Oh... I didn't know Kobayashi was recently deceased. I just saw that he's being memorialized in this months Notices.

@TedShifrin The fact that he isn't particularly active anymore after his suspension ended helps too.

oh no, @Mike ... sigh ... all my mentors/professors are dying off.

He passed away two years ago...

Maybe I did know.

OK, I think instead of doing anything useful, I'm going to go take a nap.

Night .. I'm off to dinner.

@Ted I understand most of the stuff you've sent me, and it'll probably keep me a while. Also, cya

Good ... don't hesitate to conversate (ha) with me about it :)

I won't hesitate :D

hi @anon

hi

@anon how are you?

pretty good

y tu?

so tired

do we know each other?

yes

hmm

yes anon

i used to be a cat

I'm still grumpy though

ah

:)

;)

@TedS None of the stuff that you've sent me is boring btw

@HipsterMathematician I figured out who you are from the quote in your profile rather than the "cat" hint.

@teadawg1337 Does the topic of this question concern you?

Also wikipedia is quickly turning into NPR with its fundraisers.

@Karl It's particularly concerning considering I breezed through three semesters of college calc in less than a month...

Although, I can always keep on studying on my own while taking courses I've already mastered. Hmmmmm... I might try to get credit for stuff I've learned on my own, though. Maybe taking some CLEP exams or AP exams

@teadawg1337 Yeah, I kinda did the same thing too. Since basic calculus is not very conceptually difficult to learn, if you are capable of picking up the strategies for doing the computations, then you are set. On the other hand, I have very often found myself in the trap where I believed I understood a topic just to realize I had no idea what I was doing the very next day. When you are focused, you typically will be more self-appraising than when you take a break and return to it.

@teadawg1337 Definitely try to get whatever credits you can, it saves you money.

@teadawg1337 My personal opinion is that you are not in all that terrible of a position if you find yourself in courses "that you already know". One thing, you can meet with the professor and learn more advanced topics that they would like to but are unable to cover because of the pace of the course.

Or, you can go out and learn on your own, like you said. I would prefer to try to take advantage of the resources the professor/university has to offer if possible though.

I plan to go to college within the next two years, I just need to get my mathematical path straightened out first

I spent all of elementary school through high school bored out of my mind in math classes, and I want to challenge myself

Considering that I took a couple years off, and spent a lot of that time studying math, I can't disagree. :)

I also plan to get advanced in physics as well

But this is MATH.SE, so that's irrelevant

Well, I'm heading off to bed early since I didn't sleep very well last night

@teadawg1337 Good night.

off topic: Is there a concrete example of a simple function (in the sense of analysis) that is not a step function, or is this one of those things that requires the axiom of choice?

nevermind, Dirchlet's function provides an example

@KarlKronenfeld Good Morning!

Well I slept for 11 hours again. I don't know what is wrong with me

Now I have woken up 8 hours later in the day than I should be and I have no idea how to fix it...

@robjohn I need help for this one $$\int_0^{\pi/2} x^2\cot x\, dx$$

After by parts I got $$-2\int_0^{\pi/2}x\ln(\sin x)\,dx$$

I can solve it using Fourier series of $\ln(\sin x)$, but can we use other ways to solve this integral?

I'm working through a proof that involves the double cover $ p : S^n \to \mathbb R P^n$, but haven't found a precise definition of what this map does?

Also, it later says that $f : S^n \to S^n$ induces a quotient map $\overline f : \mathbb R P^n \to \mathbb R P^n$. Does anyone know what the precise definition of that is?

@Venus That is the way that I would approach it.

I'm so sad :(

I wish I died :(

@robjohn Has this problem been posted here?

@Venus You mean on main or on chat?

On main

I only found that one @robjohn

But no attempt

@Venus I'd try fourier series with that one, too.

OK thanks

what does "Every linearly independent list of vectors in a finite-dimensional vector space can be extended to a basis of the vector space" mean?

what if the list of vectors doesnt span the space?

then you couldnt make it a basis??

"Can be extended"

or they are saying add anything that isn't in the span to it and it will span it

and if it is still indenpendnt it is now a span

and a basis

You're on the right track.

so add stuff into this list of vectors until it spans and make sure it is linearly independent and then it is a basis?

