Mathematics - 2015-03-05 (page 1 of 3)

David Wheeler

12:01 AM

@MaryStar what happens to $x$ and $y$ for a constant $\theta$?

Mary Star

They are also constant, right? @DavidWheeler

David Wheeler

No...because $r$ is not constant.

Mary Star

From the relation $\theta=\arctan (\frac{y}{x} )$, we have that $\arctan (\frac{y}{x} )$ is constant, right? But what else can we say? @DavidWheeler

David Wheeler

Instead of thinking of $\theta$ as an arctan, think of it as an angle.

To simplify for the moment, ignore $z$, and just think of what points at a constant angle look like in $\Bbb R^2$

infinitesimal

Also think about what it means for a variable to vary :-)

David Wheeler

12:12 AM

For example, suppose $\theta = \pi/3$, what "shape" do we get as $r$ varies from $0$ to $+\infty$?

Maximilian

Does anybody know about harmonic motion (in Trig)

Semiclassical

i suspect yes, but it's not entirely clear what you mean by that

Maximilian

to me right?

Semiclassical

right

Maximilian

Ok.. Well I'm sure its really easy I just didnt really pay attention and its like way off of what we've learned and a very small part of trig that were learning

The first one is "Find a function s given by s(t) = a cos wt that models the displacement of the weight

Semiclassical

12:16 AM

ahah, physics

Maximilian

and the information I get is s(0)=5in; P=1.5 sec

Semiclassical

simple harmonic motion, yeah

Maximilian

ya.. I don't know physics or anything. Its just a small section of a chapter, only 2 homework problems so its not big.. I don't like it

so uh.. what would I do?

Semiclassical

you should have some other piece of information, like the weight being initially at rest

Maximilian

Ya, only s(0) and the period

but nothing else

Semiclassical

12:17 AM

well, what's the precise statement they make?

Maximilian

oh, initial position is s(0) so its 5in

Semiclassical

actually, i need to go eat dinner. back in a bit

infinitesimal

later pal

Maximilian

I think i have to figure out the amplitude and other information to get the equation

So.. amplitude (a) = 5 I think, then reverse the period to get omega to make the equation?

Mary Star

Is it maybe a line? @DavidWheeler @infinitesimal

Maximilian

12:25 AM

I don't know if I did it right but I think its $s(t)=5cos\frac{\pi}{0.75}t$

Semiclassical

back. that works on the assumption that the weight was pulled 5 inches away from the spring, and then released at $t=0$

Maximilian

I think, the book said the initial was 5in

and then said try it at s(1)

Semiclassical

in any case, the problem isn't solvable unless you make that assumption (so pretty much have to)

do you have any familiarity with derivatives? if so i can point out something neat in this context, but without that it'd only be a distraction

Maximilian

12:43 AM

Nope

Semiclassical

kk

Maximilian

:P I don't know much, its so tiny that I doubt it will even be on a test, or if it is it will just be like the problem I just did. Ill ask my friends when they get to school since they have a answer book I guess

Thank you though!

Semiclassical

glad to help

David Wheeler

@MaryStar Not quite-it's a ray (remember $r \geq 0$).

mixedmath

1:03 AM

@MikeMiller you might be amused to know that my script is working just fine over in the Crusader's room

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Let's see what color/pattern my avatar has changed to today

David Wheeler

still looks red to me

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

I guess there is a delay of some sort for chat

Ted Shifrin

1:34 AM

@David: Only a ray in polar coordinates. She's having trouble understanding free variables ...

David Wheeler

She said cylindrical...but, sorta the same

Ted Shifrin

Sorta. It's not a ray in cylindrical. It's a half-plane.

David Wheeler

yes, I know, but I was ignoring $z$.

Ted Shifrin

That's exactly what is confusing her repeatedly. Please don't ignore $z$.

David Wheeler

Argh...taken out of context.

Ted Shifrin

1:38 AM

Teaching is hard work :D

David Wheeler

A cylindrical tube is just (essentially) $S^1 \times \Bbb R$

Ted Shifrin

LOL, you don't need to tell me :)

David Wheeler

Slices through any plane $z = c$ are circles.

Ted Shifrin

If you scroll up enough, you'll see that she doesn't understand that variables not appearing in equations are free rather than bound to be some constant that she picks (typically $0$).

David Wheeler

She didn't even understand the effect of holding $\theta$ constant in a plane, she wanted to keeping computing $\arctan(y/x)$.

