# All Questions

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### Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)?

Wikipedia claims (permanent link) without reference: Testing whether one planar graph is dual to another is NP-complete. Another claim with reference: For any plane graph G, the medial graph ...
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### Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent?

For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, ...
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### The homology of $\varinjlim SO(p,q)$

Is there a way to explicitly compute the homology of the space $$\varinjlim_{(p,q)} SO(p,q)^+,$$ where each $SO(p,q)$ is the indefinite special orthogonal group, and $SO(p,q)^+$ its identity ...
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### $K$ theory and singular homology

For cell complexes${}^1$ $X$ we have an isomorphism $$K^*(X)\otimes \mathbb{Q}\cong H^{*}(X;\mathbb{Q}),$$ which is induced by the Chern character. What is the analogous statement for $KO(X)$? ...
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### Negative Arithmetic Line bundle [on hold]

Let $X$ be an arithmetic variety and $L$ be an Arithmetic Line bundle, then how can we define a negative Arithmetic line bundle ?
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### to find topological properties under a metric on a set

we define a metric d on a set of composition operators on L2. I would like to find connected component and path connected component and other topological properties by d . Is there any book or paper ...
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### Erdös is solved and Collatz is solved too? [on hold]

I mean. Cut and Fold the "Quadrant I positiv Numbers" to a Cone with 0 in the Center. A Erdös solved Number X are a Orbit arround the 0. The Orbit are not harmoniously. The Orbit are very Fractal ...
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### Erdös is solved and Collatz is solved too [on hold]

I mean. Cut and Fold the "Quadrant I positiv Numbers" to a Cone with 0 in the Center. A Collatz solved Number X are a Orbit arround the 0. The Orbit are not harmoniously. The Orbit are very Fractal ...
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### How to solve the following bivariate recurrence?

$$F(n,r) = (1-w(r))F(n-1, r) + w(r-1)F(n-1, r-1)$$ where $w(r)$ is monotonically non-increasing in $r$ and $0 \leq w(r) \leq 1$ with $0 \leq r$ Initial condition: \begin{eqnarray} F(0, r) & = ...
This is probably a silly question and maybe in the end the answer is trivial but I can't see it. The problem is the following. Let $M$ be a Riemannian manifold, $\gamma \colon[0,l]\to M$ be a ...