yoshi
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Counting elements of a particular order in Dihedral groups
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The answer to this was to recognize that all elements were rotations or order 2 flips. So count rotations as you would in $\mathbb{Z}_{2n}$, then add $n$ more order 2 elements (the reflections).

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1 answers
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Law of total expectation and max(Y1, Y2)
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I know this is a stale post, but just in case anyone is looking (like I was) I'll add the answer here because it took me a while to understand how to compute these expectations. $$\begin{align}\Bbb E(...

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Building a Function with saddle point that has Certain Properties
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Following the commentor, I searched Monkey Saddles. I found more complicated saddles: I guess these types of surfaces are called n-th order saddles. Formulas for them can be found here: https://rivix....

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What does the Borel $\sigma$-algebra of $\mathbb{R}$ look like?
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This is a very large set. In what follows, I'm basically going to quote a discussion that can be found in the Notes and References Section of Folland's real analysis book (Section 1.6) Start with ...

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2 answers
1 votes
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Orthogonal Transformation
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Let $v=\sum v_i e_i$ (basically a basis expansion of v). Then: $Lv = L(\sum v_i e_i)$. Use the IP against some $e_j$: $$\left\langle L(\sum v_i e_i), e_j\right\rangle \\ = \left\langle \sum v_i e_i, L^...

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3 answers
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Conditional Probability and Independent Events
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I believe the awkward wording is suggesting a direct partition: $$P(\text{Male},\text{Science})+P(\text{Female},\text{Science})+(\text{Male},\text{Not Science})+P(\text{Female},\text{Not Science}) = 1$...

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Expression for the gradient using Taylor's Theorem
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This is for 1-dimension but you can get the flavor of what's occuring in (2.4), (2.5), and (2.6). Nocedal and Wright initially suppose one continuou derivative. This allows them to write (by ...

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Inner product in Sobolev spaces
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An answer I've heard follows: One way (there are others) to analyze solutions in modern PDE analysis is to: Specify a space of functions with certain regularity (e.g. all functions on a bounded ...

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