Mathematics - 2017-04-27 (page 1 of 5)

anakhronizein

12:00 AM

It's a good thing your aren't in a complete space, @Daminark

Or maybe that is bad.

Eric

Unbounded sequence of gulps

Ted Shifrin

I wonder if I can have a martini on top of the pain killer pills.

Daminark

How strong are they?

Ted Shifrin

Not strong enough to kill all the pain yet. But I only took one, about 2 hours ago.

Daminark

And lol @Eric I'm intrigued by stuff like combinatorics and graph theory so if grad schools or later on academic employers are all "Hahahahahah nope" I'll try to see if I can get the tech people to hire me to do that

Still, the anxiety is real

And @Ted it's probably not something like Vicodin then, in which case it's hard to say, but be cautious for sure

anakhronizein

12:06 AM

Combinatorics is great. I just had an exam in that today.

Ted Shifrin

No, not that powerful. Just an acetominophen-hydrocodone mix.

Eric

@Daminark combinatorics genuinely terrifies me

anakhronizein

Why is that Eric?

Ted Shifrin

Counting is hard!

@Eric: You keep using words you should probably avoid — terror, anxiety, ...

Eric

@Ted you're probably right.. midterm week is getting to me, I'll do some yoga later to calm down or something

Ted Shifrin

12:12 AM

Great idea.

Eric

@anakhronizein like Ted said counting is hard, I've actively tried to learn combinatorics sometimes and just get nowhere, for some reason it just hasnt gotten through my head

anakhronizein

What kind of combinatorics?

Generating functions and the likes?

Eric

whatever is taught in a usual first intro to combinatorics

Daminark

@anakhronizein I've only seen basic stuff in like, graph theory

anakhronizein

Should try reading Wilf's book. Generatingfunctionology or something like that.

Eric

12:15 AM

oh ive chuckled at this name before

Daminark

@Eric Talk to Laci at some point in your life. He's harsh but he really makes things clear

Eric

oh I know, I've seen him lecture a couple times and he's great

anakhronizein

Daminark, are these people you know in real life? Or on stackexchange?

Eric

when I was a prospective student visiting chicago I sat in on one of his graph theory lectures and it was fantastic

daminark and I know each other in real life

Daminark

Laci? I know him in real life, he's straight up my favorite teacher, period

anakhronizein

12:17 AM

Oh I see.

Daminark

Oh I know Eric in real life

Ted Shifrin

They're all fakes ... every one of them.

3

anakhronizein

I was just confused at all the references to exact names.

Eric

Daminark and I are the same person, running two accounts and masquerading as different people

Daminark

Oh don't listen to strikethroughtextcommandhere{me} him, he's just kidding

anakhronizein

12:21 AM

Do people do reading groups via the stackexchange chats ever?

Daminark

That actually never came to mind @anakh, sounds interesting...

Ted Shifrin

Balarka and MikeM have a joint reading project going.

Daminark

Though the absence of a chalkboard and Hagoromo is perhaps saddening (though LaTeX helps)

anakhronizein

On what subject, Ted?

Ted Shifrin

yeah, I need blackboards, Demonark.

foliation theory

anakhronizein

12:25 AM

Oh interesting I have never heard of that.

Eric

@Daminark it's all about that hagoromo

Daminark

Ah, I remember Balarka saying he's doing that. And @Ted the main thing I think I'd lose from doing it over chat is pictures

Ted Shifrin

right, Demonark, I concur completely.

That's why I told you you'd have to give me your lecture on Skype.

Daminark

Which may be a bit of a big blow in certain topics since I tend to hash out things through them, but if what's being done is not conducive to pictures, or they're available on wikipedia, I'd be up for one of those

Makes sense @Ted

anakhronizein

You can always draw on ms paint or somethin

Ted Shifrin

12:29 AM

ugh

PVAL-inactive

It takes so long to draw things..

I need to draw some things..

Ted Shifrin

heya @PVAL

Daminark

@Eric Right? And @anakh why do you say such things?

Next we're gonna start writing everything in comic sans (which I don't particularly mind for some reason...)

