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11h
comment Subgroup Decision Problem
Adam Hughes's answer shows that this claim is false without further assumptions. Take $x$ to be a transposition in $S_3$ with $p=2$, $q=3$. Then $x^q$ has order $2$ but only lies in one of the three possible choices for $G_2$.
12h
comment Euler's Formula, Square root
You are solving for $\lambda$ with a particular choice of $x$. What I meant is that the identity for general $x$ implies that $\lambda^2 = -1$.
13h
comment Euler's Formula, Square root
To be fair, it is true that if $\lambda$ is any number satisfying $e^{\lambda x} = \cos x + \lambda \sin x$, then $\lambda^2 = -1$ (by comparing $x=\pi/2$ with $x=\pi$). So in that sense one can derive $i^2 = -1$ without first taking it as a definition.
1d
comment Is there an easier way to find $\frac{d^9}{dx^9}(x^8\ln x)$ than using the product rule repeatedly?
How is this easier?
1d
comment Rank of the product of 3 matrices
Here's a hint to get you started: if $AB = 0$ then the columns of $B$ all lie in the nullspace of $A$. Is it possible for $\text{rank}(A)$ and $\text{rank}(B)$ to both be greater than $n/2$? Is it possible for $\text{rank}(AB)$ to be greater than $\text{rank}(A)$?
1d
comment Rank of the product of 3 matrices
Of course the answer can't be $n$, that would force all three matrices to be invertible.
1d
reviewed Close Probability that a randomly marked multiple choice test is all correct/incorrect
1d
reviewed Leave Open looking at the alphabet ,the letters are numbered 1-26 ,
1d
comment Sufficient condition for $n$
No attempt? Either you can try it for $n=2$ and $n=3$, or you don't understand what the question is asking. In both cases, there is nothing stopping you from adding more detail to your question.
2d
comment Irrational number “test”?
@pbs Absolutely not. Compare your first iff against $a=1$, $u_n = n$, $v_n = n+1$. The $\Leftarrow$ is very much false.
Aug
18
awarded  Nice Answer
Aug
18
comment Simulate a 7-sided die with a 6-sided die
+1 This is quite nice.
Aug
18
comment Simulate a 7-sided die with a 6-sided die
@neverlastn The idea comes from a compression technique called arithmetic coding wiki, which is sort of a continuous extension of Huffman coding. The one small twist is that I actually rescale the interval $[a, a+width)$ to lie in $[0,7)$ rather than $[0,1)$. This allows the next base-$7$ digit to be extracted by taking the floor. So the (int)(a+width) != (int)a test is just checking to see if $a$ and $a+width$ have the same floor.
Aug
18
comment A Method For Calculating Large Exponents Quickly
Indeed, this analysis shows that it is never correct unless $a=0$, and even that case blows up the $\log a$ term.
Aug
18
comment Is there a way to simulate any $n$-sided die using a fixed set of die types for all $n$?
@obinna No, of course it cannot be done in fixed number of throws, unless $n$ happens to divide into some fixed power of $k$ (replace $k$ by "product of $k$-values" if there are multiple die types).
Aug
18
comment Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite
How familiar are you with the proof that the set of fractional parts $\{\{n\alpha\}\}_{n \in \mathbb N}$ is dense in the unit interval?
Aug
17
answered Simulate a 7-sided die with a 6-sided die
Aug
17
comment Simulate a 7-sided die with a 6-sided die
A refinement of this will give an entropically optimum solution, you just have to retain the state instead of starting from scratch for each generated roll.
Aug
17
comment Solving $n! + 3n = k^2$
The second case feels about as intractable as Brocard's problem (not withstanding Mochizuki's proposed proof of ABC).
Aug
17
comment Simulate a 7-sided die with a 6-sided die
The $\log 6/\log 7$ number comes from comparing the number of bits produced by $m$ 6-sided dice rolls with the number of bits produced by $n$ 7-sided dice rolls.