Ted Shifrin

1:42 AM

I know :(

But there's a point where we need to tell people that they have to figure things out for themselves or get their teacher to help.

I know what (most) of my students don't understand because they come ask me. If they got perfect homeworks by getting people to do it here, then they'd fool themselves and fail the tests.

David Wheeler

Yes, if you say $z =c$, for example, it's hard to see that you get a translate of the $xy$-plane

Because $x$ and $y$ aren't even mentioned

Ted Shifrin

Once again, it's a question of free variables ...

Linear algebra certainly beats that into people.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Okay, so my avatar changed to green this time

Ted Shifrin

mr eyeglasses, why are you so variable?

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Am I the only one who gets randomized avatars every day?

Ted Shifrin

1:46 AM

Mine's been the same since I got here ages ago. Maybe it's time for me to disappear :P

David Wheeler

@ʙᴀᴅᴀᴛᴍᴀᴛʜ I have no idea

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Every day around 8pm EST or so, my avatar changes

David Wheeler

@TedShifrin Well you can't (easily) draw pictures here.

Ted Shifrin

you don't change email addresses, do you, mr eyeglasses?

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

@TedShifrin No

I always use my school one

Ted Shifrin

1:48 AM

One of many reasons this isn't the ideal way to teach, @David, except for symbolic manipulation (e.g., the way most people do algebra :D)

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Well, I guess I will change it when I change schools

Mike Miller

and the way some people do differential geometry, @Ted

Ted Shifrin

are you casting stones at I, @Mike?

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

I am going to take differential geometry next semester

Mike Miller

No, @Ted, at Peter.

Ted Shifrin

1:49 AM

my students were drawing all sorts of pictures all over the board today, working on asymptotic curves and lines of curvature

hell, I teach ring and group theory with lots of pictures ... I'm just weird.

Mike Miller

I learned from Milnor's Morse theory book almost everything that Peter had taught, but presented better :P

Ted Shifrin

well, I have lots of pictures in my undergraduate notes, @Mike, but I draw more in class.

Jacob told me that Peter gives my notes as a reference in his undergraduate notes because he has no pictures and I have lots.

Mike Miller

Haven't seen his undergraduate notes; can't comment.

Ted Shifrin

I like mine better :) big surprise.

weird, mr eyeglasses, on my profile page it looks like you have to click and choose to actually change the picture.

David Wheeler

@TedShifrin I get what you're saying, and I understand you had a notion of how to help her understand-frankly, I just gave up after a while.

Mike Miller

1:53 AM

damnit, i was wondering when i left this comment...

Ted Shifrin

well, there are a few people who come here for too much help, and I think we actually help most by not answering and telling them to figure it out for themselves.

Part of my unhappiness with the whole site is that there's too much just spoonfeeding of complete answers.

David Wheeler

It's a weird problem to have-a lot of people come here seeking help, and what "they" think is help is "getting the answer", and what 'we" think is help is "getting them to understand how to get the answer"

Ted Shifrin

@Mike: Well, I think it's appropriate to ask even the 10yr old, "Where did you get stuck?" and not just put the answer for him/her to copy.

The problem is that those of us who understand what teaching is form a minority here. And I get chewed out for having such narrow ideas of what teaching is.

Mike Miller

@TedShifrin: I was wondering when I left it because I didn't leave it.

Different Mike Miller.

Ted Shifrin

oh, the other Mike Miller :P

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

1:57 AM

Mike Miller trolled by Mike Miller

Ted Shifrin

Other than @Hippa's creations, I think I'm still unique.

David Wheeler

I have seen many people ask: "is my answer correct', wanting validation of their PROCESS by the RESULT.

Mike Miller

Ted Shifrin is a less common name than Mike Miller, so you can get away with it that way, @Ted.

Ted Shifrin

Yes, my students often get a correct answer the wrong way ....

David Wheeler

I see things exactly the opposite-I'd rather know my process is right, even if my answer is wrong.

Mike Miller

1:58 AM

There's a Stephen Miller coming here to speak on Monday.

Ted Shifrin

Are you going to troll him, @Mike?

Of course, @David, and so would most of my students.

Mike Miller

@Ted: It's not something I'm interested in attending. :P

Ted Shifrin

Ah

There was an algebraic geometry talk today I would have liked to have gone to (Claire Voisin, a star), but I had a mob coming to office hours, so I couldn't.

