Guys, am I allowed to ask homework-like questions in Math Exchange?

I"m coursing Calculus II at my university, and some stuff I just can't find how to solve anywhere

you are certainly allowed to... though you will much likely get a better response if you ask about the parts that you dont understand then state the whole problem

i.e. if you show you at least attempted then people generally treat the question well

also feel free to ask the chat if its something i know i'm willing to help

Alright. I know the subtitle states "Just ask, don't ask to ask", but I wouldn't be comfortable with people saying "We're not here to do your homework" :P

I'd be wiling to help as well. As far as I can tell the chat is much kinder about homework questions--it's easier to walk someone through the problem in this format.

yeah i get you. its really a meta joke. if you can ask to ask, then you can ask to ask to ask and so on.

and yeah much easier

Sure, my sincere thanks to you guys. I'll start working things out with my group, and send any difficulties here first, then on Math Stack Exchange

sounds cool, best of luck to your group

@fargle i wonder if its cyclic...

kek

i think we can call those beings that are capable of doing high math but not capable of any astechic appreciation non geniuses. and call artist geniuses. looking at the right

those truly devoid of all pleasure are not geniuses in my mind but tragedy

Hi @Fargle @shai ...

@TedShifrin Hello!

@Kerooker: Plenty of people seem to get more than a little help on their homework on MSE. I would ask you to feel free to ask questions, but don't just post a homework problem. Post what you've tried, where you got stuck, explain what concept you just don't get, and then don't just copy the answers you get. Really try to learn!

I personally try to respond with hints and incomplete solutions, but usually get sabotaged by someone who wants to get rep points by posting an exhaustive solution.

Oh, @Danu is still amongst the living ... :)

@TedShifrin Our problem atm is we get the concept of stuff the professor says, we understand the applications and what everything means

But when we try to complete the exercises.... Almost nothing comes to it

Are there not some good examples in your textbook or lectures?

There are examples, but when it comes to exercise lists (the ones we use to prepare for exams), everything is 10 times harder than the examples, and simply nothing comes out...

You should of course be doing exercises before it comes to the time to prepare for exams, @Kerooker :) I've had students tell me similar things over my years of teaching, and then numerous times I can point to exact replicas of the exercises in their notes or textbook. But you may be right in some cases. I'm curious to see examples ... :)

hey all. Is it okay if I start a philosophical discussion here? Maths is the only thing that can save us :P

@TedShifrin what is projective geometry about?

@Danu: I wouldn't do transition functions, although that will work. I would rather have you give the maps and deduce it's a SES, as you would just with vector spaces. That will suffice.

Calculus is not for me :P Thankfully Calculus II is my last subject to calculus

I study Information Systems at University of São Paulo (Brazil) if you're wondering

@towc We have them practically daily.

@TedShifrin the exams are next week, and we've successfully made all the exercises from the textbook. We're now starting to make the lists (3 lists, 10 exs each, they're old tests), and they're really hard =p

Oh, hi, @meow. So what's geometry about in general? In general terms, it's about recognizing when there is a "motion" that carries one object to another. For Euclidean geometry, those have to be isometries (distance-preserving maps). For projective geometry, they are projective motions. Two objects will be "the same" if you can project them somehow onto a different viewing plane and they look the same. That's vague, of course.

@Meow: If you email me, I will send you a chapter from my algebra book on different geometries (starting with affine geometry, but mostly projective).

what's humanity's objective?

it can't be just to expand the gene pool anymore, otherwise we'd have gene factories

@Kerooker: Do you have an example of one of those exercises that you do not have any idea how to do?

it can't be to ensure the safety of everyone, because we have daily examples against that

maybe it is to prove RH

it can't be to bring us happiness, because the whole of society would have gone towards that, as there are stable systems in which everyone is happier than the current happiest man alive

@towc Need we have one?

we must have one

we must be going towards an objective

even if not voluntarily

On what grounds do you believe that?

