Mathematics - 2015-03-04 (page 1 of 5)

but, an attempt: suppose i have an orientable surface, with some well-defined notion of NWSE. how many solutions exist for the returning explorer problem?

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

My avatar changes every day for some reason

I guess it's kinda cool

Semiclassical

12:29 AM

i imagine there's a way to make "a well-defined notion of NWSE" into a precise idea, but i don't know what it'd be :/

i'd guess it's just contained in the metric itself, but eh

David Wheeler

@Semiclassical It's an interesting question. My thought is look at a torus, first.

infinitesimal

Hmm...this could make a good question to post on the main site.

2

Semiclassical

was about to say

i'm not going to be able to do it right now, though, since i'll be occupied in about 10 minutes

so if anyone else wants to put it together, please do so! :)

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

@ɧɿρρԹʅȝՇԵՐՎԾՌ nice username

infinitesimal

Thanks for sharing @Semiclassical :-)

Semiclassical

12:42 AM

glad you like it, heh. i'd really be curious what classes of solutions could exist on a different manifold, so i hope there's some known results

i mean, the solutions for the sphere can be classified by how many times the solution winds around the south pole (with the north pole corresponding to zero winding, etc.)

does that kind of classification scheme break down when one goes to a different manifold? questions upon questions...

ok, time to go do non-math stuff

infinitesimal

Later pal.

A Beautiful Mind

1:04 AM

@ʙᴀᴅᴀᴛᴍᴀᴛʜ Bart.

Mike Miller

I love the red, @ʙᴀᴅᴀᴛᴍᴀᴛʜ

I hope you change your name soon. I don't think you're bad at math and it makes me sad to see you ingrain it

A Beautiful Mind

@MikeMiller I am going to restart meds.

Mike Miller

I hope it helps you.

I'm rooting for you just as Ted is.

A Beautiful Mind

But I need to think of what kind of therapy I need before restarting that. I don't like my previous therapist.

I think it takes a genius to understand a genius, lol.

Mike Miller

Therapy is worthless if you do not get along well with your therapist, I think. You do not feel you can share your problems with them.

David Wheeler

1:16 AM

Jasper, I hope you find a therapist who understands your desire to be productive in math, and is patient and kind.

A Beautiful Mind

Two psychologists I have seen have said some very stupid things to me.

columbus8myhw

Hello!

A Beautiful Mind

@DavidWheeler Maybe I should look for a woman. Women are usually kinder than men. Men are assholes.

Before someone flags me, I am a man too, lol.

David Wheeler

Sometimes they are. I try not to over-generalize. I mean, you're a man-you're not an asshole.

What do you do, most of the time @ABeautifulMind?

A Beautiful Mind

@DavidWheeler I walk in parks and shopping malls. I chat in SE rooms. I think about my mental problems, what caused them and how to resolve them. I read various things online. I play with conputers like installing different Linux distros. I watch some videos and listen to music online. I meet friends now and then.

David Wheeler

1:24 AM

How are you supported economically?

A Beautiful Mind

I live with my mum who is taking care of me.

I don't spend much money.

David Wheeler

Lol, I don't either. I have a possible job offer pending, but I've run into some documentation problems.

A Beautiful Mind

I am meeting my professor for lunch next Fri and my doctor next next Wed.

They just emailed me.

David Wheeler

Baby steps, Jasper, baby steps. I have faith you'll make it through.

A Beautiful Mind

My father moved out a few years ago. But the effects of what he had done remain.

To understand why it got so bad is something I have difficulty with myself.

But I even had some spatial disorientation.

David Wheeler

1:36 AM

Undoubtedly. My second step-father was terribly abusive. For years, I had feelings of rage and helplessness to deal with.

A Beautiful Mind

If my father died when I was born, my life would be very different.

He is a psychopath.

David Wheeler

I'd certainly edit my past if I could. It's probably more effective to try and edit one's future.

A Beautiful Mind

My professor has a son who is a psychologist. Maybe I will see him.

David Wheeler

When I was twenty, I was in (more or less) the "adult world", and I had to figure out "grown-up things". How to be a man? The only template I had for making my way was a terrible one (my step-father).

A lot of things took me much longer than usual to figure out.

Clarinetist

Just had my first performance review today in the business world. Eck. :/

David Wheeler

1:42 AM

They told you to give up the clarinet?

