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 Dec 19 comment A weighted measure of international diversity What exactly is your definition of a "valid method"? Dec 19 revised Zeroes of sin(x) fixed formatting Dec 19 comment Description of the universe of sets I think he is describing the von Neumann universe. In particular, I think with the "situation in which all the stages in the collection are completed" he refers to the stages corresponding to limit ordinals. Dec 19 comment Problem solving rolling dice What are the rules exactly? Does your partner tell you that you have rolled at least 9 as soon as this is true? Or could he remain silent until after you've done a few further rolls (after which you of course still have rolled at least 9)? Dec 19 comment What is ${\rm Im}(z)+{\rm Re}(z)=1$? Hint: $z=x+\mathrm iy$. What does the equation $x+y=1$ describe? Dec 19 answered List all degree one, two, three polynomials. Dec 18 comment fair die or not, from 3D printer Strictly speaking, the probability that your die is fair is zero, because even, say, a probability of $1/6+10^{-100}$ for getting a 6 would not be a fair die. If you want a non-zero probability, you need to specify acceptable deviations from the fair die. Dec 16 comment What is the maximum number of $2$-inch by $2$-inch by $2$-inch cubes Hint: If you build a cuboid out of $2"\times2"\times2"$ cubes, what property does the cuboid have? Dec 16 revised Rewriting $\sin(\arccos(y))$ and $\arcsin(\cos(x))$ replaced image of formulas by MathJax formulas, other formatting and spelling corrections Dec 16 revised Complex numbers getting too complex! added 6 characters in body Dec 16 revised Finding an invariant subspace of a linear operator. edited title Dec 15 answered 12 bit strings with more zeros than ones Dec 14 comment Is $(-2)^{\sqrt{2}}$ a real number? Actually there's no $k$ such that $\sin\left((2k+1)\sqrt{2}\pi\right)=0$ because $\sin x=0$ iff $x$ is an integer mutiple of $\pi$, and $(2k+1)\sqrt{2}$ is non-integer for any integer $k$. So there's no reason to restrict to the principal value; no possible value of $(-2)^{\sqrt{2}}$ is real. Dec 14 comment Suppose that $R_1$ and $R_2$ are reflexive relations? Suppose this post had a meaningful title. Dec 14 comment What does $\frac{d^6y}{ dx^6}$ mean? If the $\mathrm d$ in $\mathrm dx$ were considered an independent factor, you could just cancel it out in $\frac{\mathrm dy}{\mathrm dx}$ to obtain $\frac{y}{x}$. Which obviously is not a valid transformation. Dec 13 revised Can sets have derivatives? edited body Dec 13 revised Can sets have derivatives? Added something about extending the definition beyond just functions. Dec 13 answered Can sets have derivatives? Dec 13 comment Why must a topology on a set contain the empty set? It's not necessary, just more convenient. Just as for intersections, you could just rewrite the requirement, in this case to "every nonempty family of open sets has an open union". That's not inconsistent, just more complicated. Dec 12 comment Does “Doing a thing to both sides of an equation” have a name? If you define "substitution" as "any replacement on one side that keeps the (in)equality true" then your point (1) reduces to "either we modify one side or both sides in a way that preserves the (in)equality", which obviously is true.