in a stable $\infty$-category

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

Spectral perturbation theory of discrete spectra in presence of continuous spectrum

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

How to measure the optimality of the induced order by a median order of a tournament on a big subset

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 ...
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5 votes
0 answers
50 views

Criteria for tightness of Gaussian measures on Banach spaces

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 ...
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0 votes
0 answers
61 views

Topology of independence set of a vector space

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 ...
  • 1,570
0 votes
0 answers
65 views

Inequality for $q$-binomials II

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

Test function with Steklov average for Caccioppoli-type inequality for porous medium equation

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}}\...
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2 votes
0 answers
156 views

Proof of the projection formula (for cohomology of $\mathbf{P}V$)

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, ...
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