# All Questions

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### irreducible affine scheme over Z has good reduction almost everywhere

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that $f$ is irreducible over $\overline{\mathbf{Q}}$. Where can I find a proof of the following classical result: for ...
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### singularity of the solution to an integral equation

I consider a function $x\mapsto f(x)$ which is the positive solution to the integral equation $$\int_{f(x)}^\infty \frac{du}{u^\gamma-x}=1, \quad x\ge 0,$$ where $\gamma\in (1, 2]$ is some ...
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### Graph Theory - Adjacent certices in a simple graph [on hold]

Let u and v be adjacent vertices in a simple graph G. Prove that uv belongs to at least d(u) + d(v) - n(G) triangles in G.
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### Graph Theory - Question on hypercubes and cycles [on hold]

Prove that every cycle of length 2r in a hypercube is contained in a subcube of dimension at most r. Can a cycle of length 2r be contained in a subcube of dimension less than r?
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### Why is the spectrum of this matrix product invariant with respect to order of the multiplicants?

I've been chewing on the following problem for some time and I just don't have any more ideas how to tackle it. I have matrices $A_1,...,A_k \in \mathbb R^{n\times n}$ and I'm observing that the ...
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### Bounds on Product of CDF or Beta function

I have functions of the form \begin{align} I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x)~~~~i = 0,1 \end{align} $F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...
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### Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any ...
Let $\mathcal A$ be a regular surface in $\mathbb R^3$ and $P$ a point in $\mathbb R^3\setminus\mathcal A$. Suppose that $C$ is a point at minimum distance from $P$. Show that $P$ belongs to the ...