there's a future that we're getting closer to

therefore we're heading towards that

and the whole of humanity has allowed that, therefore that's humanity's aim

societies are stabilizing towards something, by doing actions to get us in a certain direction

but we don't seem to know what direction it's taking

@Danu: The interesting part is why you don't need a splitting of the original SES to define the map $L\otimes \Lambda^{p-1}F\to\Lambda^p E$.

similarly to numerical methods for the zeros of functions whose formula you don't know

That strikes me as fallacious. An aim is generally taken to mean an intent, same with objective. Just because the future is heading in some direction doesn't mean it's done with any intent. To use an example, the colonists of America who fled religious persecution did not aim from the outset to create the United States, just because that's what happened.

@Meow: The point is that (until one gets off the deep end) algebraic geometry is about algebraic varieties and the interesting ones are sitting inside projective space.

@Fargle: It looks like we need to do that again. Not just religious persecution.

@Fargle the individuals didn't. Humanity was though

the interaction between the components of humanity didn't decide that

but neither does a cell in your thumb directly decide whether it should hold onto the cliff any longer, although it may know that it will damage itself

You're essentially asking whether humanity itself--our society at large--is capable of intent--a sort of collective consciousness. My answer to that is going to be a resounding "no".

@TedShifrin Mars seems very attractive.

Maybe only a little warmer than the Arctic now.

well, is the composition of our neural cells capable of intent?

does any of the cells decide for the whole system?

Yes, but that is because there is direct chemical and electrical communication between them.

and so is between humans

it's not a direct relation, but it follows very similar complexity rules and structures

Really? I can think of no time when some grandiose notion struck me to do something for the benefit of Joe Schmo in Croatia.

Discompassionate as that may make me.

there's a delay between our communications, sometimes the communication comes off wrong, we can't really communicate with every other human at once unless there's a greater change in the whole system

Hi @Alessandro

@TedShifrin and what knowledge would reading that chapter require?

Your question of "does humanity have an objective" also assumes that anything capable of intent must have an intent.

@Fargle neither does a neural cell optimized for language parsing care much about the equilibrium cell somewhere else in the brain

although a lot of cells in that moment may be being used to help out with equilibrium

Mostly some linear algebra, @meow, and a little bit of what a group action is. Of course you need to know what equivalence relations are.

I would caution you to not mistake an analogy for an argument.

@Fargle does an amoeba have an intent? Sure

Just because we can put humanity in analogy to individual cells in a human's body doesn't make the connection between the two concepts airtight.

is it the ball's intent to fall of a cliff? In the end, it could well be

Hi @Ted!!!

Long time no talk

where do you draw the line between the amoeba and the ball?

I left you a bunch of messages, @Danu :)

I saw

I had trouble with this little lemma/exercise.

@Fargle I started with "it's not a direct connection" :P But just like vector spaces, they can have very different meanings, but behave similarly if they have similar properties

It's an important one that shows up all over the place.

I realized that transition functions are friggin terrible for any exterior power that's not top power

No, just do the vector space argument.

@towc I don't claim to know where to draw the line. I guess my next line of questioning would be: if we cannot communicate with this larger body of humanity-itself, how could we ever determine the objective of it?

@TedShifrin So I can write down a map for vector spaces, but why does it work for non-split things?

The map is just something like

No, don't split. Don't do anything unnatural.

@Fargle it's worth trying. That's also why I said that maths is possibly the only thing that can solve this ;)

To say that a ball has intent is, in my view, to broaden the definition of "intent" to such an extent as to make it useless.

Hi @Ted

@Fargle useless because appliable to anything. Therefore humanity as well

$\bigwedge^p (E\oplus F)=\bigoplus_{k+l=p} \bigwedge^k E \otimes \bigwedge^l F$

No direct sums, @Danu.

I thought about the trace thing and convinced myself it actually makes sense

Oh cool, @Alessandro.

At that point, you're just saying that "intent" is the tendency of things to happen, and your question becomes, "What will happen in the future?"