Clarinetist

Lol. They know nothing about my musical abilities or mathematical ones, for that matter :P

A Beautiful Mind

They know nothing.

Clarinetist

I got a subpar performance review because I apparently don't have enough experience to justify a good performance review, and my documentation is too detailed - despite it probably being the only source of documentation in this entire department...

Furthermore, my boss admitted to me that I have learned a certain old-fashioned programming language - despite only being there for 6 months - faster and with more knowledge of the language than those who have been in this department for 5-6 years.

David Wheeler

Uh...what kind of job?

Clarinetist

Actuarial.

A Beautiful Mind

1:45 AM

Seems to me your review should be great.

Detailed documentation and knowledge of programming are plus, not minus. Quit your job. Your boss is an asshole.

Clarinetist

It should be, but he gave no other reason to give me his subpar score other than that 1) my documentation was "not concise enough" (what do you expect from a step-by-step guide??), 2) he had only managed me for a month (my manager for the first 5 months left), and 3) I had only been there for 6 months.

David Wheeler

If your subpar performance review is based on lack of experience, their evaluation process is flawed.

Clarinetist

I'm really ticked.

Sure, the raise - 3% - it's okay, nothing impressive.

But I don't really care about that.

What really ticks me off is that my performance is being evaluated based on the 3 criteria I put above.

or so I'm told.

Criterion #1 had 1/13th of weight on the score, so honestly, #2 and #3 obviously mattered more.

David Wheeler

Here is my feeling: often new managers feel a need to "rearrange things" (because, you know, erectile dysfunction). He probably felt he could rate you low with no one objecting, so he did.

Clarinetist

And as much as I wish I could leave, I don't really feel that I have many options outside of actuarial at the moment :/

David Wheeler

1:50 AM

The bad news is, unless you have friends that out-rank him in the organization you work in, there's little you can do.

Clarinetist

Yeah, I have no one that can really support me :/

David Wheeler

Don't quit-contact an industry recruiter, and make discreet inquiries.

Clarinetist

Actuarial recruiters don't contact entry-level people.

David Wheeler

When you have another offer, THEN you quit.

Clarinetist

Oh, of course.

Searching has been very, very difficult.

I'm not quitting until I get something.

I just have no idea what my options are without a M.S.

David Wheeler

1:52 AM

Well, you have some OTJ experience. You might need more to get some offers, but you can still make inquiries.

Also, maybe he felt your documentation was "too wordy". That happens, sometimes. Less can be more.

Clarinetist

Maybe he did, but like I said... that accounted for 1/13th of the score. I'm more accepting of that reasoning than the other two reasons.

David Wheeler

Well, in another 6 months, 2 of those 3 reasons will no longer be valid. Also, that manager may not be there (he has a boss evaluating him, too).

Congratz on the raise, tho!

Clarinetist

Oh, thank you. I'm pretty satisfied with the raise; just peeved about the feedback I was given.

David Wheeler

Sometimes you just have to smile while thinking "screw you"

Jobs all have their associated BS, but, hey, ya gotta eat, right?

Clarinetist

Indeed.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

1:58 AM

@Clarinetist How long have you played the clarinet

Clarinetist

Oh man, I was serious for about... 11 years, have taken a break for a year now?

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Nice

You must be really good

Clarinetist

I'd like to get back to it someday. Studied with a really, really good teacher. :)

Hopefully in 6-8 months I'll be out of actuarial into something which 1) will encourage getting a graduate degree and 2) will not require me to do stuff outside of work...

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

If I knew how to play the clarinet, I'd play this youtube.com/watch?v=6Yl9tgkEbTg

Clarinetist

Of course, if any of YOU have any ideas... I would love to hear them. Already looked into research positions and data science positions. I could try an entry-level analyst position, but I don't think it can handle the higher-COL area I'm moving into...

Interesting clarinet solo there.

David Wheeler

2:02 AM

What kind of degree do you have?

Clarinetist

B.S. Mathematics, Statistics. I know some Python and really, really basic SQL on top of that.

David Wheeler

You might look into manufacturing estimating jobs, as well. It's a broader field, it can pay well.

Clarinetist

I have a ton of standard undergrad math books and a ton of Python, SQL, Web Design books, and two books on Java and C++. My issue is I don't have time to read these.

