/sci/ - Science & Math

>>15082258
No

Problem 3 looks pretty easy. All four of them.
Note x^4 > 6x+3 at x=2 and 4x^3 > 6 for x>=2 so x^4 > 6x+3 holds for all x>=2.
If |z|>2 then |z^4| = |z|^4 > 6|z|+3 = |6z|+3 >= |6z-3| so z^4 <> 6z-3.

There are no double roots. P(0)=3, P(1)=-2, and P(2)=7 so there are two real roots. The other roots have to be complex since from P'(z)=4z^3-6 we get P is strictly decreasing for z<cuberoot(3/2) and strictly increasing for z>cuberoot(3/2). They must be a conjugate pair.

>>15082258
Qualifying for Harvard involves having daddy pay a bribe of about half a million dollars according to the Harvard admission bribery scandal court documents, there are no intellectual requirements for matriculation other than that.

Also 1 looks pretty simple as well. You can make [0,1) out of a countably infinite number of copies of X, each shifted by a rational number in [0,1) and wrapped to be in [0,1). If the shifted and wrapped copies all have the same measure as the original, by countable additivity, [0,1) would have to have measure 0 or infinity. It's a pretty common example of a non-measurable set.

>>15082258
1. [math] \mathbb{R} = \cup_{p \in \mathbb{Q}} (X + p) [/math]. Apply the Lebesgue measure to both sides and use the fact that union is disjoint along with countable additivity of the measure.

2.
a. trivial computations
[math] \,da = \,dx \wedge \,dy [/math]
[math] a \wedge \,da = \,dx \wedge \,dy \wedge \,dz [/math]
b. a is nowhere vanishing so its kernel has rank 2 everywhere. Apply the implicit function theorem to get the smooth vector bundle structure.
c. You can show that [math] a([u, v]) = -da(u, v) [/math] everywhere. By the computation in part a, this is equal to zero at some point iff [math] (u_x, u_y) [/math] and [math] (v_x, v_y) [/math] are linearly depenedent at the point (the subscripts denote the components of the vectors here), but then at that point u and v would also have to be linearly dependent (since vectors in the kernel of a satisfy [math] u_z + \frac{1}{2}(xu_y - yu_x) = 0 [/math]), which is given to not be the case, so we're done.

3. [math] |-6z + 3| \leq 15 < 16 = |z|^4 [/math] when [math] |z| = 2 [/math]. Apply Rouche's theorem.

Trans rights are human rights btw.

>>15082258
I'd have to study for Problem 1. Don't even know what Lebesque measure is any more

Problem 2 is just basic algebra. Very easy.

[math] da= dx\wedge dy [/math]
[math] d\wedge da = dx\wedge dy \wedge dz [/math]
If
[math] X = u_1 \partial_x + u_2 \partial_Y + u_3\partial_z [/math] is a section, then [math] a(X)=0 [/math] implies [math] u_3 = \frac{1}{2} (yu_1 -x u_2) [/math] so X is in the span of [math] X_1 = \partial_x + \frac{1}{2}y\partial_z , \quad X_2 = \partial_y - \frac{1}{2}x\partial_z [/math]. The sub-bundle portion than follows from some standard theory from the sections calculated. And finally [math] da(X_1,X_2) = 1 [/math] with [math] dx(X_1)=1,\quad dx(X_2)=0,\quad dy(X_1)=0,\quad dy(X_2)=0. [/math]

This is two dimensional. Finally, for vector fields [math] u=u_1X_1+u_2X_2,\quad v=v_1X_1 + v_2 X_2 [/math]
[math] a([u,v]) = -da(u,v) = -u_1v_2 + u_2v_1 \not=0 [/math] dude to independence of [math] u,v [/math] so that [math] [u,v] [/math] is not in the sub-bundle.

>da = dx v dy
Wrong. A one form isn't equal to a two form, retard.

>>15082374
>It's a pretty common example of a non-measurable set.
Yeah this exam isn’t designed to weed anyone out, it’s just to make sure you took a certain set of classes

>>15082386
Lmao

>>15082258
Pretty sure people within the 80-90 IQ range could solve most of these. Below that I'm not sure. I wager it would come down to time constraints.
In any case one can easily surmise that IQ is superficial as most of this board isn't able to solve these problems despite being 140+ white e eye LARPing gigachads.

>>15082381
Nice clean solution for 2(c)

Is x,y,z are the euclidean coordinates then dx v dy = dz and thus da = dz. Now you have a one form equal to a one form. You're welcome.

>>15082457
New math just dropped

>>15082381
>Trans rights are human rights btw.
I concur. An oft misunderstood concept of human rights is that subhumans aren't humans. That's incorrect. Subhumans by definition are humans, and are afforded human rights. Just at a reduced level. Just like children don't deserve all the rights of adult humans, subhumans like trannies do not deserve all the rights humans have. For example, trannies do not deserve the right to vote.

>>15082462
Trannies are not subhuman. They are simply more useless, among humans. Similar to women.

>>15082466
Women are children, which are humans with underdeveloped brains. Trannies however have broken brains and are subhuman just like all mentally ill subhumans.

>>15082453
Thanks

>>15082462
>>15082466
>>15082468
>seething /pol/tards who can't into math

>>15082471
>makes political statement
>seethes at people making opposite political statements
Ask your doctor to alter your estrogen dose. It's too high right now. Basically worse than PMS

>>15082471

But I'm >>15082453 and >>15082466. We've (at Ieast I've) solved it, same as you.

And I agree, trans rights are humans rights. However, I'm rather wary of the overall push for the feminization of men which characterizes today's blue politics. But anyway that's just politics and has little to do with a discussion of math problem-solving.

>>15082476
Sorry but your opinions are worthless since you can't solve the qualifying exam problems posted by op.

>>15082478
I see, but I made that statement specifically to trigger poltards. Most discussions in threads like these won't be about the problems but instead about the various copes like >>15082340
>>15082390 anyway, so adding politics into the mix won't change much.

>>15082471
math is gay and chat GPT can do it. neck yourself tranny

This exam seems very simple, but conceptually, anyone who can't solve it, is just because they haven't been taught the formatting. It breaks down into basic mathematics, and concepts that any human should be able to comprehend, at average IQ level.

>>15082258
I fucking hate that I can't understand the questions. I need to learn more shit but literally I always find questions with words/definitions I don't know.
Fucking codemonkey workload takes too much of my time, fuck uni.

>>15082515
Go copy-paste the problems into ChatGPT then. It'll produce gibberish (not like you could tell, AItard).

>>15082869
All of math ultimately boils down to abstracting concepts until they're obvious in some way or another, you fucking retard. Grothendieck is famous for this approach and one of the most prolific mathematicians because of it. In fact, the vast majority of mathematicians prefer short, elegant, and general proofs over long and contrived ones, cause only an idiot admires complexity, and you're walking proof of that.

I'm going to be honest and say that I dont know where to begin. Can anyone tell me what I need to learn to know how to answer these questions so that I can try them?

>>15083177
Which problem? For differential geometry, the best place to atart may be Lee

>>15082258
Who's Lebesgue

>>15082258
Problem 1:
Suppose we take the quotient space [0,1] with the endpoints identified to make a circle then translation by rationals gives the entire interval, which has measure 1.

If the measure of X were k > 0 then this would imply that the measure of [0,1] is infinity.

If the measure of X were k = 0 then this would imply that the measure of [0,1] is 0;

Either way we get a contradiction.

Problem 3:
We use Rouche's theorem. The answer is all four of them.