## All Questions

2answers
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### Is deciding whether a Turing machine *provably* runs forever equivalent to the halting problem?

Assume for this question that ZF set theory is sound. Now consider the language "PROVELOOP," which consists of all descriptions of Turing machines M, for which there exists a ZF p …
0answers
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### In what rigorous sense are Sperner’s Lemma and the Brouwer Fixed Point Theorem equivalent?

I understand that one can give a proof of each of these propositions assuming the truth of the other. But this seems a bit squishy to me, since there is a trivial sense in which a …
0answers
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1answer
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14answers
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### objects which can’t be defined without making choices but which end up independent of the choice

It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data one first chooses some additional structure …
1answer
371 views

### How much of character theory can be done without Schur’s lemma or the Artin-Wedderburn theorem?

This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question wi …
0answers
112 views

### Are there any precise results about the intuition behind Morse functions?

A Morse funnction on a smooth manifold is usually intuitively interpreted as follows: Imagine the manifold to be a mountainous landscape and the Morse function as the elevation of …
0answers
26 views

### identity for number of monomials

Fix a positive integer $d \geq 2$, and let $n,k$ be natural numbers with $k \leq n$. Let b(n,k) denote the number of monomials of degree kd-(n+1) in n+1 variables x_0,..x_n with …
0answers
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### Non-crystallographic cluster algebras

Background Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Her …
1answer
98 views

### fundamental class is the sum of simplices of triangulation of the manifold?

M is an n-dimensional closed orientable manifold. I find in a book "Intuitively,the fundamental class can be thought of as the sum of the (top-dimension) simplices of a suitable tr …
0answers
44 views

### Circle segment of exact length [closed]

I need to find: a spherical line that passes through the points 0,0 and 8,8 and the distance of the line between those two points must be exactly 12 I imagine the answer will b …

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