Mathematics - 2016-08-28 (page 1 of 2)

Anderson Felipe Viveiros

00:14

I have already found a counterexample guys, thanks.

Soham Chowdhury

02:35

0

Q: Size of fibers versus degrees of residue fields in a morphism of affine schemes

Consider $\DeclareMathOperator{\Spec}{Spec} \newcommand{\F}{\Bbb F} \DeclareMathOperator{\Z}{\Bbb Z} \Spec \Z[i]\to\Spec\Z$ given by the inclusion of $\Z$ into $\Z[i]$. Now, the fiber over every prime which is 1 mod 4 has two points, e.g. $f^{-1}((5)) = \{(1+2i),(1-2i)\}$. Over primes congruent ...

@Balarka, I asked on main.

user227867

03:49

@Alessandro I will try to see 'Children of men', thanks.

user227867

@amWhy I got your latest email. I don't know what you are trying to say, but never mind.

user227867

05:57

@amWhy No, I have not.

Balarka Sen

06:16

@SohamChowdhury Doesn't seem to have any geometric motivation behind inert primes there in the answer though.

user227867

@BalarkaSen Have you decided which university to go to for your studies?

Balarka Sen

Nope.

user227867

I can't believe I got 2 stars for that stupid comment, lol.

Balarka Sen

It's a comment I also agree with.

user227867

I like some Indian authors, like V S Sunder, R Narasimhan, M Ramachandran, and S Varadarajan.

user227867

06:24

The problem is that many of their names are too long for me to remember. =)

Balarka Sen

Indeed.

user227867

At least B Sen is easy. =)

Balarka Sen

"BS" in short.

user227867

Like Brian Scott.

Balarka Sen

I had a different de-concatenation in mind.

Hi @Huy

user227867

06:30

Like Bull Shit.

Huy

hey @Balarka

Balarka Sen

@JasperLoy Exactly

user227867

I think math degrees should not be called Bachelor of Science but of Arts instead, since math is not a science.

Balarka Sen

@Huy What are you working on?

Huy

@BalarkaSen: reseasoning my cast iron skillet. just woke up, having coffee, afterwards brunch at my parents' place

and in the afternoon, I'll prepare some more geometry for my students, then probably take a break

so mathematics-wise, not much at the moment, sorry to disappoint

maybe I'll study a bit about quaternions to see if there's something interesting that I can show the kid that I didn't know yet

user227867

06:33

@Huy Also mention octonions.

usukidoll

08:07

hi

usukidoll

08:29

hiiiiiiiiiiii

user227867

Hi @usukidoll, lol

usukidoll

can someone look over one of my proofs? I think I got them but the way I worded it is crap

user227867

If the way you worded it is wrong, then maybe the proof is wrong.

usukidoll

I got this
If $a \vert b$ and $c \vert d$, prove that $ac \vert bd$\\
By the Divisibility definition, a divides b if $b=ax$ for some integer x and c divides d if $d=cy$ for some integer y. Multiplying b and d, we have, \\
$(ax)(cy)=bd$\\
$(ac)(xy)=bd$\\
Hence, $ ac \vert bd$\\

user227867

Anyway, I am only a banana.

user227867

08:34

@usukidoll You mean ac in the second line.

usukidoll

huh?

user227867

ac divides bd in the second line you typed

usukidoll

oh shit typo

user227867

Yeah, shit, lol

usukidoll

now it sounds legit

user227867

08:36

@usukidoll Just mention that xy is integer. That is it.

usukidoll

like since xy is an integer, multiplying b and d results in
(ax)(cy) = bd
(ac)(xy) = bd
Hence ac divides bd?

user227867

@usukidoll You should mention it just before the conclusion.

usukidoll

ef

errr
UGH!

user227867

@usukidoll I will show you how I would write it.

usukidoll

k

user227867

08:41

@usukidoll Since $a$ divides $b$, $b=ax$ for some integer $x$. Since $c$ divides $d$, $d=cy$ for some integer $y$. Now $bd=(ax)(cy)=(ac)(xy)$ and $xy$ is an integer. Hence $ac$ divides $bd$.

user227867

Is that alright @usukidoll?

usukidoll

yeah

user227867

Do you think you should thank me now?

usukidoll

yeah gives chocolate frog

Mats Granvik

09:57

Velvet

10:13

Hey

r9m

I am not particularly good with these stuff ... a friend asked :) so asking for help here: math.stackexchange.com/questions/1906134/… thanks! :)

hey @usukidoll .. I see you are a weird frog avatar! :P

Velvet

I think it's a Pokémon.

usukidoll

bulbasuar

bulbasaur

u there?

