Neat

I know the is more of a physics question but maybe you guys can still help me. I know that the distance something travels in the x-direction is its maximum distance given by dx=velocity in x * time, but I still don't understand how to solve this question because to find the time you need something from the y-direction either initial velocity in y or height it falls from but you have nothing?

Right now I'm still doing dynamics and complex

hi chat

@AaronM If that's the entirety of the problem, I concur with your assessment and would answer D.

I mean, if I toss a dart off a cliff it'll definitely go a lot farther than if I toss it at a wall.

@Semiclassical Thanks. The answer key says A. I have no idea how they got that

The only possibility I could imagine is that they meant "how high can it go if I toss it straight up"

but uh

that's a really dumb interpretation of the problem.

In fact, it looks like A is equivalent to v^2/g.

Oh. They might be interpreting this as a projectile motion problem. @AaronM

Yeah it is under that section

Ah. Then probably what they mean is:

If they were talking about throwing it straight up then I used the equation vf^2=vi^2+2ad. Vf=0 vi=7.76 a=9.8 so d=3.07

but not 6.14

Right. But there's a result in projectile motion that tells you the range as a function of initial angle

and the max range, which occurs at a 45 degree angle to the horizontal, is v^2/g=7.76^2/9.8 = 6.14

But why was what I did wrong

Eh. I think the point is that the problem isn't well-written at all.

What they intended was: You launch the dart, starting at a height of zero, with an initial velocity of 7.76 m/s and at some angle. If you throw it at a 45 degree angle, then the horizontal range you'll get is maximized

Ok. I see what you're talking about when you say maximum range. the equation is v^2/r *sin(2x) which is max when x=45

Right.

to be honest I never memorized that formula just saw it derived didn't think it was too important but I guess it is

But I don't like the problem regardless. For one, I don't exactly associate a dart with projectile motion.

My mental image with a dart is instead someone tossing it at some height.

thanks again for your help

np

Oh. I guess I should also point out how I figured out what they were doing. @AaronM

You're given one quantity, that the initial velocity is 7.76 m/s

additionally, you're doing a problem involving motion, so typically the gravitational acceleration 9.8 m/s^2 is involved.

And you want to get a distance in meters somehow.

And the only way to get a distance from that is to have some combination like (7.76 m/s)^2/(9.8 m/s^2)=6.14 m.

i.e. v^2/g.

Now, that doesn't automatically tell you it has to be A. B would be 2v^2/g, which is plausible as far as units are concerned.

on the other hand, C would be 0.632v^2/g. That seems quite arbitrary, and makes it dubious as an answer.

Anyways. Once I saw v^2/g gave the right answer, I remembered that the range equation was a thing.

So thinking about the units can help you understand what each option is really suggesting.

@Semiclassical Misread that as "rip" for some reason

rıp

Hi chat

Hi caht

lol

Hi $\textrm{Sym}(chat)$

yo chat

oh damn lots of ppl coming in

Lol, so it seems

After we the deluge

@Daminark why are you kiddos so concerned with catching up on lectures btw

A lot of us were just kind of annoyed at how the last two lectures went too slowly, like we could've gotten through more

that kind of seems to beat the purpose tho

The purpose of?

if ur going slow that gives you more time to do problems

the bootcamp

learning complex quickly is def not the point of titchmarsh

I mean, we're not going slowly because of the pace that was set, we're going slowly because the lectures were... undeprepared

yeah but that happens

afternoon

idt that going fast to catch up matters

or that it's even a good idea to begin with

you're gonna do more than we did anyway

So algebra tripped people up and whatnot, people getting stuck when asked questions, etc

that's growing pains though

it was expected to happen

Wait are we really covering more?

you have like more lectures per week than we did

and we had loads of people underprepared

Oh huh

like it's important to actively not rush

man lipschitz functions are freaking awesome

Are you doing complex analysis? @Dami

Perhaps. And I mean, at least the next two people lecturing are probably going to prepare reasonably well, and not necessarily go quickly. Most of the talk of getting caught up had to do with how I tend to just talk fast in general, and thus could probably pull it off if I were to sign up. Also Daniel and David are pretty good at their stuff

Yeah @Alessandro

@EricSilva How?