@TedShifrin I dunno what to do :P

@Alessandro: You can see it very concretely.

Given the maps in the original SES, @Danu, write down maps in the new SES, and then check exactness.

@Fargle ideally, but I'm asking about what will happen because of the semantic inherent behaviour of humanity as a composition of individuals

it's not a general "what will happen"

@towc We already have a branch of science devoted to studying this.

@TedShifrin Ah, that's what you meant.

Namely sociology.

I see.

As I said above, the interesting thing is why there's a well-defined map in the first arrow.

I'll try that.

it's not a physics/biology question. more like game theory applied to biology?

@TedShifrin Right, this is what I kept on struggling with

LOL @ "that's what you meant." :P

all ideas I came up with seemed to imply that the thing is split

Nope, nope, nope. :)

what were you saying about linear differential equations yesterday? @Ted

@TedShifrin The power of repetition ;)

You're right--it's a sociological question. "Can we predict what people will do based on how people tend to act?"

@Fargle oh, fair point

At least I repeated without yelling, @Danu :)

Speaking of repetition, my laptop charger has repeatedly stopped working today

I'm super scared.

I should probably order a new one right now.

My laptop is my (academic) life

I just mean to frame it in terms of "intent" is to make many presumptions on the nature of consciousness itself, when the question can be framed in a way that doesn't make mankind conscious in aggregate.

Oh, @Alessandro: Suppose you have the ODE $x'(t)=Ax(t)$, where $A$ is an $n\times n$ matrix. Do $n=2$ for concreteness. Then if $\text{tr}(A)=0$, the flow of this differential equation doesn't change areas of parallelograms.

By the way @TedShifrin I'm now almost done with Huybrechts

About 8 pages or so left :D

yeah, I'm not claiming it's anything new, but I've yet to come up with an answer to this. Most of what I was taught in philosophy classes lead to people coming up with explanations that I could never agree with. But surely most of them are quite a lot more clever than I am, statistically, so I either want to find out where I'm wrong, or where they are

@towc That's indeterminate, really. Philosophy is not a field of hard answers, but of hard questions.

Back in a minute. UPS here.

@Fargle but you see, I think it's wrong to deal only with the individuals

looking up what flow means

@Fargle but surely there's an answer

even if the immediate answer changes over time because of context, that has to be predictable as well

@towc Is there? Even mathematics contains indeterminates.

@Alessandro. Just what happens to points as time passes.

@Fargle but stable known indeterminates

Start with a parallelogram at time $0$ and look where the vertices go at time $t$.

That's great, @Danu. You won't need to speak to me ever again! :P

@towc I don't mean $0/0$, I mean facts which cannot be proven true or false.

I kind of think of the answers that philosophy gives you as the taylor expansion of the real answer, if that makes any sense

Ah, I see, that makes sense @Ted

@TedShifrin Time to start reading papers, more like it.

@Fargle sure, name one and I'll show you how it still makes sense in this context

Kotschick didn't really have a topic for me relating to the paper I picked

So I'm reading more papers now

admittedly I'm not that strong in mathematics, but what I've seen so far points in that direction. Don't get me wrong

That's great, @Danu ...

@towc I'm just saying that to claim "surely there's an answer" is to ignore the nature of our place in the world. Yeah, maybe there is an answer. But human knowledge is fundamentally limited.

@Fargle the answer is that it's indeterminable

if you will

@TedShifrin The papers I'm reading now are about characteristic numbers of manifolds with some assumption on curvature.

@Fargle and your point is that maths is limited as well, so it may not be able to give you the solution?

@TedShifrin sorry for the late answer. The list will be unlocked about now (10 pm for me, 2 min)

Oh, no problem, @Kerooker.

Sounds interesting to me, @Danu. There are lots of big theorems, e.g., about surfaces, along those lines.

