I am trying to collect up my notes from the last year and a half, and I have written the words "This is stupid obvious, derp derp derp." on the page where this is claimed.
Yup. I've both tutored for and proctored that thing a few times.
So, I use a TeX package called "todonotes" to annotate my work. I gave my advisor a copy of some notes, but forgot to first disable this package before recompiling the document. He got back to me a few days later and says "The results are nice. They are not 'horseshit'. Also, what do you mean by 'Stuff what needs to be did'?"
@XanderHenderson I'll be taking it soon, and I was wondering if anyone had any tips; I'm pretty versed in number theory, and geometry, but Questions 16-20 still trip me up
@MatheinBoulomenos I got around to writing down a solution for the problem - finding primes $p, q$ with $p^2 = 1\pmod{q}$ and $q^2 = -1\pmod{p}$ - I was stuck with (I did read your solution too). It's along the lines of, say $p = \pm 1 + kq$. Then we're asking for situations when $(q^2 + 1)/(\pm 1 + kq)$ is an integer, say $m$. Then we get a quadratic equation with determinant $\Delta = m^2k^2 - 4(1 \pm m)$. $k^2 \Delta$ becomes, by completing the square, $(mk^2 \pm 2) - 4(1 + k^2)$
Now it's a case by case analysis to see for which $k$ and $m$ it happens that $k^2 \Delta \leq (mk^2 \pm 2 - 1)^2$ (if that doesn't happen $\Delta$ is not a perfect square)
I have some physics to do, and some floor function problems to do. Should I get physically exhausted first or should I be floored by competition math first?
Eric: To an extent, they do. You have to have an objective way of judging since high school (or US college) curricula are different and standards, letters of recommendation, etc., are totally non-uniform. We would get grad school recs saying "best student in my career" from a podunk school, and generally that means a big nothing.
From an MIT prof, of course, it means a lot.
So the GRE is a way of normalizing things to be fair, ultimately.
DogAteMy: For one thing, we don't admit people to specific majors in the US (sometimes you're admitted to an Engineering school rather than the university as a whole).
I got a 790 (out of 800) on the math section of the SAT, which pisses me off because I took the test unofficially when I was 8 or something and I got an 800
Place a young child in a room with a marshmallow. Tell them you're gonna leave for 20 minutes. Tell them if they don't eat it while you're gone, they can have two marshmallows.
When I was in grad school, a friend of old friends of my family's wanted to hire me to tutor their high school son. I pointed out that they'd have to pay a premium price for a 4th year PhD student compared to a good high school student. They paid. The kid was actually smart, just bored in school. But I had no problem engaging him and he did do better just because he was more interested.
There's this psychology experiment where you give someone a shock button, ask them to press it (they get an electric shock), and ask them if they think they would ever press it again ("No"). And then you leave them in an empty room with the button for five minutes
Reminds me of an Alzheimers test I took the other night. And twice I touched the wrong button as an automatic response starting off the new segment of the test, DogAteMy. It was very disconcerting.
@TedShifrin As in, like, one button said "Continue to the next section" and the other said, I dunno, "Armageddon" or something, and you hit the wrong one?
No, they had different sections to the test and they gave you practice at whatever. Two of the five times I immediately hit the wrong key on the first question. My brain knew it wasn't right, but I just did it.
hi if i have a function f(x,y) = (y,x) i get a matrix which seems to be none of the linear transformation matrices that i know of i tried reflection and the rotation but the negatives don't match so i am a bit confused on how to state what the correct linear transformation is here =/
If the two foci of an oblique ellipse are given and its eccentricity is also given, how do i find the length of major axis? Can I still assume the distance between foci=2ae?
Let $B$ be a set of divisors of $mn$ arranged in ascending order
Thus given a set of positive integers $I$ with cardinality $k$ where $k$ is the number of divisors of $mn$, $B$ is a well ordered set: $\{b_1,b_2,...,b_k\}$
Define the half tranpose operator as follows:
If $k$ is odd $$A^\frac{T}{2} : \Bbb{F}^{mn} \to \Bbb{F}^{b_{\frac{k}{2}}b_{\frac{k+1}{2}}}$$
actually, I need to work out in more detail how to write the rules algebraically. It is very easy to illustrate by example, because the dimension of the matrix after a half transpose will be defined to be some intermediate factor between nxm and mxn
and in the special case of a square matrix, the dimensions is unchanged
In fact, I have not work out in detail how the map actually looks like, because it seemed to be quite nonlinear. We can take the 2x2 case as an example
The goal is basically to solve the following system of functional equations:
Here, we can immediately found out for the diagonal entries, $f$ is idempotent on the entries indicated while the off diagonal entries have a period of 4
Also since (A^T)^T=A, we need $f$ to have an inverse
And the only situation where f(x)=f^{-1}(x) is when x is a fixed point of f. Thus we immediately knew that for square matrices, the diagonal is left unchanged by the half transpose operator
As for the off diagonal elements:
We also found that if f(b)=x_1 and f^{-1}(b)=x_2, then we have a contradiction since we then have f(x_1)=f(x_2). To avoid the contradiction, we must conclude that x_1=x_2 hence f(b)=f^{-1}(b) which based on the previous argument, we then conclude b is a fixed point of f
As a result, the half transpose cannot be defined simply as an elementwise map of some single variable function f as otherwise, f^2(x)=x where x in {a,b,c,d}, contradict our requirement of being a half transpose.
To solve this, instead, the half transpose is defined to be a three variable function f(a,i,j) applied elementwise to the matrix
That is, given any a_ij, f^2(a,i,j)=a_{ji}
Now we often like to have maps that are continuous. Given a pair of a_ij and a_ji, there are many ways to interpolate between them. The simplest way is a linear interpolation, thus we can define:
$$\forall a_ij \in A : f(a_{ij}) = \frac{a_{ij}+a_{ji}}{2}$$
It is now obvious that the diagonal remains unchanged after application of $f$. Therefore, one natural way to define the half transpose for square matrices nxn is as follows:
actually, I think I need something more to flag that the matrix is obtained from a half transposition else the operation is not invertible. Will figure this out later
Background: The above bizzare idea is born out of thinking about the geometric meaning of transpose and wonder what happens if the transpose is halfway done
Yeah I am tidying up all my internet footprints since last year. It is a slow process since SE don't have a search function to search for only messages posted by a certian user, thus I have to crawl through all 3 years since 2015
How can I find the limit: $\lim_{n \to \infty}{\left( \frac{3n}{3n+1}\right )^{6n}}$? I did some calculation and got $\lim_{n \to \infty}{\left( \frac{3n}{3n+1}\right )^{6n}}=1$ as a result. But wolfram alpha tells me it is $\lim_{n \to \infty}{\left( \frac{3n}{3n+1}\right )^{6n}} = \frac{1}{e^2}$. Can someone give me a hint how to achieve that correct result?