Ooh, will look into that

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

What's manufacturing estimating

David Wheeler

Well, it can vary considerably, depending on the type of manufacturing. But basically, you figure out how much major projects (new equipment, purchases, expansions, etc.) will cost.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

2:07 AM

Oh

David Wheeler

The people who actually pay for these things are notoriously bad at math.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

We have an operations research class but I didn't take it because I didn't think it'd be terribly interesting

David Wheeler

Part of what I do as a truss designer involves estimating projects for bid. It's not the best paying example, though.

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

I don't even know what a truss is

Clarinetist

Thank you @David. Another thing to add to my searches! :D

David Wheeler

2:09 AM

The "techincal term" is MPCWC: metal plate connected wood components

they are things that hold up roof and floor decks

there's also "truss-like" things: bar joists, I-joists, timber/log trusses

ʙᴀᴅ ᴀᴛ ᴍᴀᴛʜ

Ohhh

Sounds...like a fun job..I guess

David Wheeler

Depends..it can be, it can be drudgery

If I was good enough, I'd be a professional musician.

Sayan Chattopadhyay

@BalarkaSen I think I have an idea about the question . since the interval is [0,1] to [0,1] the only values which x can take is 0,1 .now since f Is a continuous function f(0)=0 and f(1),=1 therefore there are values for x such that f(x) =x
@BalarkaSen if it is wrong then please tell me how to do it and pls don't start ignoring me

David Wheeler

2:24 AM

@SayanChattopadhyay Is Balarka even here?

Sayan Chattopadhyay

I have pinged it@DavidWheeler

Well @david is what I have said right?

David Wheeler

I don't understand your reasoning at all, I'm afraid

Quinn Culver

@SayanChattopadhyay what's your question

David Wheeler

@SayanChattopadhyay Try this: look at the function $g(x) = f(x) - x$. What can you say about $g(0)$ and $g(1)$?

Sayan Chattopadhyay

The value is 0 and 1 for g(0), and g(1)

Alex Wertheim

2:30 AM

If this is the classical problem "Show any continuous map $f: [0,1] \rightarrow [0,1]$ has a fixed point, then no. Why are you sure $g(0) = 0$ and $g(1) = 1$?

Sayan Chattopadhyay

Therefore @DavidWheeler what I say is right except I don't mention g(x)=f(x)-x

David Wheeler

We don't know what $g(0)$ is. What we do know is at $x = 0$, that $g(x) = f(x)$, and thus $g(0) \in [0,1]$ (it's non-negative).

Alex Wertheim

Assuming, of course, that $f(0) \neq 0$, since we're done otherwise.

Sayan Chattopadhyay

Similarly with 1 right @DavidWheeler

David Wheeler

Similarly, at $x = 1$, we have $g(x) = f(x) - 1$

So $g(1) \in [-1,0]$.

Sayan Chattopadhyay

2:34 AM

But then it dosent come in the interval right@DavidWheeler

David Wheeler

Well, let's look at the "easy cases" first. Suppose $g(0) = 0$.

What can we conclude?

Sayan Chattopadhyay

f(x)=0

David Wheeler

Almost...we can conclude $f(0) = 0$, so $x = 0$ is the point we want.

Sayan Chattopadhyay

Not 1

David Wheeler

What happens if $g(1) = 0$?

If $g(0) = f(0) - 0 = 0$, then $f(0) = 0$, so for $x = 0$, we have $f(x) = x$.

Sayan Chattopadhyay

2:39 AM

f(1) = 1 and x=1

David Wheeler

if $g(1) = 0$, then, yes, $g(1) = f(1) - 1 = 0$, so that for $x = 1,\ f(x) = x$

So in those two "special cases", we can find an $x$ that works.

Sayan Chattopadhyay

Well @DavidWheeler I was right only the fact I didn't mention the equation g(x)=f(x)-x

David Wheeler

now if NEITHER of those things is true, then $g(0) > 0$, and $g(1) < 0$.

Sayan Chattopadhyay

But they are true

David Wheeler

No, I can easily construct a function $f$ so that won't be true of $g$.

