Based on Raskolnikov's answer here, one can build an implicit Cartesian equation for a $2p \times 2q$ rectangle:
$$\left(\frac{x}{p}\right)^2+\left(\frac{y}{q}\right)^2=\sec\left(\arctan\left(\frac{x}{p},\frac{y}{q}\right)-\frac{\pi}{2}\left\lfloor\frac2{\pi}\arctan\left(\frac{x}{p},\frac{y}{q}\...
what is a 2p x 2q rectangle? ^^
Another generation question related to this chat room...anything special we've to do to render mathjax stuff here... like for e.g. $2p \times 2q$
@Oxinabox My personal opinion is that it is best to start with lowest possible bounty. Since if there is now answer and you want to put a bounty on the same question again, you have to increase the amount of the points. (You can find pointers to some relevant info on bounties here and some of the questions tagged bounty might be of interest for you.)
Hi @DanielFischer!!! Could we prove by induction the correctness of Quicksort, that is the following?
Quicksort(A,p,r)
if (p>=r) return;
q=Partition(A,p,r);
Quicksort(A,p,q-1);
Quicksort(A,q+1,r);
Last night I came up with a really good combinatorial proof that ${n+a-1 \choose a-1} = \sum _{k=0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}$. It's one of those ones that just really makers you feel smart when you get it.
Makes*
Also, here's hoping that rendered properly since I typed it on my phone
@DanielFischer @robjohn When we have the recurrence relation: $T(n) \leq T \left( \frac{9n}{10}\right)+T\left( \frac{n}{10}\right)+cn$, does the recursion ends when $\frac{9n}{10^i}=1$ or when $\frac{n}{10^i}=1$ ?
@Sawarnik I haven't read this book, but it is mentioned in this question, togheter with a lot of others number theory book that you may find interesting math.stackexchange.com/questions/329/…
@Sawarnik There are other books a lot more upvoted, but it does seem to be good. I'd wait for the opinion of someone who has actually read it though
@evinda $\log_{1/2}(1/2)=1$ and $\log_2(1/2)=-1$ so it holds also for $a<1<b$, depending on $n$. Actually I'm starting to doubt of what i wrote before!
Today,I found a interesting problem:
if
$$F(x,y,z)=\begin{vmatrix}
\cos{x}&\sin{x}&f(x)\\
\cos{y}&\sin{y}&f(y)\\
\cos{z}&\sin{z}&f(z)
\end{vmatrix}\ge 0$$
for all $x,y,z$ of an open interval $I$ for which $x<y<z<x+\pi$.
show that:
$f(x)$ is continuous in $I$ and has finite left...
Today,I found a interesting problem:
if
$$\begin{vmatrix}
\cos{x}&\sin{x}&f(x)\\
\cos{y}&\sin{y}&f(y)\\
\cos{z}&\sin{z}&f(z)
\end{vmatrix}\ge 0$$
for all $x,y,z$ of an open interval $I$ for which $x<y<z<x+\pi$.
show that:
$f(x)$ is continuous in $I$ and has finite left-handed a...
@math110 I had written the answer a while ago, but since so many others had gotten close to the answer in comments, I waited to see if any of them would answer. Since none did, I posted mine.
@Hippalectryon Do you realize how may professors will initially say it's not possible that someone without any math background publish such a book? Some will say it's something wrong there and they will try to explain that. (this message to remain for the future)
@Hippalectryon Maybe it's not the proper word in English for what I meant.
Something is rotten in the state of Denmark! I mean for many this will not make any sense. How on earth someone without any math background publishes such a book? lol :-))))) (it will be fun!)
@DanielFischer Could I ask you something about Countingsort? When we have for example the array arr[] = { 10, 6, 8, 2, 3 } we get an array count={1,1,2,3,3,3,4,4,5,5}. What do the numbers of the array count represent and how can we use them?
Hey @Huy Could I ask you something? When does it hold that $\log_a n>\log_b n$, knowing that $a<b$ ?
@evinda count[d] tells you how many numbers with last digit $\leqslant d$ are in the original array. By that, you know where to place every number in the sorted (according to last digit, ignoring anything before that) array.
@DanielFischer Maybe I know, I think $N(A)=N(AA^*)$ and $\overline{R(A^* A)}=\overline{R(A^*)}$, something like that, and than to use $AA^*=A^*A$ because $A$ is normal.
@r9m I think there's pretty obviously missing context there. "How to prove (equality)" is not a good question. (We don't know what techniques he is familiar with, whether he wants to use imaginary numbers or not, etc. There are many details he could improve the question with.)
Today,I found a interesting problem:
if
$$F(x,y,z)=\begin{vmatrix}
\cos{x}&\sin{x}&f(x)\\
\cos{y}&\sin{y}&f(y)\\
\cos{z}&\sin{z}&f(z)
\end{vmatrix}\ge 0$$
for all $x,y,z$ of an open interval $I$ for which $x<y<z<x+\pi$.
show that:
$f(x)$ is continuous in $I$ and has finite left...
