# All Questions

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### independent subset problems [on hold]

I'm interested in the following which i suspect is probably a well studied problem. Given a set $N=\{1,2,...,n\}$ and $M=\{1,2,...,m\}$ consider a map $$f:N\rightarrow 2^{M}$$ (elements of $N$ to ...
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### Embedded Contact Homology and Manifold Decompositions

Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in $\mathbb R\times Y$ and "counting" them. In the symplectic ...
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### A new method of solutions for partial linear differential equations [on hold]

Recently,I read a book on partial differential equations,which says that the solution of second order linear equation of two differentiating variables and analytic coefficients can always be expressed ...
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### totally disconnected sets and homeomorphisms

For every totally disconnected perfect subset S in the plane one finds a homeomorphism of the plane onto itself mapping S onto the ternary Cantor set. This is an exercise in a book by Engelking and ...
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### 4-regular graph with every edge lying in a unique 4-cycle

What are all 4-regular graphs such that every edge in the graph lies in a unique-4 cycle? Among all such graphs, if we impose a further restriction that any two 4-cycles in the graph have at most one ...
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### not Gauss sum with the same magnitude

Gauss sum is a sum of $p$ roots of unity with magnitude $\sqrt{p}$. Does another sum with such property exist? More exactly. Let $p$ be a prime number. $\zeta^p=1,\;\zeta\ne 1$. Causs sum: ...
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### Unitary operators on Hilbert spaces [on hold]

Consider the set $U(H)$, of unitary operators on the real separable Hilbert space $H$. Fix an orthonormal basis $\{e_i\}$ of $H$. Is the subset $S$ of $U(H)$ corresponding to finite dimensional ...
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### Do complex tori contain quasi-projective open subsets?

Complex tori are not associated to projective varieties in general. But can one find an open $U$ inside a complex torus $\mathbb C^g/L$ such that $U$ is the analytification of a quasi-projective ...
Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., \$dF(z) = ...