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Almost all of the questions on the front page these days are homework questions or textbook exercises. I think I'll be spending a lot less time here.

I hereby authorize whoever it may concern to post my comments as answers if they wish, whether as community wiki or otherwise if you like imaginary internet points.

comment Hausdorff and Fréchet distances
The Fréchet distance in this example is actually about half the horizontal length of the figure, because the point on $f$ can only move forwards while the point on $g$ goes back and forth between the left and right extremes.
comment Anyway to assert if a number is inbetween two numbers without less than or greater than operations
Can you explain what you hope to accomplish by renouncing use of less-than and greater-than operations?
comment Why do folded concentric circles and rectangles form a hyperbolic paraboloid?
I don't have an conclusive answer, but I can think of two reasons: (1) Paper is a much stiffer material, so it tries to minimize the amount of deformation. A two-curl saddle is the "lowest-frequency" mode for a disk to bend out of the plane. (2) Lettuce probably has much more excess material, that is, $c \gg 2\pi r$, so it needs more than two curls. See also: crochet models of hyperbolic space.
comment Random binary array shows patterns around prime numbers
This is known as Euclid's orchard.
comment Comma in base notation
$0.5,2$ isn't a number. There are two parameters to the $\chi^2$ distribution, which in this case are given to be $0.5$ and $2$. (P.S. It's actually $0.05$ in the PDF.)
revised Comma in base notation
deleted 62 characters in body
comment Minimum of sum of increasing and decreasing function
Probably not. If you drop the positivity conditions then any $h$ can be written as a sum of an increasing $f$ and a decreasing $g$.
comment Real life applications for logarithms
Most of the answers to this question apply here. I'm tempted to close as a duplicate.
comment What mathematical function can give me a set of curves similar to these?
Possibly relevant: Potential flow around a circular cylinder. ContourPlot[With[{r = Sqrt[x^2 + y^2], θ = ArcTan[x, y]}, (r - 1/r) Sin[θ]], {x, -4, 4}, {y, -2, 2}, RegionFunction -> Function[{x, y}, x^2 + y^2 >= 1], Contours -> Range[-2, 2, 1/4], AspectRatio -> Automatic, Frame -> False] i.stack.imgur.com/9ZStt.png
comment Cool Integral = $\pi/2$ !!
"how can we derive this same result for $n=1,2,…,7$? And why does it deviate at $I_8$?" See this previous post and Hans Lundmark's comment on it which contains a stunningly simple argument.
comment How come this grammar is unambiguous?
Those aren't parse trees. They are two derivations(?) corresponding to the same parse tree. Try drawing the actual tree and you'll see.
comment Validity of proof for surface area of a sphere
Previously: Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
comment The Least Area For a Needle to Pass Through a Curve?
No, what I'm mentioning is rotating a needle by tracing the red area instead of the blue area.
answered Why do folded concentric circles and rectangles form a hyperbolic paraboloid?
comment Is Douglas Hofstadter's version of Godel's proof utter nonsense?
Is this a rant or a question? If it is a question, it would help to provide some more context, ideally so that one could answer the question without having the book on hand. If it is a rant, it belongs on a blog or a message board rather than a question-and-answer site.
comment Seemingly-random algorithm for generating fictional static objects on a real-world map?
Use the same random seed in your (pseudo)random number generator.
comment Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]
It's just a choice of coordinate transformation, i.e. a change of variables, so it can't change the final value. The integrand changes but so do the limits of integration, and it all works out in the end. Of course, in this case, you're integrating over a disk, so what happens to the limits I leave to you to figure out... :)
comment The Least Area For a Needle to Pass Through a Curve?
Here's something interesting: Say you want to rotate the needle by an angle $\theta$. If you hold its center fixed and rotate the needle about the center, you cover an area of exactly $\theta$. But if you sweep the needle along an arc such that the center describes a circular arc of radius $r$ and the needle remains tangent to the arc throughout, then for large $r$ the area you cover tends to only $\frac12\theta$.
comment Polar coordinates for $xz$-plane: $z = r\sin\theta$ ? [Stewart P1091 16.7.25]
Actually orientation doesn't matter at all in this step, since you had to take care of it already in going from $\mathbf F\cdot d\mathbf S$ to $g(x,z)\,dA$. Now you're just picking a polar-coordinate parametrization of the $xz$-plane, and it doesn't matter if you pick $x=r\cos\theta, z=r\sin\theta$, or $x=r\sin\theta, z=r\cos\theta$, or even $x=-r\sin\theta, z=-r\cos\theta$, anything of that sort. If you're really worried about orientation for whatever reason, you can pick $x=r\cos(\theta+\pi/2), z=r\sin(\theta+\pi/2)$ which still leads to Stewart's result.
revised Set geometry and inclusion
a little easier to read now, I think