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2 votes
0 answers
32 views

What is the supremum of 1-dim Hausdorff measure of the nodal set of Neumann eigenfunction $u$ for planar convex domain

All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote $\Sigma=\{u(x)=0\}$ be the nodal set of a ...
0 votes
1 answer
17 views

Combining Dantzig-Wolfe and Benders decomposition

I'm now solving an LP that has a few coupling rows (as in Dantzig-Wolfe decomposition) and a few coupling columns (as in Benders decomposition) simultaneously; other rows and columns are block-angular....
  • 1
2 votes
0 answers
35 views

Different invariants of group actions from isomorphic subgroups

Consider $D_8,$ the dihedral group of order $8$, acting on the unit square $X=[0,1]^2 \subseteq \mathbb{R}^2$ in the natural way– essentially take the unique linear extension of the action on the ...
  • 21
6 votes
1 answer
275 views

A hypercover of profinite sets as a limit of hypercovers of finite sets

This is about a rather concrete problem that occurs in the middle of a lecture by Scholze. First I'll refer to the lecture, but then I'll state the problem. In https://www.youtube.com/watch?v=...
4 votes
1 answer
294 views

Is there any approach to solving statements about the natural numbers which are just true by chance?

I suspect that certain problems regarding the base 10 representation of natural numbers may be undecidable simply because there's no way to even start. Take any exponentially growing sequence like $16^...
3 votes
1 answer
90 views

Property of sets of positive Lebesgue measure in $\mathbb{R}^2$

Let $P\subset \mathbb{R}^2$ be a set of positive Lebesgue measure. Is it always true that a suitable rotation and translation of $P$ always contains a set of the form $\{re^{i\theta}:r\in E, \theta\...
0 votes
0 answers
22 views

Concentration inequality of the $L^2$ norm of weighted vector with moment of eigenvalues of GOE

Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ with ...
  • 128
0 votes
0 answers
20 views

Integrated risk for estimation of varying coefficient model

Consider the nonparametric varying coefficient model $$y_i = x_i'\beta(z_i)+e_i,$$ where $(x_i, z_i)$ are covariates on $[0,1]^m\times [0,1]^k$, $e_i$ are the errors, and $\beta:[0,1]^k\to [0,1]^m$ is ...
  • 211
1 vote
0 answers
54 views

Fundamental Solution to Biharmonic Equation in 3D

(This is a repost of a question posed in StackExchange that didn't get any replies.) Is anything known about the fundamental solution to the equation: $$\nabla^4 (Au) + \nabla^2 (Bu)+Cu=0$$ for ...
  • 139
2 votes
0 answers
123 views

Concrete description of “DeMorganian” open sets

Let me begin with a few definitions. My question will be basically how to simplify them to something more manageable. The motivation for these definitions is given at the end. Let $X$ be a ...
  • 23.7k
7 votes
1 answer
477 views

Name for vector spaces with two algebra structures that satisfy the exchange law

Is there a name/reference for the following object? We have a vector space $V$ over some field with two associative bilinear operations $\circ,*:V \times V \to V$ which satisfy the interchange law, i....
2 votes
1 answer
264 views

The nontrivial zeros of the zeta function and the prime counting function

The truncated explicit formula has the shape \begin{equation}\label{1} \psi(x) =x-\frac{\zeta^{\prime}(0)}{\zeta(0)}-\sum_{|\rho|\leq T}\frac{x^{\rho}}{\rho}+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}+...
  • 21
0 votes
0 answers
25 views

Numerical method for solving simple linear PDE with a grey box component

Consider the following linear PDE: $$\nabla_q V(q) - M_d(q)M^{-1}(q)\nabla_q V_d(q) = 0,$$ where $V(q)$ and $M(q)$ are known and $M_d(q)$ is a grey box function (e.x., $M_d(q)$ is fitted using a ...
  • 45
4 votes
1 answer
193 views

Complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles

Question. What are examples of compact complex manifolds whose tangent and cotangent bundles are isomorphic as complex vector bundles? A family of examples are, of course, holomorphically symplectic ...
1 vote
1 answer
43 views

Relation of MSTs in the Euclidean plane to Delaunay triangulations

It is known that the Minimum Spanning Tree (MST) of a finite set of points in the Euclidean plane is contained in the point set's Delaunay triangulation, but is that all that can be said about their ...
  • 11.4k
1 vote
0 answers
78 views

Enumerating unions of arithmetical sets

In Simpsons's excellent Subsystems of Second-order Arithmetic, we find V.4.10 which tells us the following: The following is provable in ATR$_0$. Let $(A_n)_{n\in \mathbb{N}}$ be a sequence of ...
  • 2,318
2 votes
1 answer
229 views

