Mathematics

Karim Mansour

12:26 AM

@BenDover yeah I heard good stuff about this one

Karim Mansour

12:45 AM

0

Q: Integers aren't open sets

So I am solving the following question in apostol Determine all accumulation points of the following sets in $R^1$ and decide whether the sets are open or closed (or neither). All integers: The set of integers doesn't contain an accumulation point of Z I will do it by contradiction suppose x $\...

real-analysis

Jun-Goo Kwak

1:58 AM

Does anyone have any ideas about how to simplify this expression? $$ \frac{\frac{x^3 e^{\frac{t x^2}{4}}}{8 t^{7/2}}-\frac{3 x e^{\frac{t x^2}{4}}}{8 \sqrt{2}}+\frac{t^3 x}{2}-\frac{3 \sqrt{\frac{1}{t}} x}{8 \sqrt{2}}}{\left(e^{\frac{x^2}{4 t}}+\sqrt{\frac{1}{t}}\right)^2} $$

Samuel Yusim

keep putting things over a common denominator until you cry, then take a break and get back to it

Jun-Goo Kwak

@SamuelYusim Sounds like a good plan.

Thanks.

Samuel Yusim

I'm sorry to say it looks like the kind of thing you'll need to brute force

Jun-Goo Kwak

No no I totally think you're right. That was my original approach.

I was just having difficulty finding a common denominator for the expression in the numerator.

Because it became quite lengthy and then I cried.

morphic

Use WolframAlpha

Samuel Yusim

2:03 AM

you can simplify a lot of stuff before you start combining fractions, though. For example $\frac{x^3e \frac{tx^2}{4}}{8t^{7/2}} = x^5e/(32t^{5/2})$

Jun-Goo Kwak

@Morphic Doesn't really help with this kind of expression.

Samuel Yusim

if you're worried about combining fractions just do it term by term: get the first two over a common denominator, then get this new fraction and the third term over a common denominator, etc.

oh, wait a minute

I messed up the thing I said about simplifying, I didn't realize the little tiny fraction was an exponent

Jun-Goo Kwak

xD

@SamuelYusim I'm actually trying to prove that a solution for a PDE is indeed a solution. Would it be easier instead to try and solve the PDE and arrive at that solution instead of plugging it in to verify?

Samuel Yusim

I don't know the first thing about differential equations so I couldn't say

Jun-Goo Kwak

Alright, thanks for the help. u.u this function is just brutal

Samuel Yusim

2:08 AM

for what it's worth I believe in you

morphic

We believe in you @Jun-GooKwak

Mike Miller

@Jun-GooKwak just plug it in

also, it's a PDE, how do you intend to find all the solutions in general

it's possible for some of them but i mean

i can think of essentially no case where if yuo have a potential solution it's not better to just plug it in

Jun-Goo Kwak

Yea, I was wondering about that as well.

But this solution is quite nasty indeed.

Mike Miller

2:30 AM

nothing you can do

just mash it

Jun-Goo Kwak

Alrighty. I got it so it looks a lot more manageable. I'll just have to see how the other partial derivatives in the equation work out. Hopefully something cancels out.

$$ \frac{x \left(\left(\frac{2 x^2}{t^{7/2}}-3 \sqrt{2}\right) e^{\frac{t x^2}{4}}+\frac{8}{t^3}+3 \sqrt{2} \sqrt{\frac{1}{t}}\right)}{16 \left(e^{\frac{x^2}{4 t}}+\sqrt{\frac{1}{t}}\right)^2} $$

Mike Miller

aaaa

Jun-Goo Kwak

xD

Karim Mansour

ok cool

4

A: Integers aren't open sets

To prove $\Bbb{Z}$ is not open, you could show that 1) $\Bbb{R} \setminus \Bbb{Z}$ is not closed or that 2) $\Bbb{Z}$ consists of isolated points. 1) 0 is an accumulation point of $\Bbb{R} \setminus \Bbb{Z}$ because it is the limit of the sequence {1/n}, but since $0 \in \Bbb{Z}$, it is not in ...

very nice answer

JMoravitz

3:10 AM

Question: what is an ideal way to hold something like "office hours" with someone long distance with the ability to type using $\LaTeX$ similar to here that offers a bit more privacy than simply a new chatroom? I originally thought that creating a chatroom here would suit my needs, but in the event that grades or something else personal were to be discussed, I find that it might not be private enough since anyone could see the room on the list for some time.