PVAL-inactive

Hiya @Ted

Ted Shifrin

reminds Demonark he has a midterm to study for

anakhronizein

12:30 AM

I don't think anything is particularly wrong with drawing sketches in ms paint

It's a bit more versatile than a chalkboard

Daminark

Yeah that is true, I should head out at this point

Hibye @PVAL! :P, and bye everybody!

Ted Shifrin

Good luck on the exam.

Daminark

Thanks

PVAL-inactive

@Ted Did you figure out the star-shaped thing?

anakhronizein

Indeed, bon chance.

Ted Shifrin

12:32 AM

Oh, nope.

anakhronizein

@TedShifrin what is your research in?

Ted Shifrin

I'm retired. I used to do differential and complex geometry.

PVAL-inactive

@anakhronizein Bridge and wine

anakhronizein

Heh

Ted Shifrin

@PVAL, and hopefully tennis again.

anakhronizein

12:43 AM

I guess he might be retired now.

Ted Shifrin

Actually, food is high on the list, @PVAL.

Akiva Weinberger

Axiomatic logicians of the internet, what's your get-quick-rich schema?

anakhronizein

Conservative extensions to ZFC.

Ted Shifrin

Bye all, for now.

anakhronizein

Bye!

Juan Sebastian Lozano

12:55 AM

I think it would be cool to start another room for a reading group, and it would probably work best with short things, like a single chapter of a book or a paper or something.

sthiago

hi there, i was wondering if the surface generated from the equation of this answer: https://math.stackexchange.com/questions/1621403/catenary-equation-of-a-plane-3d/1621554#1621554?newreg=cebfacf3c9d14e3eb262289eac77ce6b

has any specific name.

i mean, i thought it'd be a catenoid, but it seems it's only a catenoid if the revolution is around the directrix

Adeek

hi @TedShifrin

anakhronizein

@JuanSebastianLozano a short book would work too.

Juan Sebastian Lozano

@anakhronizein Yeah, it certainly would. What subjects are y'all interested in? I'd be happy to do one on pretty much any area of differential geometry or analysis, or some representation theory, or geometric group theory, or some introductory algebraic geometry.

anakhronizein

I am basically interested in anything. :)

Juan Sebastian Lozano

1:11 AM

I've been wanting to read this paper on measured group theory for a while, but I'm not sure who is interested in that.

anakhronizein

Looks pretty interesting.

I am weak in measure theory, but I would be up for it.

Juan Sebastian Lozano

I'm pretty good in measure theory (well, I've been working on some projects with it for a while, so I am at least used to it). How good are you with group theory?

anakhronizein

Pretty decent.

Juan Sebastian Lozano

[that wasn't meant as bragging, I meant I could help you out if we read it]

anakhronizein

I don;t know how good this paper would assume though.

I am done my undergraduate degree.

So I have had a few classes on algebra

Juan Sebastian Lozano

1:17 AM

Alright, cool. I'm currently in my undergrad, but I've had a fair bit of algebra already, so I think we will be fine.

It's a survey article kind of aimed at beginning grad students, so I think undergrad algebra should be enough.

anakhronizein

Good, good. Then I should be able to manage it easily enough.

Juan Sebastian Lozano

I started a room where anyone who wants to join the reading group can go: chat.stackexchange.com/rooms/57795/…

Will Njundong

1:49 AM

what are hyperbolic functions

arctic tern

hyperbolic trig functions. did you google the term?

Will Njundong

was looking at steps to get a definite integral and it was in there. I have no idea what im looking at

Juan Sebastian Lozano

In what context?

Oh, I see

Akiva Weinberger

$\cosh$ (pronounced either "hyperbolic cosine" or "cosh") is defined to be $\dfrac{e^x+e^{-x}}2$

Will Njundong

googled it, not sure we're supposed to be doing this yet

Akiva Weinberger

1:51 AM

where $e\approx2.718\dots$

$\sinh$ (pronounced either "hyperbolic sine" or… I actually don't know) is defined to be $\dfrac{e^x-e^{-x}}2$.

What do these have to do with the regular sine and cosine?

Will Njundong

cant really determine a definite relation from the graphs i see on wikipedia

Akiva Weinberger

Well, there's a weird connection that has to do with the hyperbola (that shape with the two pieces)

Will Njundong

but they look similar to their respective trig functions

Akiva Weinberger

But the simple version is: the addition formulas and the differentiation formulas for these look similar to the formulas for the regular trig stuff.