Sayan Chattopadhyay

2:43 AM

So I was right according to the question

David Wheeler

For example, if $f(x) = \frac{1}{2}$, then $g(0) = \frac{1}{2} \neq 0$ and $g(1) = -\frac{1}{2}$

We are trying to prove this for ALL continuous functions $f: [0,1] \to [0,1]$

There's quite a few, so looking at particular examples won't be much help.

Sayan Chattopadhyay

But if we only look at the upper limit and the lower limit of the interval then they are continuous on them

David Wheeler

I don't know what you mean by that.

The interval $[0,1]$ isn't really that important. We could use $[3,4]$ instead.

Sayan Chattopadhyay

I mean let's say g(0) is not equal to 0 then by the indeterminate value theorem

David Wheeler

Bingo!

"intermediate"

That's exactly the right idea-if $g(0) > 0$ and $g(1) < 0$ then somewhere in-between, say $c$, we have to have $g(c) = 0$.

But $g(c) = 0$, means $f(c) - c= 0$, that is: $f(c) = c$, so $x = c$ is the point we're looking for.

And that trick: of looking at $f(x) - x$ is one worth remembering, because it gets used over and over.

David Wheeler

3:08 AM

+10 for killing chat

Julian Rachman

3:34 AM

Good morning/afternoon/evening (depending on where you are) everyone!

Chris's sis

I didn't sleep this night. It's almost 5:36 AM here.

Julian Rachman

Aw Chris! Shakes head

David Wheeler

Beunos Aires!

Oh dear, that's not right, is it?

Julian Rachman

nope!

Buenas noches

Buenos dias

Buenas tardes

etcetera...

David Wheeler

Ok, Good tardies!

(I may have to work on this...)

Julian Rachman

3:44 AM

I think you may have to :)

David Wheeler

In any case, fumbled greetments!

Julian Rachman

Haha. Well. This is just like a few hours ago....

David Wheeler

I would love to talk about math. But people only want to talk about "math problems".

Julian Rachman

I want to talk about math!

Did you know that on Facebook, AMS just posted something on undergraduate topology and advice and I was like, "they sense my presence..."

David Wheeler

LOL!

Julian Rachman

3:51 AM

I was so happy in the 5 min time frame to which I saw it :/

And you?

David Wheeler

I wish topology was done "in reverse order" in schools

Julian Rachman

Like what?

David Wheeler

Typically, you learn calculus-then multivariate calculus, and perhaps even complex analysis.

At this point, you're often deemed worthy of studying "metric spaces"

Finally, after ALL THAT, you can learn topology proper.

Julian Rachman

Oh. I am on the path of calc real analysis then topology

David Wheeler

But if you had learned topology FIRST-calculus would be so much easier

Julian Rachman

3:56 AM

Well. I have decided to learn from Simmons so that i get the foundation in metric spaces

David Wheeler

Take the IVT-big calculus theorem, gives students lots of trouble

Julian Rachman

Ya, I guess that would be interesting.

David Wheeler

it's got nothing to do with differentiation, or integration

Julian Rachman

Ya. I guess

David Wheeler

it's all about continuity....but continuity gives people FITS

they're like "NO! Not another epsilon-delta proof! I can't take this!"

Julian Rachman

3:58 AM

Ya. Well should write s book together called "reversed mathematics"

David Wheeler

Whereas if students had "spatial intuition" based on "topology without numbers"

Julian Rachman

It is sad how people in my school taking AP calc are doing it for Cred instead of the love of mathematics

David Wheeler

then having "epsilons" and "deltas" would be seen as "making life easier", not "harder"

Julian Rachman

Wait. Motivating calculus with topology. Hmm, I see an expository coming along

I am going to look into it. And see if I can find a good motivation.

David Wheeler

By the time you are able to tackle differentiable geometry, you can cut out almost half the course

Julian Rachman

4:00 AM

Half of what course?

David Wheeler

See, this is what schools do-they teach math historically

Julian Rachman

They do.

David Wheeler

But some "later developments" simplify and streamline the underpinnings

I feel math should be taught in order of complexity

Julian Rachman

Ya. If I were to, would you want to write something up with me on this topic? I am actually quite interested

Also, learning abstract algebra from pre-calculus

David Wheeler

I can look something over, but I can't promise enough time for an in-depth collaboration

Julian Rachman

4:03 AM

And I can tackle diff geo after I learn multi and what else?

