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May
16
comment Is there a Continuous Multinomial Distribution??
"the parameters in Dirichlet do not have a constant sum" Sure they do: en.wikipedia.org/wiki/Dirichlet_distribution
May
15
comment Hexagon packing in a circle
Vaguely related: Gauss circle problem
May
15
comment Interesting properties of the function $(a,b)\mapsto a/(a-b)$
@columbus8myhw: It satisfies $g(a,b)+g(b,a)=0$, but I haven't found anything interesting corresponding to the other two properties. Maybe they're there but I just don't know how to find them -- after all, the second property of $f$ I stumbled on by luck, and then I had to do a bunch of blind calculations to obtain the third.
May
15
comment Interesting properties of the function $(a,b)\mapsto a/(a-b)$
That's very interesting, thanks! I know nothing about the cross ratio -- how well does it explain the three properties of $f$ I mentioned in the question?
May
15
revised Interesting properties of the function $(a,b)\mapsto a/(a-b)$
edited tags
May
15
comment Do I write $f\in C^{-n}$ for an integrable function?
By the way, I bet you will find this interesting. Repeated integration can be expressed as a single integral: $$\int_a^x \int_a^{\sigma_1} \cdots \int_a^{\sigma_{n-1}} f(\sigma_{n}) \, \mathrm{d}\sigma_{n} \cdots \, \mathrm{d}\sigma_2 \, \mathrm{d}\sigma_1 = \frac{1}{(n-1)!} \int_a^x\left(x-t\right)^{n-1} f(t)\,\mathrm{d}t$$
May
15
comment Interesting properties of the function $(a,b)\mapsto a/(a-b)$
I put in abstract-algebra because I'm wondering if $f$ forms a known algebraic structure. How did you get the second and third identities with a diagram? I'm very interested.
May
15
revised Interesting properties of the function $(a,b)\mapsto a/(a-b)$
added 395 characters in body
May
15
revised Interesting properties of the function $(a,b)\mapsto a/(a-b)$
added 395 characters in body
May
15
asked Interesting properties of the function $(a,b)\mapsto a/(a-b)$
May
13
comment Functional equation $f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}$
Does $f$ have to be defined for $1/\big(1+f(x)\big)$ whenever it is defined for $x$? Otherwise I can come up with lots of solutions...
May
13
comment Maximization of ratio of two polynomials over an interval
Counterexample: $\dfrac{(x+1)(x+2)}{(x+1.4)^2}$ on $[0,2]$. wolframalpha.com/input/…
May
13
comment If $ \textbf{Pr}(A|B) = 1 $ and $ \textbf{Pr}(B|A) = 0 $, then is it true that $ \textbf{Pr}(B) < 0.5 $?
I suppose one could argue that this shows that it is not possible that $P(A|B)=1$ and $P(B|A)=0$ simultaneously. Therefore the original implication is vacuously true.
May
13
comment Why is the velocity and accleration vector not necessarily perpendicular
Ah, but if you look carefully, $N$ is not $\mathrm dT/\mathrm dt$, it's $\mathrm d\hat T/\mathrm dt$ where $\hat T=T/\|T\|$. So $N\ne\mathrm dT/\mathrm dt=\mathrm d^2r/\mathrm dt^2$. // Also, for your larger question, consider a stone falling straight downwards.
May
13
comment How accurate the solution of over-determined linear system of equation could be using least square method?
The value of $\|Ax^*-b\|$, where $x^*$ is the least-squares solution, is simply how far $b$ is from the column space of $A$. If the SVD is $A=U\Sigma V^T$, then the value of $\|Ax^*-b\|$ is $\|(UU^T-I)b\|$.
May
13
comment The image in $\mathbb{C}$ of $\mathbb{R}^2$ under a map of counterpropagating plane waves is…?
Hypocycloids sure seem to match pretty well: i.stack.imgur.com/C3cPe.png
May
13
comment Maximum likelihood estimator on uniform distribution
The simple answer: the probability density at $x$ is not simply $1/\theta$, it is $\begin{cases}1/\theta&\text{if $0\le x\le\theta$,}\\0&\text{otherwise}\end{cases}$. Now tell me what the likelihood function is (hint: it's not $1/\theta^n$).
May
13
comment The image in $\mathbb{C}$ of $\mathbb{R}^2$ under a map of counterpropagating plane waves is…?
Neat question! Going by the picture of $I_5^1$, it looks like the real axis is pointing vertically downward in these plots -- is that on purpose? Also, have you checked whether the support of the plots matches a hypocycloid of $n$ cusps?
May
12
comment How to find expected angle between two randomly generated vectors?
The cosine of the expected angle is not the same as the expected cosine of the angle. I do think the expected angle should still turn out to be $\pi/2$, but this argument does not show it.
May
12
comment Is it ill-advised to read books casually for entertainment?
One could legitimately ask whether it is beneficial, but I can't imagine it possibly being harmful.