If $n$ is an even and $\alpha, \beta$ are the roots of the equation $x^2 + px + q = 0$ and also of the equation $x^{2n} + p^nx^n + q^n = 0$ and $f(x) = \frac{(1+x)^n}{1+x^n}$ where $\alpha^n + \beta^n ≠ 0, p ≠ 0$, find the value of $f(\frac{\alpha}{\beta})$ . I ve seen various interpretations f...
I'm teaching an undergrad course in graph theory and have just finished the proof of Max Flow/Min Cut. So far I have used Diestel's definition (more or less) of flow network as a digraph $G$ with a capacity function $c: E\rightarrow \mathbb{N}$ and a flow function $f:E\rightarrow \mathbb{Z}$ sat...
Topic: st-connectivity, st-cut ideals and path ideals of a graph My Lemma: None of the st-cut-monomials vanish iff there is at least a st-path that does not vanish. Example ST-cuts: {{1,3,5,6},{1,3,7},{1,4,5,6},{1,4,7},{2,3,5,6},{2,3,7},{2,4,5,6},{2,4,7}} ST-paths: {{1,2},{3,4},{5,7},...
The graph has tree paths IN-1-OUT, IN-2-OUT and IN-3&4-OUT between IN and OUT in the left. I want to make each path to a branch like the right. What is the name of this operation or the name of this procedure to make a graph into a tree?
On Mathematics Stack Exchange, there are several tag synonyms which haven't been merged; meaning that both the master tag and the synonym contain some questions.1 The situation when there is a synonym where the synonymized tag still contains some question is not optimal for several reasons.2 I'd ...
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