David Wheeler

Depends on what you see as "end-goal interest"

Julian Rachman

OK. No prob. Since I am free a lot and especially today, I will try to write something up.

?

David Wheeler

And I don't know what you have studied

Julian Rachman

I want to learn algebraic topology and go somewhere from there.

David Wheeler

Well, algebraic topo has become quite a large field of study

Julian Rachman

4:05 AM

I have Studied everything traditional up to topology and excluding complex analysis and multi which I will be learning formally in two years

I want to find a field that is very small and warm with a lot of topics to study and "figure out"

David Wheeler

As I understand it, alg topo on the algebra side uses a lot of what are called "abelian categories"

These are things that "act a lot like abelian groups", i.e.; ADDING is important

Julian Rachman

Ah. I know sort of what an ability group is from abstract algebra and I guess I have to learn category theory?

David Wheeler

Not "deep category theory" (at least at first), just some basic concepts

Julian Rachman

So like with what I have learned so far and what I am currently learning, how should I go from where I am?

David Wheeler

Taken linear algebra yet?

Julian Rachman

4:10 AM

Well, I have been learning stopping, learning, stopping, and sort of haven't finish 100% of a course.

David Wheeler

Because linear algebra is sort of "the town hall" of mathematics - so many things branch out from it

Julian Rachman

But i am doing a course now on edX but I am hear and there and just trying to get y certificate.

OK. So what should a "game plan" be for me?

David Wheeler

In fact, one might go so far as to say "differentiation" is the attempt to make things "linear", if only for a "little bit"

Julian Rachman

Well. I can see that.

So do linear algebra. Then what?

David Wheeler

And the constructions used in linear algebra are used again in abelian groups, just with integers, instead of field elements

Julian Rachman

4:13 AM

So do linear algebra. Then what?

David Wheeler

Well, you're going to need group theory (just a little) because the algebraic invariants of algebraic topology are usually groups

Julian Rachman

K. Then what?

David Wheeler

Well, some commutative algebra, which mostly deals with modules (any decent abstract algebra course should include a bit on rings)

Committing to a challenge

If a matrix $A$ has $A^2=0$, someone said that it means that $A$ is a 'zero divisor' and said nothing more, what is he referring to?

Julian Rachman

Any thing else?

David Wheeler

4:16 AM

Well, when you get into alg. top. proper, you'll learn about homotopies, and homology

Julian Rachman

Cool. OK. And what are the pre reqs for category theory

David Wheeler

None, but! It helps to have some "mathematical maturity"

For example, a basic concept in category theory is an "arrow"

Julian Rachman

Alright. I will look into it. I have a lot to learn

David Wheeler

If you haven't been exposed to some things that can be arrows, you won't see the motivation

Julian Rachman

So what should learn now?

Along side topology

David Wheeler

4:20 AM

But if I show you 5 things you've seen before, and then say: "these are all arrows"

you'd go: AHA!

Committing to a challenge

Are arrows things like, automorphisms, endomorphisms?

David Wheeler

You're probably set for the next 2-3 years, with what you've already outlined

@Committingtoachallenge Sometimes, but not always

Clarinetist

Do all of the studying you can before you graduate as an undergrad. That's one thing I wish I could've done, and would do if I could do it again.

Furthermore, math is nice, but I would also suggest picking up Java, C++, and Python at the very least.

Julian Rachman

@Clarinetist I am getting y studying in now :)

Committing to a challenge

My classmates are well ahead of me in programming and it is sad, so I agree with the programming statement above

Julian Rachman

4:22 AM

I know how it program in 5 languages and have produced apps

Committing to a challenge

What languages?

Julian Rachman

@David so what should I learn along side topology now?

Html, c, c++, python, robot c

Oh and latex

Obvioisly

David Wheeler

One thing you "don't" formally learn in programming, is "algorithm design" (actually a form of logic), but obviously you pick up bits and pieces in each language

Committing to a challenge

If a square matrix has $A^2=0$ is there any meaning in the statement $A$ is a zero divisor? @David (if you know)

Julian Rachman

I am most fluent in latex

Committing to a challenge

4:24 AM

What is 'robot c'?

David Wheeler

@Committingtoachallenge Yes

Julian Rachman

Some thing for robotics (it is a hobby of mine)

Committing to a challenge

@DavidWheeler What does it mean?

