# All Questions

100,843 questions
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### Is this problem in $NP$?

Where is the problem 'Given two $n$ many homogeneous system of polynomials in $\mathbb Z[x_1,\dots,x_n]$ with degree $2$ do all there integer roots agree?' in the polynomial hierarchy? Is there a ...
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### Order statistics of correlated bivariate Gaussian

Suppose $(X_1,Y_1),...,(X_n,Y_n)$ are i.i.d. bivariate Gaussian with mean zero. Each coordinate has variance 1 and correlation between coordinates is $\rho\in[-1,1]$. I'm interested in the following ...
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### Adiabatic limit of the torus bundle on the circle

Let $(S^1,g_1)$ be a circle with length $L$ and $(T^2,g_2)$ a flat torus where $T^2=\mathbb C/\{\mathbb Z \oplus \mathbb Z \tau\}$ for $\text{Im}\, \tau>0$. If $(M^3,g_{\epsilon})$ has a fibration ...
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### Example of a Manifold which has One Non-zero Component of Ric corresponding to Scalar Curvature

I am wondering if there is a simple example of a manifold such that, given a value for the scalar curvature $R$, I can find a manifold such that the Ricci tensor has all zero components except for one ...
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### Showing a differential operator is positive semidefinite

Let $R>\lambda>\chi$ be positive real constants and $\alpha$ be a real number. The following differential operator \begin{multline} \mathcal{L}g = -\frac{d}{d\xi}\left[(1-\xi^2)\frac{dg}{d\xi}\...
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### Bishop-Gromov inequality strengthened for anisotropic metrics?

The Bishop-Gromov inequality provides an upper-bound on the rate of growth of volume of a ball of radius $r$ in spaces that have a lower-bound on the Ricci curvature, $Ric \geq (n-1)K$. (I am ...
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### Minimizing union of overlapping rectangles

Believe it or not, this has something to do with making triangle-free graphs bipartite... We have a collection of $k$ axis parallel rectangles with side lengths $(a_i,b_i)$. We want to arrange them (...
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### Is transverse measure on a foliation without closed leaves unique?

Let $(F,\nu)$ be a Thurston's foliation on a surface $S$ with a non-zero transverse measure $\nu.$ Assume that $F$ has no closed leaves nor compact separatrices. Did anyone study such foliations? ...
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### Subgroup of the symmetry group of $Zer(\zeta)$ preserving multiplicity

Let $Zer(\zeta)$ denote the multiset of the non trivial zeros of the Riemann zeta function counted with multiplicity and $G$ the group of isometries of the complex plane preserving this multiset ...
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### Eigenfunctions of elliptic equations

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. ...
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### Symmetric function transition matrix and a non-conjecture by Stanley

Consider the transition matrix $$p_\lambda = \sum_{\mu} R_{\lambda\mu}m_\mu$$ between the power-sum and the monomial basis. There are plenty of combinatorial descriptions of $R_{\lambda\mu}$, it is ...
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### Construction of a $K(\pi,1)$-space?

My colleague suggested a proof of a fact which I have hard time to believe. Since I am not a topologist by training I am asking it here. Consider any CW-complex structure on the $d$-dimensional ...
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### Why is Con(ZFC) independent from ZFC? [on hold]

I am assuming that ZFC is consistent here. By Godel's second incompleteness theorem, Con(ZFC) cannot be proved in ZFC. How do we know that it cannot be disproved? Couldn't ZFC (wrongly) claim its own ...
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### Constant map from automorphic form is surjective?

Let $G$ be a connective reductive group over $\mathbb{Q}$ and $P=NM$ be a standard parabolic subgroup of $G$ and and $K$ a 'good' maximal compact subgroup of $G$. (For precise definition of these ...
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### Matrix trace minimization of quadratic and linear terms under orthogonal manifold constraints

How would one solve the following orthogonal manifold problem? $\max_{\{X : X^\top X = I\}} \text{tr}(X^\top A X - X^\top B)$ where $A \succeq 0$ I've seen one method that successively performs the ...
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### Scheme of relative connected components

Let $f\colon Y\to X$ be a morphism of schemes. Assume $f$ is finitely presented, flat, with geometrically reduced fibers. Then Romagny has proved that the "functor of relative geometric connected ...
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### Countable union of well ordered sets