David Wheeler

Of course, at a more basic level, I don't care as passionately as some people do-there's very little at stake for me, personally.

Ted Shifrin

@David: There's nothing at stake for me, other than trying to be a good person here. And I hope to influence a few people in knowledge and mathematical taste. @Mike is proof that I've failed :D

David Wheeler

2:01 AM

Well you certainly deal with more students than I do. @TedShifrin

Ted Shifrin

Ah, but I'm retiring in two months, @David.

Some of my students are jubilant :P

Mike Miller

Ah, I certainly know Voisin's name. When I was thinking of doing algebraic geometry her book was going to be my summer project...

Ted Shifrin

Yes, I was sad that I couldn't go, @Mike.

David Wheeler

@TedShifrin Are you hard on the poor lil things?

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

@MikeMiller Are you going to Talbot in April

Ted Shifrin

2:02 AM

Interesting, @Mike, my student/advisee who got into Wisconsin and Colorado (and was rejected by UCLA and Berkeley) is now being told that he might have a chance at StonyBrook. I really don't understand the game any more.

Mike Miller

I considered applying but ended up not doing so.

Ted Shifrin

Very, @David. I've been notorious for that for 36 years.

Mike Miller

@Ted: You know I have no comprehension of it.

Ted Shifrin

oh, probably including my years as a grad student, too, so more.

It just seems weirder than in my day or when I was advising students 20 years ago, @Mike.

David Wheeler

Hello, class. This is your final exam. I'll be back next semester to let you know if you passed. Good luck!

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

2:04 AM

How does one gauge which PhD programs one has a chance of being accepted into?

Ted Shifrin

I'm hoping some of my more clueless differential geometry students (who may have all C's in all their courses up to this) drop. Sadly, they shouldn't have gotten this far with a major.

Mike Miller

Doesn't seem like one can, @ʙᴀᴅᴀᴛᴍᴀᴛʜ

Ted Shifrin

A lot depends on the advanced GRE exam, mr eyeglasses, but my point is that it's become more or less a crapshoot.

Mike Miller

PS: I think you should change your name. You're not bad at math. Don't ingrain that thought into yourself.

Mary Star

I have a question... Do orthogonal coordinates mean cartesian coordinates?

Ted Shifrin

2:05 AM

No, @MaryStar.

Polar coordinates or cylindrical coordinates are orthogonal coordinates, too.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

So should I just apply to every grad school that looks interesting? Because I was thinking to just not wasting my time applying to schools like Berkeley, UCLA, etc.

Mike Miller

Cast a wide net and talk to your recommenders.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

So far I only have one professor willing to write a recommendation :(

Ted Shifrin

Well, some recommenders are not so helpful these days, @Mike, and you're seeing that even I — who want to be — turn out to have no clue.

Generally, mr eyeglasses, you should ask the ones who teach you the most advanced courses, even if you do slightly worse. Make sure you go to office hours and that they get to know you.

g'day, Committing.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

@TedShifrin The issue is that a lot of my professors don't really like to bother with students

Committing to a challenge

2:09 AM

@TedShifrin Why do you want them to drop?

@TedShifrin And hello!

Mary Star

@TedShifrin I found it at the following exercise I am looking at:

We are given the following point in cylindrical coordinates. Write it in orthogonal and spherical coordinates.
The point is $\left (2, \frac{\pi}{2}, -4\right )$.

I have done the following:

The cylindrical coordinates are of the form $(r, \theta , z)$, that are defined by $x=r \cos \theta , y=r \sin \theta , z=z$.

The orthogonal coordinates are of the form $(x, y, z)$.

$x=r \cos \theta=2 \cos \frac{\pi}{2}=0 , y=r \sin \theta =2 \sin \frac{\pi}{2}=2 , z=z=-4$

Ted Shifrin

Because they can only get the most routine exercises with my helping on almost every step, @Committing, and they cannot seem to learn the definitions. A few got solid F's on the first exam, and the next exam and final are harder.

Committing to a challenge

@TedShifrin You want them to drop for their benefit or the benefit of the other students(or both)?

(Or neither)

Conor O'Brien

Hey y'all: quick question. How might I represent x*(x-1)*(x-2)*...*(x-n)?

Ted Shifrin

I don't enjoy giving F's to students, @Committing, and I see very little likelihood of anything else. Even with one of my Honors advisees, who needs my class to graduate, I've begged him to come get help in office hours, and he doesn't.