Explicit generators from Serre spectral sequence

Let $p: E \to B$ be a locally trivial fibration with fiber $F$. If necessary, suppose that $B$ is simply connected. Suppose that the Serre spectral sequence leaves the term $H_p(B, H_q(F, \mathbb{Q}))$...
3 votes
0 answers
66 views

Short selection in the space of subsets

Given a metric space $X$, denote by $\mathrm{Haus}\,X$ the space of all compact subsets in $X$ equipped with Hausdorff metric. Further $X$ will be identified with a subset of $\mathrm{Haus}\,X$ --- a ...
1 vote
0 answers
45 views

Probability density function for the polar sine of uniformly distributed points on the sphere

If I sample three points independently, uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their polar sine? More generally, for $k<n$ if I ...
  • 111
0 votes
0 answers
88 views

How can I check whether this function is periodic?

I have a function $$F(t)=\sum_{k=1}^n{\left[\eta_ke^{-i{\omega_k}\ t}+\eta_k^{*}e^{i{\omega_k}\ t}\right]},$$ where $\omega_k\in \mathbb{R}$ and $\eta_k\in\mathbb{C}$. In fact, I don't care about the ...
0 votes
0 answers
42 views

Hemisphere containing the maximum number of points scattered on a sphere

Consider a set of points $x_1, \ldots,x_n$ on $\mathbb{S}^{k-1}$ (the unit sphere in $\mathbb{R}^k$). The goal is finding the hemisphere which contains the maximum number of $x_i$'s. Basically, we ...
  • 87
2 votes
0 answers
108 views

Expected error term in the distribution of Friedlander-Iwaniec primes

In 1998 John Friedlander and Henryk Iwaniec famously proved the asymptotic formula $$\displaystyle \mathop{\sum \sum}_{a^2 + b^4 \leq x}\Lambda(a^2 + b^4) = \frac{4x^{\frac{3}{4}}}{\pi} \int_0^1 (1 - ...
2 votes
1 answer
160 views

Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$

I've been working on the spectrum of the closure of the operator $J: \mathcal{D}(J)= \mbox{span}\{ e_n: n \in \mathbb{Z}\} \subseteq \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ defined for $x=(x_n)_{n \...
  • 227
1 vote
0 answers
211 views

Cartier and the continuity of the early history of schemes

If you allow me I would divide the early history of schemes this way _ Weil, Zariski, Bourbaki, Nagata, Van der Waerden,... up to Chevalley (you can find an interesting blog here) J P Serre varieties ...
3 votes
0 answers
63 views
+100

Interpolation on Sobolev space on $[0, 1]^d$ over finite meshes

Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e., $$ \|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^...
4 votes
2 answers
147 views

Is the conditional expectation of a Caratheodory function a Caratheodory function?

Let $(Y, \Sigma,\mu)$ be measure space and $X$ a Polish space endowed with its Borel $\sigma$-algebra. Suppose that $f:Y\times X\to \mathbb R$ is a Carathéodory function (i.e. continuous in $x\in X$ ...
  • 163
2 votes
1 answer
86 views

Inequality with decreasing rearrangement and non-decreasing function

This question is a continuation of the question here. Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$...
  • 305
2 votes
0 answers
183 views

Why unramified Milnor-Witt K-groups are unramified

Let $X$ be an integral locally noetherian smooth scheme over base field $k$. Then for every $x \in X^{(1)}$ point of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation ring. ...
0 votes
0 answers
19 views

Jacobi identity for the bracket in action Lie algebroid

I am trying to verify that the bracket in the definition of action Lie algebroid satisfies the Jaboci Identity. Let $X \to X^\dagger$ be an action of a Lie algebra $\mathfrak{g}$ on a manifold $M$, i....
  • 101
0 votes
0 answers
209 views

Intuition for comultiplication [closed]

I heard a professor say that, at least intuitively, comultiplication is, in the context of polynomials, a function which when applied to the multiplication of a polynomial $p$ and a polynomial $q$ it ...
14 votes
2 answers
1k views

Why do Grothendieck topologies used in algebraic geometry typically involve finiteness conditions?

There are many Grothendieck topologies used in algebraic geometry, with complex interrelations. Generally in one of these topologies, a cover of schemes is a family of maps which is jointly surjective ...
  • 52.6k
5 votes
1 answer
182 views

Homology of spherical $3$-manifold group

I have been studying $3$-manifolds recently and I got stuck in the following situation. For lens spaces the below fact is true. Let $G$ be a finite group acting freely and orthogonally on $S^3$ so ...
  • 159
0 votes
0 answers
25 views

The performance ratio of the approximation algorithm of maximum clique

Consider the following approximation algorithm for the problem of finding a maximum clique in a given graph $G$. Repeat the following step until the resulting graph is a clique. Delete from $G$ a ...
  • 11
9 votes
0 answers
101 views

Can the product of an exotic torus and a circle be the standard torus?