For example, $\sin'=\cos$ and $\cos'=-\sin$, but $\sinh'=\cosh$ and $\cosh'=\sinh$ (note the missing minus sign).

Will Njundong

yess

following

Akiva Weinberger

1:55 AM

Also, if I recall correctly, $\sinh(a+b)=\sinh(a)\cosh(b)+\sinh(b)\cosh(a)$ (like the regular ones)

and $\cosh(a+b)=\cosh(a)\cosh(b)+\sinh(a)\sinh(b)$ (the regular version has a minus instead of a plus)

Oh, also, $\cosh^2-\sinh^2=1$.

So like the regular version but with a minus.

I think $\sinh$ is pronounced "sinch" by the way

Will Njundong

oh god more stuff to memorize :( where in calculus is this used extensively? (if at all)

Akiva Weinberger

Useful in solving some integrals. If you've learned trig substitution - it's like that.

But I don't think it's used a whole lot

Besides, you can always go back to the definitions and rewrite everything in terms of $e^x$ rather than hyperbolic stuff if you want

Semiclassical

one place where it comes up which is sorta funny is in special relativity

Will Njundong

thankfully. I have to cram the some of the identities before exams sometimes ;(

Akiva Weinberger

Have you learned Euler's identity?

The $e^{ix}$ stuff ("what does it mean to raise something to an imaginary power?")?

Will Njundong

1:59 AM

Not really. It has been touched here and there during lectures, but I don't go to class very often

and it doesnt show up in any of the online quizes or homework either

Akiva Weinberger

Oh. OK.

To touch on it briefly:

Euler's formula says that $e^{ix}=\cos(x)+i\sin(x)$. (A cool consequence of this is "De Moivre's theorem", which also has another proof that doesn't use Euler's formula).

Another consequence of it is that we get:

$\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$

$\cos(x)=\dfrac{e^{ix}+e^{-ix}}2$

These are formulas for the regular trig functions, not the hyperbolic ones!

The fact that these look so similar to the hyperbolic definitions explains why they have similar properties.

(De Moivre's theorem, by the way, says that $(\cos(a)+i\sin(a))^n=\cos(na)+i\sin(na)$.)

Will Njundong

@AkivaWeinberger but here i would have expected: $$\frac {e^{ix}-cos(x)}{i}$$

is there another identitity/theorem involved which i dont know about?

arctic tern

why would you have expected that?

Akiva Weinberger

That works also, but the point of those is just to use $e$ on the right hand side and not any trig functions explicitly

@arctictern Solving for it

arctic tern

oh

Akiva Weinberger

2:06 AM

@WillNjundong Remember that $\cos(-x)=\cos(x)$ (it's an even function) and $\sin(-x)=-\sin(x)$ (it's an odd function).

Will Njundong

@AkivaWeinberger ahh i see

@AkivaWeinberger so I can easily derive the hyperbolic functions from these?

Akiva Weinberger

To be explicit about the connection between these and the hyperbolic functions: $\sin(ix)=-i\sinh(x)$ and $\cos(ix)=\cosh(x)$.

(How can you take the sine of an imaginary angle? Through those formulas, that's how.)

(Well, I suppose there's another way:

arctic tern

one can also talk about hyperbolic rotations of points on a hyperbola (analogous to rotations of points on a circle)

Akiva Weinberger

Use the approximations $\sin(x)\approx x$ and $\cos(x)\approx1$ for small $x$, which come from the tangent lines to those functions.

Decide that these should work for complex numbers as well, so that $\sin(0.0001i)\approx0.0001i$ and $\cos(0.0001i)\approx1$.

And also decide that the angle sum formula should work for complex numbers, and use the angle sum formula ten thousand times to arrive at $\sin(i)$ and $\cos(i)$.

To get better and better approximations, take the limit as the $0.0001$ part goes to zero.)

