# All Questions

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I am currently an undergraduate junior. I have taken most of the standard undergraduate math courses and a few introductory graduate courses (measure theory, algebraic topology, complex analysis, ...
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### Trapping a convex body by a finite set of points

In $\mathbb{R}^n$, let $K$ be a convex body and $T$ a finite set of points disjoint from the interior of $K$. Say that $T$ traps $K$ if there is no continuous motion of $K$ carrying $K$ arbitrarily ...
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### trick question: how to construct the centre of a given circle? [on hold]

This is a trick question I heard from a high-school teacher today: find the origin of a given circle. You can use any mathematically correct tool and method. Give me the shortest, the most elegant or ...
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### Smallest distribution of points with genuinely different clusterings

An hierarchical clustering algorithm for (finite) sets of points in a given metric space is essentially determined by its linkage criterion, which defines the distance between arbitrary (finite) sets ...
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### integral involving nth order incomplete gamma function [on hold]

\begin{eqnarray} \int_0^{\infty}\,x^{k+r+\xi-1 }\,e^{-\lambda ^{-k}\,x^k}\, \left( \Gamma\left(1+\frac{\xi }{k},\,x^k \lambda ^{-k}\right)\right)^n\,{\rm d} x\,. \end{eqnarray}
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### The largest size of a boolean subgraph (a hypercube) of a given graph

Let $G(\mathbb{F}_2^n)$ denote the graph that represents the lattice of all subspaces of $\mathbb{F}_2^n$ (also called a Hasse diagram). I am interested in knowing if there exists a large hypercube ...
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### Chow ring of two varieties

Suppose we are given two smooth projective varieties $X$ and $Y$. Maybe this is elementary but what is the Chow ring $CH(X\times Y)$ in terms of $CH(X)$ and $CH(Y)$?
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### Direct image of an ideal sheaf along a blow-up

Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
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### Number of representations of an integer as an (arbitrary) sum of products

If $n$ is a positive integer, let $r(n)$ denote the number of representations of $n$ as a sum of products of pairs of positive integers. (Here, the order of the terms in the sum does not matter, but ...
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### endomorphisms of the Jacobian of a curve

Let $C$ be a smooth, projective curve over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. Precisely, ...
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### “Paradoxes” in $\mathbb{R}^n$

One may think of this question as a duplicate of this one. I see it more like an extension. The "inscribed sphere paradox" discussed in the aforementioned question states that if you inscribe a ...
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### real algebraic geometry software?

Does anyone have suggestions/experience for any software packages to study real algebraic varieties (for example, counting connected components of hypersurfaces, figuring out the topological type of ...
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### Graph Theory is the slum of Topology (?) [on hold]

(Edited in accordance with suggestions in comments.) I remember once I read a quote that sounded like "graph theory is the slum of topology" (please approximate). I can not find it on the web, and I ...
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### Homotopy with non piece-wise linear boundary

in the middle of a long proof I encounter the following problem. Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand ...
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### Is Laplacian matrix singular? [on hold]

I'd like to ask is Laplacian L matrix singular? Than if it is singular, how it is possible do inverse and lu factor of Laplacian?