Mike Miller

@JMoravitz: mathim.com

2

pjs36

That's amazing, @MikeMiller! And quite a good question, JMoravitz

Mike Miller

3:28 AM

I learned about that website from Pedro, I think

user147690

@SohamChowdhury The challenge wasn't very well planned unfortunately. Well it was as well planned as it could be, given the length. But that was the problem, the length was excessive. You just can't plan that far ahead without more direction, and without it being a larger part of your life(I.e. uni had priority).

user147690

@Rememberme I have been busy with non-math things unfortunately - you know, life getting in the way :P. That's not meant to imply I haven't still been doing math of course, just haven't been doing it like I normally do.

user147690

@SohamChowdhury Also that drum track was indeed pretty intense xD.

Karim Mansour

3:46 AM

hi @AlexClark

James Pirlman

How might I show that $x = \cos(t) + \sin(t)$, $y = \sin(t)$, where $-\pi \leq t \leq pi$, is the equation of an ellipse?

Semiclassical

@james note that your first equation can be rewritten as $x-y=\cos(t)$, and in that form this should remind you of the equation of a circle.

James Pirlman

Err, I'm still a little confused.

That reminds me of the equation for a circle?

pjs36

Perhaps verifying that it satisfies $\frac{(x-h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ for some constants $a, b, h, k$ (it helps if you know the center and axis lengths in advance).

I think @Semiclassical is hinting that you could think in terms of circles and linear transformations. I'm not sure though, it's a fairly high-brow hint :)

Semiclassical

nah, i'm hinting at the fact that $\cos^2 t+\sin^2 t=1$ :)

pjs36

3:59 AM

D'oh!

James Pirlman

Well this problem is coming from me trying to calculate what the shear transformation does to the unit circle, by the way!

Oh, well, $(x - y)^2 + y^2 = 1$ :)

Semiclassical

note, though, that it's not an ellipse whose axes align with the $xy$ axes

Mike Miller

@pjs36: I don't quite remember the formulae, but I think your formula only works if your longest axis is vertical or horizontal. This won't be

Semiclassical

which is consistent with it being a shear of a circle

pjs36

Ah, which means my hint wouldn't work, double d'oh.

You guys take all the fun out of me pretending to know things, you know!

James Pirlman

4:03 AM

Oh, I like that idea.

Semiclassical

it's not as good as i thought, actually

James Pirlman

Oh … darn!

Semiclassical

you'd have to figure out some way to compensate for $y=\sin(t)$ not having that phase angle

James Pirlman

Well, anyway then, I get $x^2 - 2xy + 2y^2 = 1$.

Semiclassical

though i think there's a rather cute way with linear algebra alone. lemme think a moment

James Pirlman

4:05 AM

Which is certainly the equation for an ellipse

Semiclassical

right-o

James Pirlman

I need to find its area next

Hmmm

Semiclassical

what you probably need to look for is coordinates $(u,v)$ such that it's the standard form of an ellipse

Mike Miller

now that is a job for linear algebra

Semiclassical

right. and it's actually best to work entirely in terms of linear algebra:

pjs36

4:07 AM

That's what I was thinking, Semiclassical. I checked WA and it's not obvious (to me) what the major axis is.

James Pirlman

I think the major axis is $2$ and the minor axis is $1/2$

I'm open to being wrong though

Semiclassical

first, note that the equation of a circle $x^2+y^2=1$ can be written in matrix form as $\mathbf{x}^T I_2 \mathbf{x}=1$ where $\mathbf{x}=(x,y)^T$

pjs36

Er, I guess I mean the orientation of the major axis; how much it needs rotated to line up with the $x$- and $y$-axes.