I see your point, but I don't think it's enough to stop us from trying to answer this

@towc No, my point is just that even mathematics, the field many would call closest to pure, axiomatic truth, is inherently incomplete. To demand that sociology be solvable is to constrain a field with much more subjectivity to being more well-behaved than mathematics itself.

yeah, that's what I was agreeing with

At the very least, we wouldn't be able to perfectly predict the future, even something as small scale as our future, until we entirely solve physics.

but surely there's a trend to the behaviour of humanity

@TedShifrin The main theorem of the current paper I'm looking into is that the only rational linear combinations of Pontryagin numbers that are bounded on manifolds with nonnegative sectional curvature are multiples of the signature.... I think.

ideally there's a trend to the stock market as well, so there's that...

The proof is via an explicit construction of some bundles over $S^4$, computing the total Chern class etc.

@towc There are trends, yes. But there are also moments that defy those trends.

What does that mean, @Danu? Are we looking only at 4-folds? Bounded means what?

But this sounds quite interesting, if only I knew what it meant :P

@Fargle which I don't see why would they not be in a trend that shows the defiance of the original trend, and so on

hence the taylor expansion comparison

Our picture is incomplete until we have some way to distill all societal interaction, all internal decision-making, and everything else into a solvable physical system.

@towc Almost all functions don't have Taylor expansions.

In a very well-defined sense.

@Fargle but they still have a lagrange error formula. And those are assumptions, that I'm happy to make to a certain degree

@TedShifrin Arbitrary dimension. Bounded means... ehh :P Well here is the paper.

wait, they do, right?

not sure

@towc No, they don't.

oh

well, that analogy fails then

Almost all continuous functions aren't even differentiable.

There is apparently some bound on the signature due to Gromov.

In the same sense that almost all real numbers are irrational.

Apparently one of his Big Deal Papers :P

Interesting, @Danu. I'll look more in a bit. :)

@TedShifrin There's another closely related paper by a Chinese mathematician on the $\hat A$-genus, that I should also look into soon. If you like what you're seeing on this paper I can link you to that too.

either way, you're saying that you strongly don't believe we could find a trend to the "choices" humanity makes, if we only have a small degree of assumptions?

@towc Trends, sure. A hard-and-fast answer, no

what's meant by fast?

By "hard-and-fast" I mean "predicts with 100% accuracy".

Ok, so one of the exercises is as follows: (I'll try to use latex, but never did ;p)

oh heh, never wanted that :P

just kind of want to know what is humanity for

Well, you wanted mankind's objective. That seems to beg for something definitive.

and why we're not all dead and we all lie to each other about it

@towc Because one of many trends in humanity is to act against humanity's best interest if it benefits the individual.

Another exercise that I do think I managed to solve, Ted, was to give a nondegenerate pairing $\bigwedge^k E\times \bigwedge^{\text{rank}E-k}E\to\det E$, by sending e.g. $(e_I,e_J)\mapsto e_I\wedge e_J$ (where $I,J$ are those great multi-indices we discussed at length; $J$ being complementary to $I$ should give something nonzero)

@Fargle that's the individual. Like a cancer. But then either the organism dies, or the cancer is removed, stabilizing the system either way

Sure, @Danu. In fact you can make that $\otimes$ rather than $\times$, methinks.

To bring back the body analogy: the body works because all the cells have no self-interest. They are, for lack of a better term, designed to make the body work.

@TedShifrin Yeah, right?! Huybrechts didn't and it weirded me out :P

@towc But all individuals have that tendency, in however small a measure. That would be akin to saying the entire human body is made up of tumors.

@Fargle except they do. They mostly wouldn't last long outside of ourselves

@Fargle every cell has a chance to become cancer. Possibly bad analogy. More like, any cell is going to take as much oxygen/ATP as it can take. It just happens that the system doesn't allow them to take more than they need

@towc Do you think this is because any cell has "made the decision" to stay, or just because this is the nature of a determinist chemistry?

By the way, you wouldn't happen to be interested in chess, right?

Listen to chat about the purpose of humanity, or look up how multi-indices for differential forms work...