Mike Miller

2:11 AM

that seems like a great representation

Committing to a challenge

@ConorO'Brien Big pi product

Ted Shifrin

LOL @Mike

@MaryStar: I have no idea what 'orthogonal' means there. They should say cartesian.

Mary Star

Ok...

Committing to a challenge

@TedShifrin An honors student who doesn't care???

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Is there a popular month when applications to PhD programs are usually due by? (For the Fall semester)

Ted Shifrin

2:13 AM

He cares, @Committing. I think there are some mental health issues, but I've told him every week that I'm here to help him. His homeworks are terrible (as if he tries only a half hour and gets 1 or 2 out of 5) and he failed the first exam.

Committing to a challenge

@ConorO'Brien $\pi (x-n)$ and set your n?

Ted Shifrin

Usually by mid-December, mr eyeglasses.

Conor O'Brien

@Committingtoachallenge er... "set your n"?

Ted Shifrin

That's bad, @Committing. You mean $$\prod_{j=0}^n (x-j).$$

Committing to a challenge

@ConorO'Brien It was meant to be a hint :P

@TedShifrin It was meant to be a little hint haha(not the answer, and I don't have latex)

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

2:14 AM

Okay...there's 2 potential professors teaching classes I will take next semester, but I don't know if they'll be willing to write letters before the semester ends in late December

Ted Shifrin

Your use of $n$ just made it confuzling, @Committing.

Committing to a challenge

@TedShifrin Fair enough

Conor O'Brien

@Committingtoachallenge OH Gotcha ;) something like $\prod _{a=n}^xa$?

Ted Shifrin

Well, for many schools that's enough time, mr eyeglasses.

Committing to a challenge

Oh whoops I just rendered the latex and yeah it looks like something else completely aha

Ted Shifrin

2:15 AM

Plus little $\pi$ looked like number theory, not product, @Committing. But I'm done criticizing you ... for now :P

Committing to a challenge

@TedShifrin $$\large\color{green}{\checkmark}$$

Ted Shifrin

LOL

Committing to a challenge

Someone was adding that to their answers, and I love it

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

@Ted Should I just try to impress those 2 professors during September, October, and November, and then ask them if they would write me letters by December?

Committing to a challenge

I'll be back in 5 minutes if noone logs my computer out (>.< at uni)

Ted Shifrin

2:18 AM

If you feel comfortable with them, mr eyeglasses, you should ask them for advice on applying altogether ... and ask if they'd be willing to write, as well.

Clarinetist

Hello everyone. :)

Ted Shifrin

heya @Clarinet

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

I thought it would be an issue if I asked a professor for a recommendation letter before they input final grades

Ted Shifrin

well, it's only a problem if you bomb the final and they can't write a good letter, mr eyeglasses

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

But what if the final is after applications are due?

Clarinetist

2:19 AM

Working on a complete career change. Been fishing and hoping for a bite!

Mike Miller

I got a recommendation the quarter I applied from someone I met that quarter

Ted Shifrin

well, some apps are definitely do beginning of January, mr eyeglasses. You need to start exploring for options ... and since you can't afford too many schools (it isn't cheap), you need to be wise about your choices.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

We usually get our final grades by the beginning of January

Ted Shifrin

You're becoming a fisherman, @Clarinet?

Clarinetist

Lol. :P Nah, looking into statistics positions and am awaiting a grad school decision (M.S.). Saw my undergrad professors last Saturday. The whole Wisconsin situation is really hitting them hard.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

2:22 AM

Well, if all the schools I apply to don't think I'm good enough for some sort of stipend (not expecting a fellowship obviously) then I don't think it will be worth it to go to grad school, I guess

Ted Shifrin

Yes, @Clarinetist, much about this wonderful country is so disappointing these days.

Yes, mr eyeglasses, of course. You need to apply to schools where you'll have a good chance to get supported (so perhaps no top 25). But I meant that the applications themselves are not cheap. :(

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

@Ted I thought I could get application fee waivers

Ted Shifrin

Most likely you will have to TA. Very few fellowships around.

I actually don't know how that works, mr eyeglasses. Perhaps you can. Do websites say what you have to do to qualify?

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Yea, I found some and I meet the requirements

Ted Shifrin

Awesome news.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

2:29 AM

How can I tell which schools I have a shot at getting supported?