@EricSilva This wrt GMT?

@Daminark with respect to any time you need a class of functions that satisfies nice properties

Ah, yeah Lipschitz functions are pretty convenient

@Akiva they're basically regular enough that you can do normal $C^{1}$ calculus but also analytically a much more well-behaved class of functions. It's way more natural to get estimates for Lipschitz functions and prove compactness type results.

Lipschitz functions always make me think of special relativity

Oh, nice, that's a coop topic @Dami

What are you doing more precisely at the moment?

@Semi do they show up a lot in special relativity?

Last class we ended with proving that differentiable functions are analytic

Next class we should be doing Cauchy's bound, Liouville, FTA, Morera, that sorta thing

Nah. But the Lipschitz condition for a real-valued function $f$ is $|f(x_1)-f(x_2)|\leq K|x_1-x_2|$ for arbitrary $x_1,x_2$ and some positive $K$.

well not just for real valued things

I see. FTA from Lioville is an easy proof, but there a few ways to prove it via complex analysis

@Daminark that result is great tbh

now prove it for harmonic functions :D

If you think of $x$ as being time and $f(x)$ as position, then it can be interpreted as saying that the average speed over any interval is at most $K$.

right

Okay so harmonic functions are solutions to Laplace's equation, right?

Which sounds a lot like the special-relativity statement that nothing can go faster than the speed of light.

that is their definition yes

oh tru @Semi

$\sum \frac{\partial^2 f}{\partial x_i^2} = 0$

Right, a lot of results also work for harmonic functions. Harmonic+bounded implies constant and the maximum principle works for harmonic functions as well

I don't know if anyone draws that connection in practice, but it's a neat comparison.

personally i think it's more natural to think of holomorphic functions as being harmonic objects than that harmonic things extend complex analysis if im honest, but that is maybe weird

Hmm, well you know that said functions are at least $C^1$, so if you could somehow find some way to make it satisfy C-R, you could probably use analyticity of holmorphic functions

Did you show that real and imaginary part of an holomorphic function are harmonic and/or talk about harmonic conjugates? @Dami

Something like

Like, $f + if$ or something

@Daminark how do you know they're $C^{1}$?

The second partials exist

Or is that insufficient to conclude?

a priori solutions to "Laplace's equation" don't need to be classical solutions

typically one wants $f+i g$ to have $f,g$ identically real.

(oof, bad typo)

in fact any harmonic functions is real part of some holomorphic function i believe

(they turn out to be but this was a Hilbert problem)

in which case $f+if=(1+i)f$ would be bad.

What do classical solutions mean?

@Balarka some version of this statement is true

Okay @Balarka that's convenient

@Daminark $C^{1}$ solutions

Okay wait hold on maybe try something like $f_1 - if_2$

(I'm using subscripts to denote partials, so first partial, second partial)

Ah.

Or no, minus

it actually proves a lot of the properties of harmonic functions eg the averaging property

i mean, the mean value property

or the Liouville's theorem

you don't need the fact that harmonic functions are real parts of holomorphic functions to prove that

@BalarkaSen you need it to be defined on a connected domain I think

it's actually easy anyway

Even simply connected actually, unless I'm mistaken

I mean I'm basically worried that just because you satisfy C-R locally doesn't automatically guarantee that your function is holomorphic, but if you can guarantee continuity you're good

@EricSilva sure. I just always prove it that way

also max value theorem etc etc

well maximum principle is the starting point

you can prove that from subharmonicity without trouble

and then etc etc

basically my point is once you know the relevant complex analytic theorems all the corresponding props of harmonic functions are ez

@Daminark you only need the existence of partials everywhere + C-R is satisfied + $f$ is continuous

oops, right, maximum principle not max value theorem :P

that's a name for a different thing

@AlessandroCodenotti yeah probably

Well, I'm trying to guarantee continuity of the function $f_x - if_y$.

also the Poisson integral formula just falls apart from Cauchy integral formula

so you only need pointwise things to prove analyticity which is insane

@Daminark is $f$ harmonic?