James Pirlman

Oh, okay.

Semiclassical

and with $I_2$ being the 2-by-2 identity matrix

then, notice that your shear amounts to replacing $(x,y)\to (x-y,y)$ which can be neatly expressed as a matrix transformation on $\mathbf{x}$.

James Pirlman

4:10 AM

My shear does $(x, y) \mapsto (x + y, y)$, actually.

Semiclassical

psh, details :P (yes, woops)

James Pirlman

It has the matrix $$\begin{pmatrix}{1, 1}, {0, 1}\end{pmatrix}$$

Haha, okay.

Woops that matrix didn't typeset properly....

Semiclassical

that'll give your equation in the form $\mathbf{x}^T A^T A \mathbf{x}=1$

Mike Miller

@pjs36: maximize the norm; to do this find where the derivative of the norm squared is zero; this ends up being when $2 \sin(t) = \cos(2t)$

which might be solvable i guess?

Semiclassical

@MikeMiller $\cos(2t)$ is quadratic in $\sin(t)$, so yeah

Mike Miller

4:13 AM

oh, fair point

James Pirlman

Could I also just take the equation $x^2 + 2xy + 2y^2 = 1$, rotate it appropriately to remove the $xy$ term, and then calculate the area of the rotated ellipse that aligns with the axes?

And I mean $x^2 - 2xy + 2y^2 = 1$, woops.

pjs36

What do I look like @MikeMiller, a supercomputer??

I mostly kid, but it's too late for me to function even adequately :)

Mike Miller

i figured it's hard to be worse at solving equations than i am

Semiclassical

you can. and what that amounts to is finding a rotation transformation $\mathbf{x}\mapsto R\mathbf{x}$ such that $R^T A^T A R$, the matrix between the two vectors, is diagonal

Mike Miller

so i shared my progress with all so that someone who knows stuff could finish it

James Pirlman

4:15 AM

Okay.

Thanks for all the help!

pjs36

How many (aspiring?) mathematicians does it take to rotate an ellipse?

Fargle

There's a joke about eccentricity in there somewhere.

Mike Miller

ooh

i like that potential joke

pjs36

Bam, solid entrance, @Fargle!

Fargle

^_^

Semiclassical

4:17 AM

and probably one about eccentric anomaly if one was to push deep enough

oh, just thought of one. what do you call someone who really likes conic sections?

pjs36

How much do they like conic sections? No, I mean, "What?"

Mike Miller

Semiclassical

snerk

Fargle

I prefer "nerds" as the punchline to that one.

4:21 AM

Hi @Semiclassical

Semiclassical

hi @karim

Karim Mansour

do you know of john baez @Semiclassical ?

yeah

he is really good !

he knows a ton, yeah

4:26 AM

was reading some of his stuff math.ucr.edu/home/baez

yeah

Fargle

@Karim: have you seen the Visual Insight blog?

Karim Mansour

@Fargle no

Fargle

It's also him, as I recall.

@Karim blogs.ams.org/visualinsight

Karim Mansour

awesome !

very neat stuff !

Fargle

Indeed! I've been following it for a year or so now.

Karim Mansour

4:34 AM

that is what I want to do in the future @Fargle mathematical physics

along with pure math

Fargle

A noble pair of pursuits, I'd say, @Karim

Karim Mansour

5:47 AM

thank you @Fargle

Karim Mansour

6:00 AM

0

Q: Argument verification

Determine all of the accumulation points of the following sets in $R^1$ and decide whether the sets are open or closed or neither. e)All the numbers of the form $2^{-n} + 5^{-m}: (m,n = 1,2,...,n,....) = X$ Claim: The accumulation points are the following $\{\frac{1}{2^n}: n \in \mathbb{N}\} \...

real-analysis general-topology

@TedShifrin

iwriteonbananas

8:23 AM

@BalarkaSen hello

Transcript for

$Mathematics$ Mathematics