David Wheeler

@JulianRachman Finish linear algebra, and abstract algebra for now

Julian Rachman

OK. I'll check back with you later. :)

And if I get something with the topology-calculus thing, I will TeX it and send it to you.

@David

David Wheeler

4:27 AM

@Committingtoachallenge Well it means one of 2 things-technically 0 is a zero divisor

Paul Plummer

Say $a$ (left) divides $c$ provided there is a non zero $b$ such that $ab=c$. So if $A$ is not the zero matrix and $A^2=0$ then $A$ divides $0$ in the sense above.

David Wheeler

but since in this case, we are saying $A^2 = 0$, to say that $A$ is a 0 divisor, means $A$ is non-zero, and thus nilpotent.

For an example: $A = \begin{bmatrix}0&1\\0&0 \end{bmatrix}$

Committing to a challenge

I had a look and found $A = \begin{bmatrix}0&b\\0&0 \end{bmatrix}$ or $A = \begin{bmatrix}0&0\\c&0 \end{bmatrix}$ or a few other things

Paul Plummer

I am not 100% sure if my definition is the one that is being talked about, definitions in ring theory tend to be subtly different depending on the source

David Wheeler

@Committingtoachallenge Yes, those work, too

Committing to a challenge

4:31 AM

@PaulPlummer It could be, it hadn't been defined yet, but there are 4th year students in the class (with us third year students)

David Wheeler

@PaulPlummer Yes, $\text{Mat}_n(F)$, with $F$ a field, for example, is a ring.

Usually matrices have entries in commutative rings, but non-comm examples are possible (just tedious to calculate with)

An obvious example being "block matrices" where the matrix entries are themselves matrices.

Paul Plummer

@DavidWheeler I am just not sure if it is universal to exclude $b$ from being zero

David Wheeler

@PaulPlummer As I noted before, 0 is technically a zero divisor, just not very interesting.

Paul Plummer

I think it must be universal since then it would be meaningless to have the term zero divisor now that I think about it

David Wheeler

One often sees things like "fields don't have zero-divisors", which is (strictly speaking) untrue, unless one makes an exception for 0.

Mike Miller

4:36 AM

i've never heard a definition in which zero is a zero divisor, for the sake of making "integral domains are rings without zero divisors" concise.

a zero divisor is a nonzero $a$ such that there's a nonzero $b$ with $ab=0$.

David Wheeler

@MikeMiller As Paul pointed out, formal definitions tend to vary subtly, depending on the source.

Paul Plummer

@Committingtoachallenge How is the challenge?

David Wheeler

Wikipedia counts 0, other texts I have don't.

Mike Miller

someone should fix the wikipedia article, then!

Paul Plummer

Jacobson had 0 as a zero divisor

Mike Miller

4:39 AM

someone should punch Jacobson in the face.

Committing to a challenge

@PaulPlummer The challenge is confused at the moment xD. I am working through the text a little fragmented at the moment, which is making it hard to track progress.

Alex Wertheim

@Mike: this opinion is more pervasive than one might expect. math.stackexchange.com/questions/1144452/…

David Wheeler

@MikeMiller Read the article-especially the section "zero as a zero divisor"

Alex Wertheim

(Of course, it references the same wikipedia article, but you get the idea)

David Wheeler

Either convention you pick, some statements need to be made more carefully.

Mike Miller

4:40 AM

I understand perfectly that the convention works just as well, and one is just stupider than the other :)

David Wheeler

Lol, stupider

Paul Plummer

:)

Alex Wertheim

I personally have never cared for zero as a zero divisor, but oh well.

Committing to a challenge

Thanks @David @Paul @MikeMi

David Wheeler

Hey, while we're at it, let's settle the $0^0$ problem once and for all!

Mike Miller

4:42 AM

It's one.

Alex Wertheim

What's the issue with $0^0$? I've usually seen it treated as $\lim_{x \to 0} x^{x} = \lim_{x\to 0}e^{x\ln x}$, which evaluates to $1$.

Mike Miller

If one writes it as $\lim_{x,y \to 0} x^y$, then you get something undefined. Of course, $0^0$ is the number of maps from the empty set to the empty set, which is one. Everybody's happy!