The limitation of the complex-variables POV is that it only works in 2D.

Yeah

@Semiclassical yup

@Daminark you basically NEED a regularity result somewhere

( <- master of the obvious)

this is the reason i dont think of holomorphic functions as the primitive things and harmonic functions as the extension basically

That said, it's still really cool that the content of harmonic functions in that 2D case can be realized via holomorphic functions.

@Eric you're a geometric analyst, you don't count

to me Laplace's equation is like THE most natural thing to think about like in math in general

complex analysis rules man

i wouldn't call myself this quite yet

don't wanna jump the shark

oh i looove complex analysis

i guess one can boil that down to the fact that $\partial^2_x+\partial^2_y=(\partial_x+i\partial_y)(\partial_x-i\partial_y)$

it's just more natural for me to think of $\Delta$ first personally

@BalarkaSen we proved that in the complex analysis course, but I don't remember the details

but this is hindsight basically

Wait hold on isn't it on one side taking second derivatives, and on the other hand squaring? @Semi

The copy of Rolfsen they have in the library is falling apart and horribly typeset :( @Balarka

Eh?

@Daminark Semiclassical is writing product in the sense of operators

$\partial_x \partial_x=\partial_x^2$, yeah.

@Daminark he's thniking about them as operators

sniped

rip me

Oh okay fine

get rekt

but yeah

As is my wont as a physicist

@AlessandroCodenotti I kinda like old and falling-apart books like that; I also kinda like the typewritter font typesetting

I think it's badass

In physics the places I see Laplace's equation would be...

printer font?

fixed

ah like the courier font

it's one of my favorites

electrostatics in vacuum, with the potential as a harmonic function

@Balarka I mean I like paper pdfs as much as the next guy but like, non-LaTeX and bad condition is just a nope for me

basically why I liked Milnor's red book at first glance

incompressible fluid mechanics, which also has a potential function

Btw hey @PVAL

I borrowed it anyway, the content is more important than the typesetting

Actually, I was going to say quantum mechanics, but

hey chat

@Alessandro I've recently checked out this one book by Narasimhan which I find to be pretty quality

that'd be the (time-dependent) Schrodinger equation $-\Delta \Psi=i\dot\Psi$

A book about what?

I think there are some mathematicians out there who write papers in MS word

Complex analysis (I guessed that Rolfsen was one as well)

@Semi i think long ago someone told me F= ma was everything and it stuck with me that second order diff operators are really important and then because $\Delta$ is the simplest second order diff operator it just struck a chord with me

Rolfsen is a knot theory book

yeah, $\Delta$ is ubiquitous in physics.

and geometric analysis

Oh, whoops

but this is because of physics i guess

@PVAL-inactive How goes it?

alright

I woke up later than I expected so I have to work later than I expected

oof

Darn

Do you have stuff to do over the summer beyond writing and whatnot?

no nothing beyond writign

are you going to beat me to the arxiv?

I don't know.

If you said next two weeks probably not.

Nah I probably need a month.

At least.

I think you are still ahead of me.

I'd like to be close sometime in september

We'll see. I suspect Ciprian does more revisions with students than Bob does.

and submit maybe a little after that.

Well I probably will want somebody to do one.

So I need to get it to that point.

Ya of course.

My advisor is getting a little tired of waiting I think.

Hey everyone

Something strange happened in one of my lectures yesterday, the lecturer of my Real Analysis course, who was taking us for the first time (our uni reopened yesterday), saw a student standing up (the student was waiting for another student to move), and shouted at the student (so as to make a point), "Sit down or f*** off" and then began the lecture

That's oddly aggressive

Among other things covered in the lecture was his bragging about how he supposedly can't get fired

ugh.