Assume I have a sequence $(A_i)_{i<\omega}$ of well-ordered subsets of an ordered set $S$. Assume that $A:=\underset{i<\omega}{\cup}A_i$ is also well-ordered. Let $\alpha$ be an ordinal upper ...
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### Reference Request: $n$-edge-coloring bipartite graph $K_{n,n}$ such that monochromatic parts are isomorphic

I am finding references for the following problem: We call a $n\times n$ 0-1 matrix permutation if there are exactly one $1$ in each row/column. Suppose $A$ is a 0-1 matrix of size $n\times n$ in ...
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### Two models for the classifying space of a subgroup via the geometric bar construction

Let $H$ be a topological group which is a subgroup of two other topological groups $G$ and $G'$. It follows (from Rmk 8.9 in May - Classifying spaces and fibrations (MSN, free)) that there exist weak ...
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### Example of non-trivial foliation

Let $M$ be a closed oriented manifold, an oriented foliation $F$ is said non-trivial, if $F$ is not fibration of $M$, i.e. there does not exist a closed manifold $B$, such that $M\overset{F}{\to} B$. ...
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### solving fractions results in 2 different answers [on hold]

Question - 156 biscuits to be shared amongst 18 of us. How much does each get? How many remain? 156/18 = 8r12, so answer is each gets 8 while 12 remain. If I simplify the fraction like 156/18 = 26/...
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### Open Questions about Wasserstein Space and PDE

While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...
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### Reconstructing almost known polynomial from a system of polynomials with common roots

We have $n$ algebraically independent degree $2$ homogeneous system of polynomials with $\mathbb Z$ coefficients in $n$ variables with exactly $t$ primitive (gcd of coefficients is $1$) integer roots ...
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### Is Ackermann's set theory minus class comprehension equal to ZF?

Ackermann in 1956 proposed an axiomatic set theory. Reinhard proved that Ackermann's set theory equals ZF It's clear that Zermelo set theory can be interpreted in Ackermann's set theory minus class ...
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### Do analytic functionals form a cosheaf?

Let $X$ be a complex-analytic manifold. Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
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### Is this a new way to make Collatz variants? [on hold]

Summary: https://drive.google.com/file/d/1Esf-mEYWnzkBxV-UVVre2kUpri3t2IQR/view?usp=sharing Since I discovered the Collatz conjunction this month I got hooked by it. When I first saw the function f(...
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### Lower semicontinuous and convex envelope

L.Ambrosio, in paper [1] writes: Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...) for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse ...
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### Domain of exponential map [on hold]

Let $M$ be a Riemannian manifold and $exp_p:T_{p}M\rightarrow M$ be the exponential map. Let $\gamma_v$ be a geodesic starting at $p$ with the $\gamma_v'(0)=v$. Also define $I_v:= Domain (\gamma_v)$ ...
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### Lambek calculus, linear logic, and linear algebra

In his 1958 paper, The Mathematics of Sentence Structure, Joachim Lambek introduced the Lambek calculus. In modern terms, it could be understood as a syntax for biclosed monoidal categories, and he ...
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### $\omega$-nilpotent cover of a recurrent surface

Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville. $\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to ...
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### Unions and sums of well-ordered positive subsets of ordered groups

Let $G$ be an ordered group. I am looking for a proof of the follwing facts : Let $S,T\subseteq G$ be well ordered subsets of $G$ with order type $\alpha,\beta$, respectively. Then $S\cup T$ is well ...
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### Isometric embedding of a genus g surface

Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $\mathbb{R}^4?$
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### Find representation set of orbits when group acts on a set

Let group $G$ acts on a set $S$. Burnside's lemma gives as how to count numbers of orbits. I am interested how to find the orbits. By finding orbits I mean how to find a representative from each orbit....
Let $\textit{F}$ be the family of $C^1$ curves in $\mathbb{R}^2$ of fixed length $\bar{l}$ and fixed tangent's turning angle $\bar{k}$. What are the curves of positive curvature in $\textit{F}$ ...
I'm looking for a reference to Witten's conjecture that the free part of the (Kauffman bracket) skein module of a closed 3-manifold is finitely generated, i.e. the dimension of $K(M)$, where $M$ is a ...