Me, @Danu? Nah. Bridge is my game.

@Semiclassical There is an obvious choice here.

@Semiclassic: Humanity has no purpose.

@Fargle it's because they simply couldn't exist outside of our bodies. Individuals are generated by humanity, just like cells are generated by bodies. If either of the components went outside of its group, it would die

Did somebody say chess? @Danu

so the only ones we know of are the ones who didn't die

@towc Are there not historical examples of hermits?

@Fargle that still falls within society

@Fargle Yeah; you following the world championship?

@Danu No, unfortunately.

in humanity, the "outside of the body" has no spacial meaning

@TedShifrin Sadface. Though I heard bridge is mathematically interesting.

@Fargle :(

I'd tend to say it as: humanity only has a purpose to the extent that we create it for ourselves

@Semiclassical which tends to be...?

Well, probability is involved tangentially, @Danu, in terms of trying to analyze what the likelihood is that a strategy for playing the cards will or will not work.

@towc Self-interest. Look at the world today.

Considering for each n $\in $\mathbb{N}, the improper integral f(x) = $\int_{0}^{$\infty} (t^(n-1)/e^t - 1) dt$

But, honestly, I do that mostly by the seat of my pants. The real big-whigs know all the probabilities by heart.

Uh... I tried using latex, I failed T_T

Or at any point in human history.

@Fargle that's the individual, just like in the cells. But the group follows a different trend

My having an answer to that would suppose a definite answer

@TedShifrin :)

Chess is super awesome.

Chess always struck me too much like work.

@Semiclassical again, there must be a trend

It's the only non-academic thing that gives me the same sense of awe that mathematics does.

There is so much known; just like in math.

You need dollar signs to do math formulas, @Kerooker.

or rather, I strongly believe for there to be a trend

@towc This again presumes the answer is even knowable. Humanity in relation to individuals is far more complex than an individual in relation to its cells, because near as we can tell, cells are not capable of abstract thought.

Uh... Can't do it hahah

Dollar signs only around the math parts, @Kerooker.

I'll explain what we tried when I can get the thing done

So like this, Kerooker $ [math here] $

Ah

Uh oh..

yeah sure, we can't understand what a brain thinks about the mexican wall by analyzing the weighs in the connections and the structure of the connections, but by observing the brain in operation we can still get an idea

to the extent that said trends would represent a 'definition' of humanity's purpose, I'd reject it

@Semiclassical sure. I'm not looking for a definite answer, but a trend

The question doesn't even make sense, @Kerooker, but I can see why this is freaking you out.

I can't make the latex work, nevermind...

Is the improper integral perhaps $f(n)$?

you can edit your posts within a time window, you don't have to post it over and over again

Sorry @arctictern

i have no objection to the application of stats/probability to human behavior

You want $$f(n)=\int_0^\infty \frac{t^{n-1}}{e^t-1}dt,$$ don't you?

Maybe... But I can't see it as latex hahha

That's empirical.

see "$\LaTeX$ in chat" in the room description, upper right corner ---->

But questions of meaning and such lie in the realm of metaphysics, and so cannot be answered by observations alone.

btw, unrelatedly, n^(1/n) is divergent as n → infinity, right?

no, it converges to 1

@towc My point the whole time has been that identifying a trend in any way with some "objective" of humanity is fallacious. To say that because there is a tendency, it is in some way indicative of an intent is to assume the conclusion.

The situation I'm in is: I'm not really sure how to solve the integral. And with that, I'm not sure if it converges

You can't solve the integral.

it's zeta times gamma, so involves special functions

oh whaaaat. It does, but why...

Tern, @Kerooker is a Brazilian Calc II student with a bit crazy of a teacher, we now deduce.

imgur.com/a/x5EUj

This is the integral

how do you even differentiate n^1/n?

Note what I put up there ^^^^ :)

He is insane... He's Spanish and uses his own book... hahah

@towc You get a sequence $(1, \sqrt{2}, \sqrt[3]{3},...)$, and any neighborhood around $1$ has infinite intersection with this sequence.