Alex Wertheim

:D

David Wheeler

@AlexWertheim That is the majority view, but people still debate on it. The best answer is, it's "context sensitive"

In "most cases", 1 "plugs the hole", but it can arise in situations where that isn't the obvious choice

twirlobite

hey guys, can you have a look at a very simple fourier series problem that i am stuck on? i have been stuck on it for a while, and have included all my work (including a picture!) math.stackexchange.com/questions/1174642/…

Alex Wertheim

4:55 AM

Sure, I could see that.

twirlobite

@AlexWertheim thanks! i really appreciate it!

Committing to a challenge

5:18 AM

@twirlobite $a_n$ uses $\cos $ . $b_n$ uses $\sin$

Is the proof that a bounded set with infinitely many points has a limit point easy?

(e.g. could I prove it by myself without a large amount of time)

Kaj Hansen

@Committingtoachallenge, I believe you'll need compactness, not just boundedness.

Consider, e.g., $\{1/n | n \in \mathbb{N} \} \subset (0, 1]_\mathbb{Q}$.

Committing to a challenge

@KajHansen Compactness, hmm I can't remember the definition. Compactness was requiring an open cover with a finite subcover?

Oh good example!

Kaj Hansen

Indeed @Committingtoachallenge. If your space is complete, then compactness <--> closed and bounded.

Committing to a challenge

Is that an iff? or just an if?

Kaj Hansen

It's iff provided space is complete.

Committing to a challenge

5:28 AM

Awesome, thanks!

Kaj Hansen

It's "if" always.

Committing to a challenge

That open cover section of rudin was not very clear to me, so I will have to rework it

(+ it will be covered in classes)

Kaj Hansen

It's not too bad when you get the hang of it. Compact set with infinitely many points has a limit point can be done by contradiction using open covers.

Committing to a challenge

@KajHansen I'll give it a go tonight.(3:33pm now, got classes from 4-7pm)

Alex Wertheim

How goes topology, Kaj?

Kaj Hansen

5:37 AM

@AlexWertheim, just working on my problem set that's due tomorrow

Pretty well though. Lots of cool ideas

Alex Wertheim

The "No More Lulz" set?

Lots of good problems in there. :)

Kaj Hansen

Yep, that set

It's been rather enjoyable indeed

Committing to a challenge

How old is the lecturer/tutor who gave out the 'no more lulz' set?

Alex Wertheim

I think Pete is in his early 40s.

My mistake. He's turning 39.

Mike Miller

what's this now?

Alex Wertheim

5:44 AM

math.uga.edu/~pete/4200HW_four.pdf

Committing to a challenge

@AlexWertheim Why are banach spaces at the end of this? Surely they are covered prior to this all?

Alex Wertheim

Hehe, Kaj could probably give you a better answer. I'm not in the course myself, so I couldn't say.

Committing to a challenge

I am worried now about functional due to that link and the placement of banach spaces :P

Alex Wertheim

But Banach spaces strike me as one of those flexible topics that can appear at various points in early analysis/topology courses.

Committing to a challenge

Oh okay, that is good to hear

I must go to class now[complex analysis], thanks for answering my questions!

Alex Wertheim

5:47 AM

In a functional analysis course, certainly Banach spaces should come first. I don't know how one does much functional analysis outside the setting of Banach spaces. But I don't know much.

Sure. Have fun!

Committing to a challenge

Fair enough, cya later!

Mike Miller

@AlexWertheim: Functional analysis can be said to be the study of topological vector spaces and the maps between them. The next most reasonable thing is, say, a Frechet space.

Alex Wertheim

Interesting, @Mike. I'd heard the words Frechet space before, but I never knew what they were until just now.

Mike Miller

I have had some context to think about slightly worse spaces (the space of distributions, say), but rarely. Frechet spaces are the most common decent thing.

(The space of smooth functions on a manifold, say, is Frechet.)

(you've got infinitely many seminorms - the $C^k$ norm for all $k$.)

Alex Wertheim

Neat! That's all reasonable. I'm surprised I haven't seen this before. I guess I need to expand my horizons.

Where have distributions come into your work, out of curiosity? (I may not know enough geometric topology to appreciate the answer)

Mike Miller

5:59 AM

I'm taking a PDE class, which is where they show up. And PDE shows up in topology because of gauge theory - a tool for studying 4-manifolds up to diffeomorphism, rather than homeomorphism or even homotopy equivalence like many invariants you know.

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