Tim The Enchanter

Of course small here would refer to the modulus, as there is no order defined on $\mathbb C$

Akiva Weinberger

2:11 AM

Yeah

Tim The Enchanter

@WillNjundong You could also use the power series expansion of sin and cos over $\mathbb C$

Akiva Weinberger

(I said "decide" because, technically, we could define the sine and cosine of an imaginary number to be anything we want. But if we want a useful definition, we'd probably want it to satisfy those properties.)

Tim The Enchanter

Although that is the same thing as exponentials when you think about it

Akiva Weinberger

(And we'd also need to prove that those properties aren't self-contradictory. They aren't, though, so everything's fine.)

Meow Mix

Hi

Akiva Weinberger

2:15 AM

In any case: Doing that process eventually leads to $\sin(i)=1.175\dots i$, and luckily $1.175\ldots=\dfrac{e-e^{-1}}2=\sinh(1)$.

I made a typo above. $\sin(ix)=i\sinh(x)$, not $-i\sinh(x)$.

Will Njundong

wow...the world of math...

Akiva Weinberger

Also, I'm kind of getting off on a tangent here, but have you ever wondered what $\arcsin(2)$ could mean?

Tim The Enchanter

Another example I like is solving $sin(x) + cos(x) = 2$

Akiva Weinberger

Well, you'd think it's meaningless, since $\sin$ is always between $-1$ and $1$.

But now we've just defined the sine of a complex number, which changes everything.

Will Njundong

@AkivaWeinberger no, lol

Akiva Weinberger

2:19 AM

In fact: $\sin\left(\dfrac\pi2-i\ln(2+\sqrt3)\right)=2$.

Will Njundong

@TimTheEnchanter solving it?

Akiva Weinberger

That is, the sine of complex numbers need not be contained between $-1$ and $1$.

Tim The Enchanter

@WillNjundong You can rewrite it as $sin(x + \frac{\pi}{4})= \sqrt(2)$

And take the arcsin of $\sqrt 2$

Which is complex like akiva said

Akiva Weinberger

TL;DR: Hyperbolic functions are things with properties similar to the trigonometric functions, and their names end in "h" (like "sinh" and "cosh"). They are defined in terms of $e^x$ and stuff. There's a deeper connection between them and the trigonometric functions that involves the complex numbers and Euler's formula.

Will Njundong

understood

Akiva Weinberger

2:23 AM

They can be useful in solving some integrals, but they're never necessary since you can always replace them with the $e^x$-and-stuff in the final answer if you want.

Will Njundong

@AkivaWeinberger just wondering. why was it useful to look into these by whoever established these relationships?

Akiva Weinberger

Not sure. Googling.

Dan Brumleve

@Akiva O/H = 2 means the triangle is laying flat on the ground and you're looking down at it slightly in a line of sight defining a vertical plane with its hypotenuse

can you believe i figured that out without leaving my desk :)

Akiva Weinberger

@WillNjundong It seems to suggest that part of its history was motivated by the following question: "What shape is a hanging chain?" (It looks like a parabola but not quite)

Turns out, the answer is "the graph of the $\cosh$ function".

Will Njundong

wow....someone really went through the math to figure that out XD

Akiva Weinberger

2:30 AM

It was a famous problem for a while :P

Will Njundong

thanks for looking into it

Akiva Weinberger

Lots of people couldn't figure it out.

I think one of the Bernoulli brothers first solved it?

In addition to that, if you know what a power series is, it's very natural to go from $\sin$ to $\sinh$ (the power series of $\sin$ has alternating plusses and minuses; if you change them all to plusses, you get $\sinh$). Similarly for $\cos$ and $\cosh$.

PVAL-inactive

I don't know what the physical problem is, but the hyperbolic functions are soultions of very nice ODE's similar to the sine functions.

It's easy to imagine these ode's occuring in nature.

arctic tern

(cosh,sinh) parametrizes a hyperbola

also, catenary (hanging chain problem)

Akiva Weinberger

Yeah, "catenary" is the name of the shape.

Turns out that an upside-down catenary makes a good shape for an arch, also.

Dan Brumleve

2:35 AM

or a road for a square wheel

Akiva Weinberger

Huh, never knew that those were catenaries.

Will Njundong

@AkivaWeinberger any useful application for this?

