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location Seattle, WA
age 43
visits member for 3 years, 8 months
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Software engineer and long-time dabbler in mathematics; 11th in the Putnams forever ago but I've long since atrophied.


Apr
16
comment “Here's a cool problem”: a collection of short questions with clever solutions
Nifty! Quick (hopefully not too spoilery) check: does the answer (or at least an answer) use a 'bricklayer' style coloring?
Apr
16
comment “Here's a cool problem”: a collection of short questions with clever solutions
That seems pretty simple! Three points determine a circle, so $1, z, z^2$ determine the circle - which must therefore be invariant under multiplication by $z$, since $z, z^2, z^3$ determine the same circle. If $C$ is the center then this implies $C=Cz$ which immediately gives $C=0$, and the fact that $1$ is on the circle gives that it's the unit circle.
Apr
16
comment Topological groups, why need them?
I feel like this answer leaves open OP's core question: what are they good for - or, alternately, why do we care that a group is topological? Unfortunately, showing a bunch of examples doesn't really give a lot of motivation as to how this extra structure helps.
Apr
15
revised What's the value of $n+\cfrac{n}{n+\cfrac{n}{n+\cfrac{n}{\vdots}}}$ for $n\in\mathbb{C}$?
added 22 characters in body
Apr
15
answered What's the value of $n+\cfrac{n}{n+\cfrac{n}{n+\cfrac{n}{\vdots}}}$ for $n\in\mathbb{C}$?
Apr
15
comment Why can $2^3$ be defined but $0^0$ cannot
A very good question! (Someone will step in with a proper answer, but the short version is 'because log and exp are continuous functions, and because the limit of a product is the product of the limits', which you should be able to prove yourself if you know the definition of a limit)
Apr
15
comment Does Shor's algorithm work for noncommutitive or nonassociative algebras?
Comment rather than answer because it's little more than a Wikipedia link, but: no efficient quantum HSP algorithm is known for non-abelian groups, and such an algorithm would imply efficient quantum algorithms for graph isomorphism; see en.wikipedia.org/wiki/Hidden_subgroup_problem.
Apr
15
asked Nontrivial relations in rotation groups
Apr
14
comment How to show $\int_{0}^{\infty}e^{-x}\ln^{2}x\:\mathrm{d}x=\gamma ^{2}+\frac{\pi ^{2}}{6}$?
Given that the indefinite integral isn't elementary, it's hard to see how this approach could be made to work.
Apr
14
comment Generalized Cross Product
It's worth noting that there's one subtlety in this definition, one that's bitten many a graphics programmer: technically, the cross product of two vectors in $\mathbb{R}^3$ (or of $(n-1)$ vectors in $\mathbb{R}^n$) isn't a vector; it's a covector: that is, a linear map from vectors to scalars. There's a one-to-one correspondence between vectors and covectors, so we often speak of 'the vector' cross-product, but the catch is that the cross-product doesn't transform linearly under the same transform that vectors do; i.e., $(Uv)\times(Uw)\neq U(v\times w)$ if $U$ is a linear transformation.
Apr
13
comment Please explain why this isomorphic class has 30 graphs and 4 automorphisms
4 choose 2 comes from picking either the two non-$a$ vertices that are connected to each other, or (equivalently) the two non-$a$ vertices that aren't connected to each other. If you think about the classes of vertices a bit, you should see where the automorphisms come from...
Apr
12
revised Is $i \in \mathbb{Q}[\sqrt[4]{-2}]$?
Corrected the spelling on 'Eisenstein'.
Apr
12
comment “World's Hardest Easy Geometry Problem”
I don't have a reference to hand, but I'd swear I remember an article either in the AMM or the Intelligencer about generalizations of the problem with 'nice' angles. Digging around on the web suggests that this is called "Langley's Problem" but doesn't turn up what I'm thinking of...
Apr
12
comment “World's Hardest Easy Geometry Problem”
You use $x^\circ$ for two distinct angles in your diagram...
Apr
12
comment The Limit: $\lim_{x \to \infty}\frac{e^{f(x+a)}}{e^{f(x)}}$
You should also be able to use the MVT here, since there's some $x_0$ in the vicinity of $x$ (specifically, in $[x, x+a]$) with $\frac{f(x+a)-f(x)}{a} = f'(x_0)$.
Apr
11
comment some problems with evaluating $\int_0^{\pi}\ln(\sin x+\sqrt{1+\sin^2x})dx$
@hammood arcsinh(sinx) is not the indefinite integral of your integrand. Remember that if $u=\sin x$ then $du=\cos x\ dx$; you're missing that $\cos x$ term in your integrand.
Apr
11
comment Question regarding the golden/silver ratio
The silver ratio doesn't 'approximate' $1+\sqrt{2}$; that's exactly what it is. If you rewrite your ratios non-homogeneously using $x=\frac{a}{b}$, you get $x=2+\frac1x$; multiplying by $x$ turns this into the quadratic equation $x^2-2x-1=0$ with solution $1+\sqrt{2}$.
Apr
11
comment Suppose we toss a coin 5 times and define Y as number of runs of heads. How do you find expectation and variance?
+1 for providing a workable scheme, but I strongly suspect the math of the recurrence works out a bit more clearly if you work with enumerating outcomes and divide by $2^k$ if you need probabilities.
Apr
11
comment How prove this result $\frac{x}{y}=\sqrt{\frac{\sqrt{5}+1}{2}}$
"Regular" is usually used to indicate that all faces are equilateral and equivalent; by definition, a tetrahedron with right-triangular faces can't be regular.
Apr
11
comment Determine if $\sum\limits_{n=1}^{\infty}(1+\dfrac{2}{n})^n$ converges or diverges
@CameronWilliams (pedantry: given that there's a limiting process involved, it's probably worth noting that this means that the limits are at least one - but the important part, of course, is that it's bounded away from zero regardless.)