Ted Shifrin

As I was saying earlier to @Mike, I don't read the game as well as I used to. But it depends on your grades, what courses you've taken, your GRE scores, and the content of your letters. So there are a lot of basic unknowns.

Clarinetist

I highly recommend the Mathematics GRE Forums in your searching.

Alex Wertheim

What Ted said. Professors who know you well can also help.

Ted Shifrin

mr eyeglasses is less fortunate than you were in terms of "useful" professors, @AlexW

Alex Wertheim

Oh really? :(

That may present other problems in the graduate applications process as far as letters go...

Committing to a challenge

2:32 AM

Why the name mr. eyeglasses? Is this an old username?

Or the text is hard to read

Ted Shifrin

yeah, we've been discussing that, @AlexW

Committing to a challenge

So you need glasses

Conor O'Brien

G'night, all!

Ted Shifrin

no, his avatar used to look like eyeglasses (although Mike insists it was eyebrows)

night, @Conor

Committing to a challenge

@ConorO'Brien Night

Mike Miller

2:33 AM

He used to be ಠ_ಠ

Committing to a challenge

@MikeMiller Ahhhh Nabla

Mike Miller

the things curled up on top of the eyes are eyebrows

Alex Wertheim

That's a tough spot. The application process is difficult enough as it is... it doesn't need to be made more difficult.

David Wheeler

@TedShifrin when we were talking about "where we put the inverse" on change-of-basis -this is what I meant by what I do: math.stackexchange.com/questions/1176027/…

Committing to a challenge

@MikeMiller LOL

Ted Shifrin

2:33 AM

the eyes = eyeglasses :D

@David: To be honest, I don't recall the conversation :)

Mike Miller

Ted Shifrin

@Mike, just as you're stubborn about your bad syntax, I'm stubborn about mr eyeglasses. :)

David Wheeler

@TedShifrin Well, memory is funny thing-some things stick, some don't

A Beautiful Mind

Hi!

Committing to a challenge

I have a lecturer who is also insanely intense in workload

Ted Shifrin

2:36 AM

I'm old, @David. My brain is too cluttered :)

A Beautiful Mind

I am old too.

Ted Shifrin

In hindsight, you usually end up liking such professors the most later on, @Committing.

no, Jasper, you're still a young'un.

David Wheeler

alright, alright, I suppose me being only 53 gives me an unfair advantage

Clarinetist

So it was just depressing, going back to WI to visit profs for the first time in about 9 months: every single one of them, it was all budget talks. :(

Ted Shifrin

damn right, @David :)

Committing to a challenge

2:36 AM

@TedShifrin He gave us a 6 question assignment due in 6 days today

With multiple parts each

Ted Shifrin

Depends on how involved/hard the 6 questions are, @Committing. Could be horrific, or could be a few hours.

Paul Plummer

3

Q: The sum of orbit size of some element over the image of group "polynomial"

$\DeclareMathOperator{\orb}{orb}$ Say I have a group "polynomial", $p$, on $S_n$, that is $p(x)=a_1 x^{\epsilon_1}...a_n x^{\epsilon_n}$ for all $x \in S_n$, fixed $a_i \in S_n$ and fixed $\epsilon_i \in \mathbb{Z}$. Let $p S_n$ be the image of the polynomial. Generally, I am looking for $$ X(S_n...

Alex Wertheim

Ain't that the truth.

David Wheeler

@Clarinetist srsly, f- money

A Beautiful Mind

If you want to say fuck, just say it. This is a free chat.

Committing to a challenge

2:39 AM

@ABeautifulMind Not true, you can get banned(I have been[for saying s***])

Ted Shifrin

well, the Board of Regents in my wonderful state has just decided (now that I'm retiring) that they will no longer provide secondary health insurance once I get on Medicare. That is just changing, policy not yet truly announced. Pfeh on politicians welching.

David Wheeler

I do, but once I start, I'm afraid I'l never stop

A Beautiful Mind

@Committingtoachallenge So be it.

Clarinetist

Not to mention that there are talks of removing tenure :/

Ted Shifrin

I was banned for "bitch," as I recall. Used as a verb, not as a noun.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

2:39 AM

@JasperLoy Hi

Committing to a challenge

Incoming ban for Jasper I imagine :P

Ted Shifrin

We have had post-tenure review for 20 years or so here to appease them. It has gotten rid of the most incompetent.

Mike Miller

You can search the number of times I've said fuck in this room; it's highly nonzero.

