# All Questions

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### 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 ...
• 121
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### 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....
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### 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 ...
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### 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=...
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### 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 ...
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### 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 ...
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1 vote
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 ...
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### 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 ...
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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....
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### The nontrivial zeros of the zeta function and the prime counting function

The truncated explicit formula has the shape \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}+...
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### 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
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### 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
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### 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 ...
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1 vote
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### 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 ...
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### 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 $H_p(B, H_q(F, \mathbb{Q}))$...
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### 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 ...
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1 vote
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 ...
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### 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 ...
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### 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 ...
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### 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 $x\in X$ ...
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1 vote
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### 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 &...
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1 vote
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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 ...
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### 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 ...
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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. ...