That's distasteful.

lmao

But yeah this generally is looking like an iffy pattern

that is wildly inappropriate

Fair to say I won't be attending any more of his lectures anytime soon

Hopefully there's a book which more or less covers the class?

He also mentioned how he's sick of his undergrad students

(Or is there another section?)

Perhaps it'll be a smoother one

Get it? :P

Nah but really though do see if there's an open time at which someone else is teaching that same class

@Daminark You and your diff topology jokes :p

The other section has a lot of zeroes.

They're using Bartle and Sherbert, but I already went through Baby Rudin last year so I'm not really worried about the lecturer

@Daminark There's only one class per year for Real Analysis in my uni and I'm stuck with this lecturer (sighs)

How many people are there in class? If he's got that attitude you don't won't want to stand out as someone who doesn't show up

The Residue Theorem: From Zeroes to Heroes

how can someone who doesn't show up stand out

As long as you are getting the content out of the lectures, I would suggest continuing to show up.

@Eric come on a test day and be a new face

I was sitting in the lecture with Guillemin and Pollack open ready to do my own thing, then he started 'interrogating' students for answers to questions he was asking

Maybe you could think of it as free entertainment

You may feel like you are taking a stand against the way he is treating people.

@Daminark I guess, I've done this a lot but no one cared so I've been lucky I suppose

But really you are just denying yourself education.

(Depends on exactly how he conducts his classes, I guess)

So my plan of quietly doing Diff Topology during his lectures, just flew out the window, because he started to single students out for answers

@PVAL-inactive I've already gone through the content of the course (and quite a bit more) on my own through self-study

@Daminark There's about 50 students in our class

@Eric I mean, at least in college I don't get the vibe that either of us had too many professors who would mind, but also we haven't had too many like the one in this description

@AkivaWeinberger random question. Would I be wrong in assuming that if a number is an element of the quadratic rationals and it is a solution to a monic polynomial with integer coefficients, then there exists a second order monic polynomial with integer coefficients that it is a solution to? Don't prove it. I want to try to do it.

Interesting video on motivating people

what are the quadratic rationals?

@AlessandroCodenotti something me and akiva were discussing

Quadratic irrationals, you mean

that's not a very helpful definition to me :P

ok. give me a sec to write it out.

@AlessandroCodenotti Roots of quadratic things with rational (or integer) coefficients

NO

No?

Generally my professors are all pretty chill, this one is just an outlier

@AkivaWeinberger I guessed so when you said irrationals, I was confused by the "rational"

A quadratic field is any field Q[x] such that x^2 + ax + b = 0, where $a$ and $b$ are integers. The elements of any quadratic field are quadratic rationals.

@AkivaWeinberger no....

similarly, there are quadratic rings and quadratic integers

So you're asking if every element of a quadratic field is the root of a second order polynomial?

That's equivalent @Typhon

@AlessandroCodenotti no.

that is false for sure

@AlessandroCodenotti He's asking if it's the root of a monic one with integer coefficients, I think

nope not that either

@Typhon Example?

@Typhon I disagree

well

ah, wait, I'm using polynomials with rational coefficents

if an element of Q[sqrt{2}] satisfies a monic polynomial with integer coefficients, then the element is an element of Z[sqrt{2}]

same for all sqrt{d} where d = 1 (mod 4)

I think

ok, so you have an element of a quadratic extension of $\Bbb Q$ which satisfies a polynomial of some degree in $\Bbb Z[X]$ and you want to know wether it must also satisfy a degree $2$ polynomial in $\Bbb Z[X]$ if I understood correctly

@AlessandroCodenotti I'm asking if the quadratic rationals that do satisfy monic polynomials with integer coefficients also satisfy second order monic polynomials with integer coefficients.

@AlessandroCodenotti essentially. You just missed a couple requirements of the polynomials. See above.

wait no