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 Oct 22 comment Local Extrema (Weird Cases) Yes, that is what I mean: Dom$(k)=\{0\}$, CoDom$(k)=\mathbb{R}$, and im$(k)=\{1\}.$ Although, now that I am pondering it, I don't see how changing the codomain to $\{1\}$ would change the extrema. Oct 21 revised Local Extrema (Weird Cases) deleted 23 characters in body Oct 21 comment Local Extrema (Weird Cases) A function is a subset of the Cartesian product of two sets. Perhaps I should rewrite as $k=$ rather than $k(x)=$ to be proper. Oct 21 asked Local Extrema (Weird Cases) Dec 14 comment Is $(0,0)$ a solution to $x^y-y^x=0$? This answer and comment by Ross most closely represent my expected answer. Namely, it is a matter of how we are interpreting $x^y$. Dec 14 accepted Is $(0,0)$ a solution to $x^y-y^x=0$? Dec 14 revised Is $(0,0)$ a solution to $x^y-y^x=0$? edited tags Dec 14 comment Is $(0,0)$ a solution to $x^y-y^x=0$? @StefanSmith The question stemmed from a discussion of whether the graph of $x^y=y^x$ contains the origin. Knowing the process of showing $0^0$ is an indeterminate form, I hoped to follow a similar procedure using paths/limits. Dec 14 awarded Editor Dec 14 revised Is $(0,0)$ a solution to $x^y-y^x=0$? deleted 13 characters in body; edited title Dec 14 asked Is $(0,0)$ a solution to $x^y-y^x=0$? Oct 19 comment Centroids of triangle Better yet, do you see that $G_1G_2G_3$ and $G_4G_5G_6$ are equilateral? Oct 19 comment Centroids of triangle Can you solve the case when ABC is equilateral? Oct 19 answered Centroids of triangle Oct 8 awarded Supporter Sep 30 comment Permutations of n beads on a string. Oops. Just logged back in and noted that I answered the wrong question. Rotations aside, the reflective symmetries part still applies and hence provides the desired solution $n!/2$ for the $n$-bead case. Sep 28 awarded Teacher Sep 28 answered Permutations of n beads on a string. Sep 14 comment Area of twisted torus That is the exact explanation I was hoping for. It is counterintuitive to me that twisting the torus can increase the length of a fixed band, but twisting a region does not change its area. I guess this is similar to the idea that a shear transformation on a rectangle preserves the area. Thank-you for an excellent explanation. Sep 14 awarded Scholar