Peter Taylor
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 Sep 10 comment Find an expression for the $n$-th derivative of $f(x)=e^{x^2}$ More generally, $$\frac{d^n}{dx^n}e^{x^p} = \left(\sum_{k \in \mathbb{Z}}a_{n,k;p}x^{kp-n} \right) e^{x^p}$$ where the coefficients $a_{n,k;p}$ form the exponential Riordan array $A_p = [1, (1+x)^p - 1]$. This can be proved using some of the techniques from math.stackexchange.com/questions/18284 and is given as an exercise (although not in those terms) in Comtet's Advanced Combinatorics. Aug 30 comment Calculate $\pi$ to an accuracy of 5 decimal places? -1 Incomplete. You need to show that the terms smaller than 1E-5 don't sum to more than 1E-5. In fact, actually it's even more complicated than that, because the answer should be correctly rounded. You also potentially have loss of significance due to adding from the largest elements in the series rather than from the smallest. Aug 11 comment What does asymptotically optimal mean for an algorithm? @Geek, depends on what you average over. And worst case is what's relevant unless otherwise specified. Jul 5 comment Quantization of angular momentum: is Dirac's proof wrong? With step 2 as written (and BTW it would be easier to follow without overloading $j$ - it was the first element of the ket earlier, and now it's the second), isn't the existence of the eigenket $\lvert \beta, j\rangle$ true by definition of $j$ as "the maximum allowable value of $m$ for fixed $\beta$"? Jul 4 comment Seeking formula for a custom game mechanic Ross' answer gives the exact solution if you let the time step tend to 0. The advantage of using an exact solution is that you can vary your time step without affecting the results (modulo rounding errors on the order of a couple of ULP). However, if you want to complicate the equations by going to solid bodies, handling collisions, etc. then incremental calculation with fixed time step is likely to be more straightforward. There's a separate gamedev.stackexchange.com where you'll find plenty of questions and answers about building physics engines. Jun 29 comment Does this ratio converge to the Golden Ratio? It may be a useful observation (provable by induction) that each column becomes constant $(k-1)!$ after a triangular-number $k(k-1)/2$ of smaller values. Jun 27 comment Getting the max/min longitude/latitude within a distance from a point @sandis, the best approach probably depends on your restrictions on d and plausible latitudes, but it's really a data structures question rather than a mathematical one. A relatively simple approach would be to use oct-trees; for more sophisticated approaches look at papers from the GIS community or search for terms like k-nearest. Jun 14 comment Bat and ball calculations Perhaps a more pertinent question is "Why would you work it out in MS Excel?" Jun 7 comment Find the remainder when $12!^{14!} +1$ is divided by $13$ Note that the original problem can also be solved by Fermat's little theorem: $12!^{12} \equiv 1 \pmod{13}$. Jun 6 comment Finding a grammar for a formal language Do you realise that this language is just [ab]*? May 21 comment Change of coordinate system on a sphere $L \times O$ is intended to be a cross product in the Euclidean embedding, i.e. one of the two poles of which $LO$ is (part of) the polar. The descriptions of $LP'$ and $PP'$ are intended to clarify what you mean by local lat and long. $LO$ and $NP$ cannot possibly be parallel, because on the surface of a sphere two lines are parallel only if they are collinear, and $N$ is not on $LO$ by construction. May 21 comment Why do the Fibonacci numbers recycle these formulas? Putting the last row into the OEIS search turns up some sequences with a few references. They would be a good place to start looking for an answer. May 19 comment Change of coordinate system on a sphere To clarify what you know: let $L$ be the local origin, $O$ be the true origin; $N$ be the "North Pole" of the local coordinate system (i.e. $L\times O$ normalised); let $P'$ be the intersection of $LO$ with $NP$; you know the lat and long of $L$ in the global system, the (signed) distance $LP'$= local longitude, and the (signed) distance $PP'$= local latitude? May 15 comment Complex roots of $z^3 + \bar{z} = 0$ You only find 3 solutions, but you haven't explained why you're discarding two (x, y) pairs. May 15 comment Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $∞$? I don't get it. What's "English" about a temperature scale invented by a German who was born in Poland and lived in the Netherlands? May 14 comment Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $∞$? Celcius to English? What English temperature scales are there? Kelvin would be the closest (although he was Irish by birth and arguably Scottish by adopted), I suppose, but I don't think they teach absolute temperature at that age. May 8 comment How to merge two poly Bézier curves? If they were simple polygons, would you be able to do it? Can you find the intersection of two Bézier curves? Can you generate a Bézier curve which is a continuous subset of another Bézier curve? May 2 comment In how many ways can a number be expressed as a sum of consecutive numbers? An equivalent statement to that main result is given in the comments on A001227, although without reference. May 1 comment Graph theory - A type of undirected simple finite graphs What does the notation $\{a; b\}$ mean? Is that a two-element set, or an uninterrupted range of integers? Apr 29 comment Intuition and derivation of the geometric mean And as a point of interest, there are at least two other means which are $f^{-1}(\textrm{arithmetic mean}(f(x_i)))$, namely the harmonic mean ($f(x) = x^{-1}$) and the RMS ($f(x) = x^2$).