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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.
May
12
comment Solve: $f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x).f(\frac{1}{y})$
I'd start by substituting $z=1/y$ to get the less weird-looking equation $f(x+z)+f(x-z)=2f(x)f(z)$.
May
11
comment A weird Calculus of Variations problem
Well, we do a different substitution for each integral. Afterwards we can add them up because $\int f(x)\,\mathrm dx+\int g(y)\,\mathrm dy=\int\big(f(x)+g(x)\big)\,\mathrm dx$.
May
11
comment A weird Calculus of Variations problem
It might help to substitute $x=x_n+\xi$, and thus write $(2)$ as $$\begin{multline} \sum_{n=1}^N\int\big(y(x)-t_n\big)\nu(x-x_n)\eta(x)\,\mathrm dx\\ =\int\left(\sum_{n=1}^N\big(y(x)-t_n\big)\nu(x-x_n)\right)\eta(x)\,\mathrm dx\\ =\int\left(y(x)\sum_{n=1}^N\nu(x-x_n)-\sum_{n=1}^Nt_n\nu(x-x_n)\right)\eta(x) \,\mathrm dx \end{multline}$$ and then you can maybe see why $y(x)=\sum_{n=1}^Nt_n\nu(x-x_n)/\sum_{n=1}^N\nu(x-x_n)$.
May
8
comment Aproximating the volume of a sphere by dividing it into infinitesimal cubes
@Circonflexe: That's Schwarz's cylinder area paradox. Computing volumes by subdivision is still fine though.
May
8
comment Is there a sample of a $f(x)=y$ multivalued function whose inverse $f(y)=x$ is also multivalued?
For (3): Multivalued functions are just relations. An algebra based on relations underlies most database systems today.