Paul Plummer

Some advice on my question, or some upvotes so it get more attention, or a critique so I can make the question better would be appreciated.

A Beautiful Mind

Maybe they give me more leeway since I am mad.

Committing to a challenge

2:40 AM

wt*, I got banned for S*** and you guys don't get banned for f***

Mike Miller

It's bullshit sometimes, @Committingtoachallenge

David Wheeler

One of my pet peeves is the unrelenting greed I see all around me. As if nothing had value BUT currency.

A Beautiful Mind

Like I said above.

Clarinetist

Post-tenure review... hmm. I've never actually heard of that

Ted Shifrin

Only the rich deserve human rights, didn't you know, @David?

A Beautiful Mind

2:42 AM

Do professors in the US say fuck in the classes usually?

Ted Shifrin

NO, Jasper.

David Wheeler

@TedShifrin Humans deserve human rights. It's practically a tautology.

Alex Wertheim

I don't think so, unless they have a particular reason to...

Mike Miller

But maybe they should start.

A Beautiful Mind

OK. I thought yours is a free country.

Ted Shifrin

2:43 AM

I got called out on student evaluations many years ago for saying "damn." We live in the religious South.

David Wheeler

@ABeautifulMind Right-wing propaganda

A Beautiful Mind

Needless to say, mine is a very unfree country, just one level above Muslim countries.

Alex Wertheim

It's funny, @Ted. The research triangle is relatively progressive.

But go just a few miles out of it...

Ted Shifrin

Yeah, @AlexW, one reason I'm returning to my home state, although it's full of plenty of conservatives and bigots, too.

A Beautiful Mind

Here in the army, we are not supposed to say fuck, but it is very common. So it is not too bad.

David Wheeler

2:45 AM

@TedShifrin I'm surprised math doesn't get cut from your curriculum on the basis that you teach $\pi \neq 3$, contrary to scripture.

Ted Shifrin

Alabama passed a bill declaring $\pi = 22/7$ some years ago.

A Beautiful Mind

@TedShifrin What is this?

Ted Shifrin

our brilliant lawmakers, Jasper

David Wheeler

oh, geez, don't get me started

Committing to a challenge

I thought $\pi = 3$

A Beautiful Mind

2:48 AM

Now pi = Pizza.

David Wheeler

@ABeautifulMind Haven't you read the internets? Everyone knows $\pi = 4$.

Ted Shifrin

well, I'm gonna call it a night. You all take care. Keep me posted, mr eyeglasses.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Night @TedShifrin

Alex Wertheim

Goodnight, @Ted.

Mike Miller

Morning, @Ted

The Emperor of Ice Cream

2:52 AM

I know a function from $\mathbb R $ to $\mathbb R$ restricted to $\mathbb Q$. Can there be more than one way to extend it to $\mathbb R$ so it is continuous?

Is there a name for this result?

A Beautiful Mind

@TheEmperorofIceCream No.

Committing to a challenge

@TedShifrin Night!

David Wheeler

@TheEmperorofIceCream Not under the usual (metric) topology. It's a consequence of the density of the rationals in the reals.

A Beautiful Mind

@TheEmperorofIceCream Take a sequence of rationals that converges to the real number. f(lim x_n)=lim f(x_n) by continuity.

@DavidWheeler Your answer is too hard for the OP, lol.

The Emperor of Ice Cream

thanks jasper

David Wheeler

2:55 AM

@ABeautifulMind what answer?

A Beautiful Mind

@DavidWheeler What you just said, lol.

David Wheeler

It's like this-every real number has a rational number arbitrarily nearby.

anorton

Hi everyone!

David Wheeler

continuity ensures preservation of nearness

lately, though, I've been having doubts about the ontological foundation of real numbers

The Emperor of Ice Cream

How can I prove INT(CL(INT(CL(A))))=INT(CL(A))?

in any topology

A Beautiful Mind

3:03 AM

@TheEmperorofIceCream This is a very interesting question. I did it over a decade ago, lol. Prove that each is a subset of the other, lol.

anorton

I have a quick group theory question if anyone is interested... I'm trying to find how many distinct $a$ satisfy $a^{65} = 1$ in the cyclic group of order 52. I believe it is three: the element of order 13, the element of order 4, and the identity. Is this correct?

(I'm preparing for a test, and myself and a classmate have gotten ourselves confused...) :-)

The Emperor of Ice Cream

Thanks, I think it also works if I switch CL for X-INT(X-A)

@anorton No

an element of order $4$ won't satisfy what you want.