Manolis Lyviakis

10:30

hey

fellow numberfriends

Velvet

What if you only like topological spaces?

usukidoll

what's the gcd(3,5)?

Velvet

1

usukidoll

really?! well I was dividing a lot and got 1

but when I did the first part I got 2

Velvet

Isn't it quite obvious? They're both primes.

usukidoll

10:34

like 3 divided by 5 but the remainder was 2...
then I had gcd(2,3) did this again and got a remainder 1

well I was using EUclidean ALgorithm

Show by example that if $au+bv=d>1$, then $(a,b)$ may not be d.\\
I let a =3, u=4, b =5, v = 6
(3)(4)+(5)(6)=12+30=42
but gcd(3,5) is 1

Velvet

And?

usukidoll

that's it?!

42 is greater than 1 but I highly doubt that gcd(3,5) produces 42

Velvet

What do you mean with (a, b)?

usukidoll

ya

._.

Velvet

?

usukidoll

10:39

my d turned out to be 42 when I let a =3, u = 4, b =5 v = 6

Velvet

But what do you mean with the notation (a, b)?

usukidoll

but gcd(3,5) is 1 so (a,b) isn't d which was 42.. right?!

like gcd(a,b)

it's the notation from abstract algebra an introduction 3rd edition

they left out the gcd

Velvet

Not everyone is familiar with the notation used in your book. It's good to be unambiguous.

But doesn't Bezout's theorem contradict what you are saying?

usukidoll

O_O!

so am I wrong?

Velvet

Well take $a = 2, \; b = 4, \; u = -1, \; v = 1, \; d = 2$.
$$(2)(-1) + (4)(1) = gcd(2, 4) = 2$$

usukidoll

10:58

Show by example that if $au+bv=d>1$, then $(a,b)$ may not be d.
MAY NOT BE D THOUGH!!!!

so I'm proving that by example

Velvet

Ooh, misread that, my mistake.

usukidoll

my D was 42 but when gcd(3,5)= 1
$ 1 \neq 42$

Velvet

You can easily construct a solution that fulfils your condition.

usukidoll

a=3,u=4,b=5,v=6

12+30=42 but gcd(3,5) = 1 whoops not equal

Velvet

Let $a, b \in \mathbb{Z}$ and let $u, v$ be integers such that Bezout's relation $au + bv = gcd(a,b)$ is fulfilled. Let $d = 2 \times gcd(a,b)$ then $$2au + 2bv = 2 \times gcd(a,b) = d$$ Then your solution is the quadruple $(a, \; b, \; 2u, \; 2v)$.

usukidoll

11:04

but I'm wondering if what I did was right?! since 42 is d but the gcd(3,5) was really not d

Velvet

It is clear that $d > 1$.

usukidoll

yeah

Velvet

Yes, it's correct.

usukidoll

yay ^_______^

usukidoll

11:18

the sum of three squares
$n \equiv 3(mod4)$
I think $3^2+3^2+3^2=27$ which is 3 in mod 4

Balarka Sen

@Huy Ah, I see.

Busy with teaching, I see.

Huy

quite a bit, mostly because I've never taught 9th graders before, so I need to get some experience how much will be enough for how long

I kind of underestimated them at first, so now I'm trying to adapt

Balarka Sen

I have this strange experience in here that a generic 9th grader is more mature (in the sense of being more serious about learning, actively thinking more and asking questions, quick at understanding, etc) than a generic 11th grader.

Huy

11:34

no, it's completely the opposite here

Balarka Sen

Might be a coincidence because I have only actively interacted with 3 batches (one being my own, when I was a 9th graders, the next one, and the one now - I am in 11th now)

Huy

mostly because the school where I work at is for 9th till 12th grade, so 9th graders are new and very self-confident and believe they are the smartest kids alive, but after 1-2 years at the school, they realise how much they still have to learn and become much more humble

Balarka Sen

Oh.

Huy

and since most teachers just do revision at the beginning to make sure everyone's on the same level, they are convinced they know everything already

Balarka Sen

Yikes, yes, dealing with overly excessive self-coincidence is a bit of a trouble.

Huy

11:36

yes, I'm glad I don't only teach 9th grade this year

Balarka Sen

Is the quaternion guy from 9th?

