# All Questions

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### Must the coordinates of a polynomial iteration have about the same size?

The following statements seem plausible (not to say intuitively obvious), but I do not know how to prove them. I would like to know whether these or similar problems have been considered anywhere. ...
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### Can infinte dimensional algebras be smooth?

Is it possible for an infinte dimensional algebra to be smooth? That is can an infinte dimensinal algebra have finite global dimension? For example is the polynomial ring $R[x_n]_{n \in \mathbb{N}}$ ...
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### Tensor products [on hold]

I'm currently trying to teach myself about tensors but I'm having some trouble with understanding what's going on. I've managed to come to a basic understanding of the ranks of tensors, rank n ...
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### Are the asymptotics of A003238 known?

Sequence A003238 of the OEIS counts rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins 1, 1, 2, 3, 5, 6, 10, 11, 16, ... and it is ...
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### Internal logic of the topos of simplicial sets

I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, ...
22 views

### Characterizations of the GOE/GUE family of distributions

For a random symmetric matrix of size $n\times n$, with entries drawn from a Gaussian Ensemble, the joint probability of eigenvalues can be written as: ...
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### Undergraduate Mathematics Research [migrated]

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 ...
19 views

### 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 ...
112 views

### 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 ...
67 views

### 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 ...
118 views

### 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 ...
408 views

### 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 ...
16 views

### 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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