Akiva Weinberger

I've ridden on that once :)

@WillNjundong Pretty sure no

Dan Brumleve

i think we just found one

Will Njundong

heh, answered my question befor ei asked ;)

Akiva Weinberger

2:37 AM

Yeah, makes a fun exhibit I guess :P

Will Njundong

@AkivaWeinberger but thats one right there

PVAL-inactive

I'm not convinced that there aren't lots of shapes, which a square wheel works on.

Will Njundong

well a square certainly rolls best on something oval like or circular :)

Dan Brumleve

what if the square wheel is allowed to slide as long as its center stays level

Akiva Weinberger

google.com/search?q=square+wheel+exhibit&tbm=isch

@PVAL-inactive I think it probably would be unique if you want it to roll without slipping

Semiclassical

2:39 AM

@PVAL-inactive One place in physics where hyperbolic functions show up is special relativity, amusingly.

Dan Brumleve

is there another road for a greasy square wheel?

and can you still drive that thing on the catenary with no friction?

PVAL-inactive

@semiclassical I suspect they show up there quite directly because they are the solutions to $f'' =f , f(0)=0,1$.

Dan Brumleve

it seems like maybe it would slide backwards if both wheels are rolling up the same time

Semiclassical

Actually, no.

Akiva Weinberger

This seems to be the original article

"Square-wheeled bicycle", YouTube

Semiclassical

2:44 AM

The point is really that a rotation takes a spatial vector gives a linear combination of two spatial vectors, whereas a boost (picking a reference frame that's moving faster in some direction) would give a combination of a spatial vector and a temporal vector.

Akiva Weinberger

Props to whoever came up with the idea of making it a cone-shaped thingy

PVAL-inactive

props to whoever came up with circular shaped wheels

micahscopes

@arctictern hey, really nice answer, thank you. I'm still reading it through

PVAL-inactive

They seem to work a lot better.

Akiva Weinberger

Well, I guess it's probably the natural response to "Maybe we should make the tracks go in a circle, how do we do that"

Semiclassical

2:45 AM

And just as the rotated spatial vector have cos,sine components, so too will the boosted spatial vector have cosh,sinh components.

It boils down to time being equivalent to an imaginary spatial direction.

("What about rapidity?" "Quiet, you.")

Akiva Weinberger

Special relativity is kind of "special"

Ted Shifrin

3:03 AM

That's a fun exercise, DogAteMy — it's in the first chapter of my diff geo text — getting the road for the square-wheeled bicycle

really interesting exercise is: given the path of the back wheel of a bicycle, what's the path the front wheel is following?

(That's for a regular bicycle.)

Akiva Weinberger

(Sorry, misread)

Ted Shifrin

Oh, I missed whatever you done said.

Akiva Weinberger

I confused it for the one you had in your linear algebra book (figuring out which of the two paths was from the front wheel and finding out in which direction the bike went)

Ted Shifrin

oh, not linear alg, but multivariable — yeah, that classic problem led to my writing the other one.

Akiva Weinberger

What I meant, sorry

Ted Shifrin

3:10 AM

It turns out that except for two basic cases, you can't solve the diff eqn in closed form. But it's a great question.

Akiva Weinberger

It has lots of linear algebra-y stuff in it

Ted Shifrin

It's amazing how quiet it is when certain people are actually studying for midterm exams ...

Leaky Nun

hi everyone

Ted Shifrin

hi Leaky

Leaky Nun

Per this and this, considering that I will not use my alternative account again, and considering the fact that there has been (and will be) no interactions between the two accounts, I don't see any problem with having a (now inactive) second account.

Ted Shifrin

3:24 AM

this is too complicated for me

Little Rookie

Sigh :( didnt finish my real analysis paper

Leaky Nun

@TedShifrin I'm just doing a disclaimer

Ted Shifrin

oh oh @Little

Leaky Nun

@TedShifrin All I mean is, I have a second account that I will never use again.

Ted Shifrin

well fine, then, leaky

Little Rookie

3:25 AM

Basic topology and sequence/series of functions were not tested, sigh

Leaky Nun

@TedShifrin I guess you can figure out who I am

Ted Shifrin

I have no idea

Little Rookie

Instead, countable sets were tested, and i spent 30min doing the 10marks question

Leaky Nun

@TedShifrin you will. I don't feel safe publicly announcing my second account despite what I said above.

