# All Questions

100,159 questions
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### Is there a notion of tunnel number for 2-knots?

Given an embedded circle $K$ in $S^3$, the tunnel number of $K$ is the minimum number of embedded arcs one needs to add to $K$ so that the complement of $K$ and the arcs is a handlebody. For an ...
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### A special connected subset of the Cantor fan

Is there a dense connected subset $X$ of the Cantor fan $$(C\times [0,1])/(C\times \{1\})$$ such that for every two connected subsets $X_1,X_2\subseteq X$, the intersection $X_1\cap X_2$ is connected? ...
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### how to compute the number of possible trees in a tree graph? [on hold]

Let's suppose that I have a tree with n nodes. The root of my tree does not change in time. It is the same. However, the rest of nodes change their positions (...
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### Minimization Proof of Conditioning on Gaussian is Gaussian

It is well known that $E[X|X+Y]$ is Gaussian if both $X$ and $Y$ are, and the result can be derived using standard density arguments. However, how can one prove it by only resulting to optimization ...
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### Reference request: invariants/tableaux functions for 4 lines in $P^3$

Does anybody have a reference for invariants of configurations of linear subspaces in the projective space? In particular I would be curious to see an explicit expression of the invariant functions ...
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### Why is the set of lifts of a p-divisible group canonically the same as the set of lines that span $M(G)/FM(G)$?

Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the ...
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### William Thurston's quote?

Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. Is this from Thurston? If yes, where and when it has been said. I've checked "ON PROOF AND ...
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### Are complete minimal submanifolds closed?

Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset? What about the case in which the ambient manifold is an euclidean space?
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### Explaining the consistency of PRA and ZF from predicative foundations

Recently I got interested in predicative foundations, mostly because of Laura Crosilla's work and because Agda employs a predicative type theory. From the point of view of a predicative foundation to ...
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### Chromatic polynomial and the circle

In https://arxiv.org/pdf/1208.5781.pdf It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$. My ...
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### Differential equations satisfied by quasi modular forms?

It is known that modular forms are solutions of differential equations. More precisely, let me cite the statement from the following question. Differential Equations Satisfied by Modular Forms ...
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### Infinite spectral norm of linear mapping [on hold]

Suppose we have a linear mapping $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$. We define its $k$-spectral norm as: $\sigma_k(A)=\sup_{x} \frac{||Ax||_k}{||x||_k}$. We know that when $k=2$, $\sigma_2(A)$ ...
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### Finding a curve through a zero-cycle and a dense subset

Let $k$ be a field. Is it true that for any smooth irreducible projective $k$-variety $X$ and a dense open set $U\subset X$, for any zero-cycle on $X$ one can find an irreducible curve containing its ...
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For an application in kinetic theory I have to apply the Hardy-Littlewood-Sobolev(H-L-S) for $q=\infty$. The dimension is $3$ and H-L-S inequality says that for $1<p<q<\infty$ and $0<\nu&... 0answers 59 views ### On the exponent of a certain matrix$A$in characteristic$p > 0$Let$A$be a square matrix in characteristic$p > 0$with both column and row having length$(1 + p + \cdots + p^i)$, where$i > 0$. Suppose that further the$(m,n)$-component$a_{m,n}$of the ... 0answers 59 views ### Euclidean Geometry [on hold] In Euclidean Geometry we can create a square with side length of$\sqrt2$. as we know,$\sqrt2$is a number which has no end. so it is physically impossible to have a square with this length in the ... 0answers 31 views ### Find and prove optimal solution for a linear programming problem Given$C \in R^+$,$\epsilon \in R^+$,$vec \in R^n$and$eps = [\epsilon \dots \epsilon]$vector of n$\epsilon$I have to$ \textrm{minimize } [vec, eps]^T [x_1, x_2]$with$x_1, x_2 \in R^n$... 0answers 50 views ### Component Groups of Reductive Groups Suppose$G$is a reductive group that is not necessarily connected and$Z \subset G$is a central subgroup. Suppose$G^0$is the identity component of$G$. Is it true that$G/G^0Z= \pi_0(G/Z)$? I can ... 0answers 33 views ### Estimate on Covariant Derivatives of Coordinate Derivatives I am currently reading Topping's lecture notes on Ricci flow. At one point in the narrative (page 65) he says that using the fact that$\bigg(\frac{\partial}{\partial t}\nabla - \nabla \frac{\...
Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. ...