Moreover the quantity of elements of order $n$ is not always one.

Try to prove there are $\varphi(n)$ elements of order $n$.

David Wheeler

65 - 52 = 13, so every element of order 13, and plus the identity.

There is only one subgroup of order 13, so that accounts for all of them.

13, 12 elements of order 13, and 1 of order 1.

anorton

Oh! thanks. I can't believe I did that...

David Wheeler

A cyclic group has one, and only on,e subgroup of each divisor of the order of the group.

If the divisor is prime, every non-identity element of that subgroup will have the same prime order.

In this case, if the generator of your cyclic group is $x$, then the powers of $x^4$ are the elements you seek. It's pretty easy to verify there's 13 of them (including $x^{52} = e$), and that the order of $x^{4k}$ is, in fact, 13, for $k = 1,2,\dots,12$

anorton

3:16 AM

That makes sense, thanks. For some reason, I wanted to extend the special case that we discussed in class (which was elements such that x^2 = 1) to the general problem of (x^a=1). In the former, there is a unique element of order 2; I forgot that it isn't always a unique element. (And why I wanted to include elements of order 4 I have no idea.)

:)

David Wheeler

you knew that 52 = 13*4, you just forget which number went where.

anorton

^ yeah.

David Wheeler

it's sort of tricky: cyclic groups are abelian, integer multiplication is abelian, but...divisibility is not.

so it matters which is the divisor, and which is the quotient-something often trips people up.

The Emperor of Ice Cream

@ABeautifulMind INT(CL(INT(CL(A))))=INT(X-(INT(X-(CL(INT(A))))))=INT(X-(INT(X-(X-(INT(X-INT(A)))‌))))=INT(X-INT(INT(X-INT(A))))=INT(X-INT(X-INT(A)))=INT(CL(X-INT(A)))=INT(CL(CL(A‌)))=INT(CL(A))

anorton

The meta problem I'm having right now is that I'm in a number theory course where the professor likes group theory, so he takes the (ofttimes elegant) group-theoretic approach to prove problems in number theory.
But, I've forgotten way too much group theory since last semester! :)

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

3:20 AM

@anorton mine does that too

He proved the fundamental theorem of arithmetic from group theoretical point of view

The Emperor of Ice Cream

@anorton thats awesome

anorton

Don't get me wrong--I really like it, and I think the group theory approaches are really insightful. It's just that when it comes time for me to prove something on my own, my group theory starts to fail. :P

David Wheeler

I think for some people groups are "too little structure". They're more used to rings and fields.

The Emperor of Ice Cream

3:55 AM

Studying groups is certainly usefull without asking more structure,

Bib

Can someone confirm for me that all finite index subgroups of $\mathbb{Z^n}$ are $(m\mathbb{Z})^n$? Or am I wrong?

David Wheeler

@TheEmperorofIceCream Yes, it is, but the "looseness" of the structure makes some things "harder to prove"

David Wheeler

4:21 AM

pin drops

Mike Miller

4:35 AM

In what sense do you mean that, @Bib? Does $\langle (1,0), (0,2)\rangle$ count as your sort of thing? Does $\langle (1,1), (0,2)\rangle$?

There is nothing of the form $(n,0)$ with $n$ nonzero in the latter subgroup, but it's certainly finite index.

Wait, I lied.

$(2,0)$ is in it. But $\langle (2,0), (0,2) \rangle$ is not that subgroup. It's index four; that one is index 2.

Bib

I knew something was fishy!

Mike Miller

The only subgroups of index 2 of the form $\langle (n,0), (0,m)\rangle$ are $(n=1, m=2)$ or $(n=2,m=1)$, and this is neither.

Bib

By "this", what do you mean?

Mike Miller

$\langle (1,1), (0,2)\rangle$.

Bib

Oh, I see what you mean now.

So there are more subgroups of finite index.

Mike Miller

4:41 AM

Right. Of course, quotient of such things are still of the form $\bigoplus \Bbb Z_{k_i}$ for some finite sequence of integers $k_i$ (by the classification of f.g.a.g.s).

Bib

f.g.a.g.s... you mean the special case of structure theorem?