As discussed in this question from last week, if $M$ is a closed manifold such that $M\times S^1$ is homeomorphic to the torus $T^{n+1}$, then $M$ is homeomorphic to $T^n$. Is the corresponding ...
7 votes
1 answer
757 views

Definable set in ZF that cannot be proved to be Borel

Is there a predicate $P(x)$ such that $\mathrm{ZF} \vdash \exists! x. P(x)$, and $\mathrm{ZF} \vdash \forall x. P(x) \to (x \subseteq \mathbb R)$, but $\mathrm{ZF} \nvdash \forall x. P(x) \to \mathsf{...
  • 517
-1 votes
0 answers
73 views

Grouplike elements is a group in a Hopf algebra [closed]

I have been looking at this question and two questions arose for me: (1) A simpler one: Why is $(g \otimes g)(h \otimes h)=(g h \otimes g h)$ ? My guess: $$(g \otimes g)(h \otimes h)=g^2(1 \otimes 1)(...
0 votes
0 answers
40 views

A question on shrinking bases

Let $(x_{n})_{n=1}^{\infty}$ be a basis for a Banach space $X$ with the biorthogonal functionals $(x^{*}_{n})_{n=1}^{\infty}$. Recall that $(x_{n})_{n=1}^{\infty}$ is called shrinking if $(x^{*}_{n})_{...
39 votes
2 answers
6k views

Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture

Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635 In that preprint, Kirti Joshi claims that he agrees with Scholze and ...
2 votes
0 answers
75 views

How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?

It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
3 votes
0 answers
142 views

Is every tree a deformation retract of the disk?

I apologise if this question is not suitable for MathOverflow. We define a graph here to mean a disjoint union of points and copies of $[0,1]$ quotiented so that the endpoints of any interval lie on a ...
6 votes
1 answer
249 views

The center of $\mathbf{hTop}$

What is the center of the homotopy category $\mathbf{hTop}$? I strongly believe that it is trivial, but it is hard to prove since $\mathbf{hTop}$ is not concretizable and hence has no small separator. ...
2 votes
0 answers
66 views

Closed form for coefficients related to excedance set of permutation

Working on suitable closed form for A329369, I discovered very useful coefficients, which have the following recurrence relation: $$T(0,1)=T(0,2)=1$$ $$T(n,1)=1, n>0$$ $$T(0,k)=0, k>2$$ $$T(2n+1,...
1 vote
0 answers
59 views

Invariance signature in infinite dimension

Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that: $g_0$ is positive-definite $g_t$ is non-degenerate for ...
  • 385
2 votes
0 answers
89 views

Canonicity in split sequence in cotangent spaces

Let $X$ be a locally ringed space. We have for a point $p$ the exact sequence $$0\to \mathfrak{m}_p^2\to \mathfrak{m}_p\to \mathfrak{m}_p/\mathfrak{m}_p^2 \to 0$$ where $\mathfrak{m}_p$ is the maximal ...
  • 21
0 votes
0 answers
57 views

Operators decomposition in pseudo-Hilbert space

Let $(H,g)$ be a pseudo-Hilbert space, i.e. $H$ is an infinite dimensional vector space endowed with an indefinite symmetric product $g$. Suppose we have a linear operator $D:H\to H$ and let $D^*$ be ...
  • 385
3 votes
0 answers
98 views

A new arranging of discrete sine transform

Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix $$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$ Let us denote by $s_{-,l}$ the $l^{\text{...
  • 3,421
1 vote
0 answers
28 views

Error bounds for Sobolev space norm approximation on a finite grid

Suppose that $f : [0, 1] \to \mathbb{R}$ is an element of the order-$k$ Sobolev space, \begin{multline} f^{(k - 1)}~\text{is absolutely continuous},\quad \|f\|_{W^k}^2 := \int_0^1 f^{(k)}(x)^2 \, dx &...
1 vote
0 answers
49 views

about codimension two foliation

Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold I am curious about examples of codimension Are there any previous studies or lecture notes of foliation ...
0 votes
0 answers
43 views

VC dimension of axis-parallel boxes

Let $A$ be the family of axis-parallel boxes in the $d$-dimensional unit cube $[0,1]^d$ having one vertex at the origin. It is known that the VC dimension of $A$ is $d$. Let $B$ be the family of all ...
  • 101
0 votes
0 answers
114 views

Extension of action in algebraic group

I was asking this on stack exchange but I didn't get the answer. Borel's book Linear Algebraic Groups contains the following result 10.9 Theorem. Let $G$ be a connected affine group of dimension one. ...

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