I'm on the right track baby, I was born this way!

hi twink

hi @beginner

i think i see you on here every day

but you dont talk

I do talk

ok

:)

:)

(:

(:

¬¬

:(

i cant make those

:3

E:

@beginner are you really 13?

no i am 11 but it didnt let me put in but dont tell please

lol

a user said it in a comment on my question but they didnt ban me :)

I won't say but you better remove that comment

before an admin reads it

i cant remove comments on my mums tablet :(

Then you'll be banned

Hmmm, I wonder what is the problem with having younger mathematicians on this site.

this is a 13+ site

noooooooooo

Oh is it 13+?

yes

Interesting. I can understand one danger......

if we talk it will go off the page so it is fine

Though, I would think a 11 year old mathematician on this site - is smart enough to understand the dangers....at the very least, haha.

@GustavoMontano There are US laws requiring that in online forums (I'm not sure of the exact wording of the laws), the users are required to be 13+.

Ahhh, I see.

for example, my username should only be seen by 13+

"Twink"....

Yes

i am not in the us

Oh. I just looked it up.

:)

That's cool.

my country has no rule like that :) so i am fine

Intresting.

one of my brothers is gay

@beginner The point is that it is a US-based website. They enforce it on all users (and they have to, by law, if they know you're <13, no matter what country you're from)

Luckily, I'm no snitch

From a scientific point of view, I wonder if there is an increasing number of homosexuals, OR if there is an increasing number of homosexuals announcing their sexual preferences publicly.

i am 13 than

^. Good guy Mike.

i was born on may 12 2001 now

I am an only child. When I was 11 I would hang around all sort of chat rooms. Had to find a way to keep myself occupied. Always lied about my age.

so what is a direct sum again?

@beginner I admire you, you're talking about linearly independent vectors at age of 11, at that age i was learning the multiplication tables

oh thank you hehe, my brother taught me lots and he lends me his books. but he is so busy now he doesnt help me anymore

I agree with Twink. It's amazing.

my friends think it is lame

so i dont tell them anymore

Knowledge is beautiful. Especially in the form of mathematics.

yeah i think math is cooler than everything else

no he doesnt like math he is doing chemistry

The act of learning mathematics is beautiful. I see it as adding value to yourself.. Engaging in other pointless activities is stupid. That's not to say that I don't engage in pointless activities.

All of the sciences are beautiful. I have been learning computer science of late. It's amazing.

when i get to highschool i have to see if i like chemisty biology and physics

I'm sure you will.

thats what my math brother says

he doesnt want to push math on me

so a direct sum means that the span of a vector space has to be made by the subspaces of the direct sum?

$U_1=a(1,0), U_2=b(0,1)$ and direct sum of these makes $\mathbb{R}^2$

@beginner maybe you should read a linear algebra book, you can learn de definitions from books better than from us

i am learning from a book but i try to verify

user Kaj was helping me but he has finals i think

what is $a(1,0)$?

$a$ is a constant so it spans the x axis

maybe i wrote it wrong but i think that is how kaj wrote it

i think maybe i should write $U_1 = \{(x,0)\in F^2 : x \in F\}$

and $U_2 = \{(0,y) \in F^2 : y \in F \}$

so $U_1 \bigoplus U_2 = \{(x,y) \in F^2 : x,y \in F \}$

so the direct sum of the two spans of x and y axis is the span of $F^2$ where my field is reals

sorry if the math doesnt show my mums tablet doesnt show it

yes that's right @beginner $F^2$ is the direct sum of $U_1$ and $U_2$

yay i think i get direct sums or just the simple ones maybe

and a normal sum of two subspaces is just every combination of their vectors

which is sometimes the same as a direct sum, like the one above

I don't know what a normal sum is

oh i meant that to mean addition

$U_1 + U_2$

ok

Two planes in $\Bbb R^3$ would be an example of $+$ where you don't have $\oplus$, @Twink @beginner

@TedShifrin I believe you

@beginner: Kaj's finals start in a week ... But he has plenty to do :D

I finished my finals today

and I failed

lol

:(

@TedShifrin do you have a smartphone?

At least you are countably old, @TedShifrin

two planes so $U_1=\{(x,y,0) \in F^3 : x,y \in F\},U_2=\{(x,0,z) \in F^3 : x,z \in F\}$ and $V \ne U_1 \bigoplus U_2$ because both can change $x$ so we don't have unique representations of each point, but we do reach every point so the + holds?

No, @Twink ... I only gave in and got a dumb phone out of necessity about 8 years ago.

@TedShifrin why?

why did you fail twink??