The Emperor of Ice Cream

no, he just wrote the first g by accident

I think it should be FTFGAG

Fundametal Theorem for Finitely Generated Abelian Groups

Bib

I suppose you could also refer to it as the classification of finitely generated abelian groups

Mike Miller

I meant the second thing you said, @Bib

but as long as everyone knows what I meant, who cares :)

David Wheeler

I read it as: fear the f- gag, which made absolutely NO sense

Mike Miller

4:54 AM

are you suggesting that one should NOT fear the f-gag?

The Emperor of Ice Cream

what is the f-gag?

Mike Miller

something worth fearing to be sure

The Emperor of Ice Cream

like 9-gag but R-rated?

David Wheeler

@TheEmperorofIceCream insult a moderator, and find out :P

The Emperor of Ice Cream

@PedroTamaroff = bad

David Wheeler

4:57 AM

lol, nice knowing you

The Emperor of Ice Cream

^^^^

Howcan

5:29 AM

Hi guys, how would I interpret {(x,y,z): x=7t-1, y=2-3t, z=t | t is real}? I know it's a set of lines, but I have to use this to check whether the set is a subspace of R^3.

David Wheeler

you should see it as $(x,y,z) = (-1,2,0) + t(7,-3,1)$

Howcan

Ahhh okay, I am familiar with that form, thank you!

So now I would check if the set of lines is closed under addition, scalar multiplication, and contains the zero vector for all real numbers t?

David Wheeler

5:47 AM

Check to see if it has the 0-vector first-it's the easiest check.

And it's not a set of lines, it's the subspace $\langle (7,-3,1)\rangle$ translated (offset) by the vector (-1,2,0).

A Beautiful Mind

6:09 AM

@DavidWheeler Still not sleeping? I went out for lunch and a walk.

infinitesimal

How are you @ABeautifulMind?

A Beautiful Mind

@infinitesimal So so. I may need to do certain things to get well, but I am not sure I have the courage to do them. I won't elaborate here, but you get the idea.

@infinitesimal It doesn't require me to walk naked in public, or to do anything illegal or dangerous, but lots of people will know that I am crazy if I do those things.

infinitesimal

I see @ABeautifulMind

A Beautiful Mind

@infinitesimal Don't say just do them. Most people won't be able to do them. So either I must do them somehow or convince myself I don't have to do them.

@infinitesimal So I will wait till I have a very stable mind and then decide what to do, and if I have to do them, I pray for the strength to do them. This is mostly OCD related by the way.

infinitesimal

Okay ...

... why wait @ABeautifulMind?

A Beautiful Mind

6:16 AM

@infinitesimal Well, because I cannot think clearly now.

@infinitesimal I know this all sounds very strange, but if you know everything that happened in my whole life, you would understand it, but it is not possible to describe that now.

infinitesimal

"Clear" in what sense? @ABeautifulMind

A Beautiful Mind

@infinitesimal In the normal sense, lol.

infinitesimal

I don't think the "normal" sense applies to mental illness, but I could be wrong :P @abe

A Beautiful Mind

@infinitesimal Maybe you are right. You asked a very difficult question.

infinitesimal

I know pal @abe that's why I asked

Food for thought.

A Beautiful Mind

6:22 AM

@infinitesimal When you have OCD, you suffer tremendously from strange thoughts people don't understand, so they don't know you are suffering.

Howcan

Sorry to interrupt, and I know I just asked a very similar question, but I was absent for a long time and am catching up now. How would I interpret {(x,y,z): x=s, y=s+t, z= -2s+2t | s,t are real}

infinitesimal

I would interprete what is inside the braces as a set.

Howcan

Oh I know that, I mean similar to the answer David Wheeler before.

A Beautiful Mind

@Howcan Write (x,y,z)=(s,s+t,-2s+2t) first.

infinitesimal

@Howcan^

A Beautiful Mind

6:26 AM

@Howcan Then write that as s(1,1,-2)+t(0,1,2).

Howcan

Ahhhhh okay it all makes sense now. Thanks so much!

Can't believe I didn't see that.

A Beautiful Mind

@Howcan This is a plane passing through the origin and spanned by (1,1,-2) and (0,1,2).

@Howcan Learn the technique and you can do all questions of this kind. This is a simple question which you should master.

Howcan

Yeah I'm fine with proving that something is a subspace, I just completely missed class when we did things with lines, planes, etc... in that form. Thanks again!

A Beautiful Mind

@infinitesimal I am going to bed, good night.

infinitesimal

Later pal

There's no need to apologize for interrupting @Howcan this is a math chatroom after all

:-)

Transcript for

$Mathematics$ Mathematics