Did he at least write the book in Portuguese, @Kerooker? :P

Yes, it's Portuguese.

@Fargle neighbourhood?

Our main difficulty is knowing if the integral converges

@Fargle what you just said is not sufficient to prove convergence, only convergence of a subsequence :-)

Of course, my students have complained about my writing my own books, so I have to be careful.

@arctictern That's fair.

Oh hey, Bose-Einstein integrsls

OK, can you tell me whether this integral converges or diverges? $$\int_0^\infty \frac{t^2}{e^t}dt?$$

Yes Ted, but his book is in an alpha stage right now hahah

@towc A neighborhood of a point is an open set containing that point--in this case, open intervals work.

why can't I see this in latex?

You need to do what tern said and do the LaTeX in chat thing over there >>>^^^^^

@Fargle doesn't that mean that the set of all numbers contains 1 as well, so it's in the neighbourhood of 1?

or do you mean that it's an arbitrarily small open set containing 1?

Can't do it, @TedShifrin

in which case it would be the same as saying that it converges to 1

@towc This. I'm saying that every neighborhood of 1 contains some $n^{1/n}$--in fact, infinitely many.

Well, better to spend time learning the math than dealing with this, @Kerooker. Good luck! :)

No, i meant can't do latex

@Fargle technically, neighborhoods don't have to be open, they just have to contain an open set that contains the point

I know.

@arctictern That's fair as well.

can't read it in math

I have my own technological issues at the moment, as it happens.

@Fargle all of them

either way, how do you prove it?

@Kerooker huh?

First day of winter here in Minnesota

@TedShifrin it would be great if you could send me that chapter

@towc I'll leave that to someone who actually knows how to do analysis proofs.

sure

@meow: I'm trying to re-typeset it and having some issues I don't understand. But I will if you email me (see my profile).

Pretty much every day of November was nice, until today

@Semiclassical Why so?

Weather-wise

It's snowing here

@towc do you agree ln(n)/n tends to 0?

@arctictern yes boss

Prior to that it's been nice weather all month

n^(1/n) = exp( ln(n) / n)

wait wut

n^(1/n) = (e^ln(n))^(1/n) = e^(ln(n)/n)

oh wow

yeah

and now it's trivial

ok, cheers

that's so clever

does the technique have a specific name?

so I can google for more of it :P

well, same trick used to differentiate x^x. typically taught in intro calc at some point as an exercise

heh, never even tried that before

used a lot in doing things with (algebraic expression)^(algebraic expression), e.g. asymptotics or calculus

but I guess now i know how to do it

doesn't really have a name I'm aware of

the whole idea is that a = e^ln(a)

well, you could do logarithmic differentiation, tern

and (a^b)^(c) = (a)^(b*c)

that's fair, and generalizes better

and I just need to figure out when to apply that

when ln(a)*c has any meaning that could help me

cheers

Hmm, I have an old LaTeX document (my first book) that will no longer typeset past 36 pages. Memory overflow with the "improvements" to typesetting ...

is there theory to solving systems of Pell equations? For example $b^2-6a^2=c^2-12a^2=1$

again almost unrelatedly, but as I'm here it's probably worth asking about: I'm attempting a cambridge interview and the csat in early december. Been practicing on MAT (Oxford's own Maths Admissions Test) and a few other high-level tests that I've found there. Any Oxbridge people in here?

applying for computer science at Queens'

"hence or otherwise", any resources that you think I could find useful?

@TedShifrin That's very odd.

@Fargle Indeed. I've never had this happen before.

@TedShifrin Could it be a problem with the particular interpreter?

No, there's something in that particular book file that's causing issues. I chased down a bunch of graphics rearticulations, but it's still happening.

Hmm.

Well, that was my only idea. I am bereft.

Worry about your own work :)

But thanks :)

@Semiclassical I didn't understand anything because I know nothing about algebra, but thanks