Right @beginner :)

yay i get it !!

you understand direct sums but you don't understand a smartphone? xD

Why did I get a dumb phone at all?

Hahahha "DumbPhone"

@GustavoMontano!

Hey buddy! How are you?

this might sound silly but a sum is like a span and a direct sum is like a basis

Fine. What's up with you?

if I understand how to use a smartphone, you definitely can

This is the second Gustavo ;) Howdy.

Not much! Exams over! 3 months of holidays! Making the most out of it :).

Ah cool.

a sum just wants to span it and a direct sum wants to be a span that is unique

@BalarkaSen. What about you? Finals done?

Yep.

@TedShifrin. Howdy Ted!

A month's worth of holidays.

Now I can sleep in :D.

@TedShifrin If you had a smartphone you could use some very good apps

Very astute, @beginner. You're already better than a lot of our math majors!

^^

oh so that is somewhat right??

Youngsters here are quite adept at mathematics, it seems.

Except me.

you are really good balarka i want to get to your level by your age

I am still stuck at covering spaces.

=(

Covering spaces?

@Gustavo Algebro-topological mess.

espacios cubrientes

Sounds scary.

thank you for your support ted!

My pleasure, @beginner. But don't forget to be a kid and have fun, too!

p : E \to X such that for any point x_0 in X there is a nbhd U around x_0 such that p^{-1}(U) is disjoint union of V_\alpha for V_\alpha \in E such that p restricted V_\alpha is a homeomorphism, @Gustavo

$p$ is a covering space.

yeah i cant do math with my friends so i will when i am not alone hehe

Ted is ignoring me for saying algebra > geometry. bah.

@TedShifrin I'm still a kid and have fun :)

Good for you, @Twink :)

bad for me :(

where is @JasperJoy when Ineed him?

spans and basis and stuff are algebra but they are geometrical in understanding are they geometry as well?

Get your fun in now, when you have time.

@Mike I got smacked by my prof for describing my analogy of monodromy representations with 2-dimensional representation of Gal(\bar Q/Q)

such geometry.

I view/teach linear algebra quite geometrically, @beginner. Many do not.

Don't blame me.

@BalarkaSen. Interesting. I had to think for a while T_T.

omg, I think @JasperJoy deleted his account :(

I didn't encourage it, I just said "sure" repeatedly while you talked. :P

@GustavoMontano Take the example $\Bbb R \to S^1$ by $x \mapsto \exp(2\pi i x)$

@MikeMiller :P

seeing math in your head is the only way to understand it properly though so i think all math should be geometry somehow

He's back as Will Hunting.

@Twink He just changed his username to Will H

smacks @beginner

why beaten? @MikeMiller

@Twink i.e. Ted beat me to saying it

why are you smacking me balarka?

lol

@beginner not all math is geometry!

and neither it should be!

I like depressive people because they make me feel I'm not the only one who is depressive

where are you @WillHunting?

i guess i do multiplication and stuff without seeing it so i guess i was wrong

Basic arithmetic is definitely something memorised.

However, it helps when you can "see" mathematical objects.

@beginner just compute lattice points on a square

@beginner: For fun you should use vector methods and dot products to prove some geometry results.

there's your geometrized version of multiplication

Maybe I should put you on ignore too.

why @Mike?

:(

me mike?? ok ted i will try that :)

That's completely unhelpful to the person who doesn't know what the words involved mean, which is the average person.

No, @beginner.

I'm so sad

I wish I died

@Twink: Don't say that.

OK. @beginner mark n equidistant points on one side of a rectangle and mark m equidistant points on another side. Now fill up the square by equidistant points.

:(

Compute the number of points.

You have multiplication.

Math is not a good career for people who get depressed.

@Mike Happy?

oh that makes sense but i didnt mean i couldnt use geometry i just meant i didnt in my head

No. I'm not engaging on this...

like 20*33 you would times 33 by 10 and then double it, you woulnt fill in 33*20 dots

@MikeMiller But you're not going to ignore anymore, right?

Ok, past my bedtime. Take care, @Twink, @beginner, @Mike.

:P

nite @TedShifrin

have a good sleep ted, thank you for your help and ideas

k night, @Ted, although you are ignoring me :P

@TedShifrin why isn't math a good career for people who get depressed?

@robjohn Did this account get suspended because I had an argument with him in David's answer? I remembered I had flagged his rude comment to this site.

@AshwinGokhale I'm good

Hi math people!

Anyone in here motivated to help with a triple integral problem? :D