Michael Lugo
Reputation
11,066
Top tag
Next privilege 15,000 Rep.
Protect questions
 Jul 7 comment The “beach problem”: does anyone know it? or know how to solve it? I don't read German. Is this basically just a brute-force search or is there something more going on? Jul 6 comment How accurately can huge factorials be calculated? The last term in your sum should be $\log \sqrt{n}$ or ${1 \over 2} \log n$. Jul 5 comment What technique would be suitable to solve this: $\int \sin ^{5}\left( x^{2}\right) \left( x\cos \left(x^{2}\right)\right)\mathrm{d}x$ $x^2$ is "ugly". So we call it $u$, the first letter of "ugly". Jul 5 comment Recursion equations: $T(n)=T(\frac{n}{4})+T(\frac{3}{4}n)+1$ The next natural question is to ask for $\lim_{n \to \infty} T(n)/n$. Since $n/4$ and $3n/4$ are not necessarily integers this depends on how you round; the correct way to do this presumably comes from the algorithm that gives rise to this recurrence. But it appears that for any combination of rounding up, rounding down, and rounding to the nearest integer that limit doesn't exist; rather $T(n)/n$ oscillates between two constants $a$ and $b$. Jul 5 comment Five Fridays and Sundays on October It actually showed up in my e-mail. (From someone who seems to think I'm someone they know and has the wrong e-mail address for me. Everyone I know knows better than to send me things like this.) Jul 4 comment Probability metric spaces for which $r \leq R$ implies $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$ The first thing that comes to mind is that if you're working in the $d$-dimensional unit cube, then (except near the edges) $Pr(B(x,r))/Pr(B(x,R)) = (r/R)^d$. So in some sense you want spaces which are of dimension one or less. Jul 2 comment Shortest sequence containing all permutations 9 is correct for $s^\prime(3)$ where $s^\prime$ is the substring version of the problem. It's obvious any string of this type has to be at least of length 8, so that it has enough substrings. But if it's of length 8 then every substring has to be different, so you start constructing the string 12312ohcrap. 123121321, of length 9, works. Jul 1 comment sum of independent random variables In general you want to find the density of $Z$ and invert it. How to do this efficiently is going to depend very strongly on the specifics of the problem; what specifically are you interested in? Jun 30 comment Equality of outcomes in two Poisson events I thought of this but got tripped up on what the bounds on the integral should be; that's why I made the second transformation, from Poisson to binomial. Jun 30 comment Equality of outcomes in two Poisson events If it helps, assume throughout that $\lambda$ is an integer. Since we're worrying about the limit as $\lambda \to \infty$ it seems quite likely that we can just find the answer for integer $\lambda$ and wave our hands about the solution being smooth. That being said, I don't intend this to be a fully rigorous proof. Jun 30 answered Equality of outcomes in two Poisson events Jun 29 comment Enlightening misunderstandings of test questions Joel, I knew a guy in high school who, when asked on a physics test to "name the force acting on this object", named it "Bob". Jun 23 comment Formula for occurrence of leap years in the Jewish calendar I haven't thought about it, but I would imagine that for the sets that are likely to occur in calendar applications -- that is, where the leap years are roughly evenly spaced -- you're more likely to get "lucky" than you would than for sets chosen uniformly at random. Jun 21 awarded Nice Question Jun 21 awarded Scholar Jun 21 accepted What's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$? Jun 20 comment Is the Iterated Continued fraction from Convergent​s for Pi/2 exactly 3/2? So I don't know much about this, but is it possible that this process always converges to $3/2$, or at least does so for starting points in a fairly wide range? That seems more likely than that this is some special property of $\pi/2$. Jun 20 comment What's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$? Well, there is the same mysterious constant of 12. I may give it a shot. Jun 20 asked What's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$? Jun 20 comment Three consecutive sums of two squares The sequence of numbers n such that n, n+1, n+2 are all sums of two squares is given at oeis.org/A082982 . (As you'd expect, they're all multiples of 8.) This sequence doesn't look to me like there's a nice formula. Of course this doesn't prove there isn't one.