We have had quite a few interactions.

PVAL-inactive

if i wasnt me

I'd guess me

Mike Miller

3:28 AM

lol

Little Rookie

How can i be so unlucky, the only part that i didnt revise is countable sets!!!

Ted Shifrin

guesses @PVAL

that's Murphy's Law, @Little

Leaky Nun

@LittleRookie for example?

Little Rookie

It was a rather straightforward question, but i took longer than requored because i didnt revise countable sets since recess week

Ted Shifrin

From countable sets to uniform convergence is a huge amount of stuff

Akiva Weinberger

3:31 AM

Can I share a thing?

Too late

Little Rookie

Suppose given a finite set A and a countable set B where both A and B contain distinct elements and the intersection of A and B is empty

Ted Shifrin

For obvious reasons, I don't believe in Ted talks unless I'm talking

14

Little Rookie

Prove the union of A and B is countable

Its a straightforward question

Ted Shifrin

OK, that shouldn't have taken too long

Little Rookie

I took 30mins!

Ted Shifrin

3:32 AM

oh dear

1, 2, 3, ...., k, k+1, k+2, k+3, .... k+n, ....

Leaky Nun

just relabel the elements

Ted Shifrin

Do you understand what I just wrote, @Little?

Huh?

Leaky Nun

Never mind, I read it wrongly

Akiva Weinberger

How do I biject $\{1,2,3,\ldots\}$ with $\{1,2,3,\ldots\}\cup\{a,b,c,d,e\}$?

Little Rookie

Yes i did that

Akiva Weinberger

3:35 AM

(All of my finite sets contain at most 26 elements)

Leaky Nun

I guess I've been saying too much

Akiva Weinberger

Oh, but you've already solved the problem now, yeah? @LittleRookie

Do you know how to show that countably infinite plus countably infinite equals countably infinite?

Little Rookie

But took me 30min of my exam time

Akiva Weinberger

Oh, this was on an exam!

Oh, no :(

Ted Shifrin

It means you need to practice writing basic proofs more quickly ... practice is the key for most everyone.

Leaky Nun

3:38 AM

@AkivaWeinberger did you read what I said above?

from here

16 mins ago, by Leaky Nun

hi everyone

Little Rookie

We had a question on countable sets for midterm and i believed other topics like basic topology is important n will be tested for finals instead

I am wrong :(

Ted Shifrin

This was the final?

Little Rookie

Yes

Ted Shifrin

Finals are usually comprehensive, but I would have expected more on the part of the course that had not been tested yet.

Little Rookie

Yes, thats what i thought so too

Apparently my Professsor dont believe in this

Ted Shifrin

3:40 AM

I would tell my students to expect certain things if they'd messed them up before, though. Sometimes they showed up; sometimes I figured scaring them into studying was enough.

Little Rookie

:D

I compensate some of the 30min spent on countable sets by doing the question on proving the bolzano weirestrass theorem quickly

But it was not enough :(

I left part c) of my last question undone

Little Rookie

5:59 AM

Omg, the question that i had no time to attempt
https://math.stackexchange.com/questions/197400/if-sum-a-n-converges-then-sum-sqrta-na-n1-converges?rq=1

SoumyoB

6:22 AM

@LittleRookie was it an exam?

Naveen Balaji

6:59 AM

3

Q: On reducing complex ODE's to Bessel's form

I am trying to reduce the following ode to bessel's ode form and solve it: $$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1}$$ I tried to solve it via the standard method,i.e., by comparing it with a generalised ode form and finding the solution from then on. The general form (as giv...

Little Rookie

7:15 AM

@SoumyoB yes T_T

Real analysis I

SoumyoB

8:07 AM

@LittleRookie that's why always have a glance at all questions before answering them

mtheorylord

11:16 AM

Hello my peoples

Tim The Enchanter

Yes oh lord

Benjamin

11:32 AM

Does it take time after registration to get access to the MAA e-Library?

Secret

12:11 PM

theconversation.com/…

Speaking of which, do we mathematicians generate big data as well for not so applied fields?

e.g. I imagine we the maths community should already have libraries of algebraic structures somewhere given how much effort we have put in classifying finite groups alone...?