Huy

yes

Balarka Sen

I have a bad feeling about his eagerness to learn quaternions then

Huy

same

I'll find out soon enough

Balarka Sen

pro-tip: just don't be the guy who's teaching us physics and says eagles fly at escape velocity

just a random suggestion. might be useful

Huy

11:43

I'm not teaching them any physics any time soon anyways

Balarka Sen

I need a light bulb. Can't solve the problem I want to.

Huy

buy Philips Hue. they make you smarter.

Balarka Sen

you should send an ad-proposal to Philips on that theme.

user227867

12:10

@Huy You should not teach quaternions to a 9th grader. If he is so inclined, he can learn it himself, lol.

Balarka Sen

@JasperLoy Why not?

If someone is genuinely interested, one should not discourage - rather encourage - based on intellectual maturity, IMO.

user227867

@BalarkaSen If a 9th grader is talented enough to learn quaternions, he can learn it himself. If he isn't talented enough, then he has no business learning it.

Balarka Sen

I do not agree with that point of view. One doesn't need to be talented to be interested in learning something.

user227867

If one cannot understand 1+1=2, he should not try to understand 2+3=5.

user227867

I feel that too many people are trying to learn things beyond them these days.

Balarka Sen

12:20

Maybe he does understand complex numbers and so on. Maybe he does understand the definition of quaternions but wants to know what it's good for.

I don't think one should deny outright to help someone learn something beyond the textbooks. This is the kind of attitude which discourages most people to learn anything at all, especially in high school.

user227867

Then he can learn it himself, like I said.

Balarka Sen

No, I don't think this is true.

user227867

For me, I learnt everything on my own.

Balarka Sen

E.g., I am interested in learning quantum mechanics. I probably can't learn it myself without significant guidance.

user227867

Another problem is finding the wrong guidance.

user227867

12:25

@BalarkaSen I think you can. You are genius.

Balarka Sen

I tried to put some effort in understanding the basics. I don't understand everything satisfactorily well enough in the little of the book I have read.

user227867

Yeah, we need to read a few books sometimes to see things. One needs a good book, and one needs a good teacher.

user227867

Not all books are good, and likewise not all teachers are good.

Balarka Sen

Yes, one needs a good teacher. A book doesn't explain how to think - a teacher does.

user227867

Some people learn to think themselves.

user227867

12:30

I think we don't disagree at all. Maybe I just feel that it is too young for someone to know about quaternions.

user227867

Maybe I have seen too many cranks trying to learn that 2+3=5 when they don't know 1+1=2.

Huy

@BalarkaSen I already work for them.

Balarka Sen

should have known

Huy

I'm an excellent advertiser.

user227867

13:14

@usukidoll If you know the meaning of gcd it is clear that it is 1.

Agawa001

13:34

@r9m interesting

will switch to this when i m done working

r9m

14:16

@Agawa001 indeed :) I was stumped with this problem ..

0celo7

14:36

@Huy quaternion guy?

Huy

?

0celo7

who is the quaternion guy

Huy

kid who wants to learn about quaternions. pretty sure you've been here when we discussed it

0celo7

maybe but I was probably doing this god damn analysis problem set

Agawa001

@r9m have u tried graphs ?

Huy

14:38

lol

0celo7

@Huy is this the guy you were wondering about what to teach

Huy

y

r9m

@Agawa001 I ain't good with graphs .. :|

0celo7

@Huy did he say why?

I find them totally boring

Manolis Lyviakis

hey im given that my matrix is 3x4 and the nullity is 2 so the rank of my matrix is 2 how do i find a basis for the row space?

im not given any matrix

Huy

14:47

@0celo7 that was pretty much exactly my thought

Manolis Lyviakis

only the null space

Huy

hehe

my theory was: probably just "other kids only know rational/real numbers, some smart kids know complex numbers so I must know quaternions !!"

Manolis Lyviakis

i kno that the row space has dimension 2 subsdet fo $R^4$

0celo7

@Huy him not knowing calculus is crappy

Huy

idd

0celo7

14:49

idd?

Huy

indeed

Agawa001

@r9m me niether but being not good is better than not experiencing it at all

i have long history with graphs

the biggest graph structure i mad was the air of my room wall

0celo7

@Huy how are the areas of spheres related to the volumes of the balls they bound

Huy

are you asking me or telling me to teach him

0celo7

asking

there is a neat physicist proof of the volume of an $n$-sphere

but it uses calculus

user227867

14:52

@user1618033 Hi!

Huy

yes

is there a relationship without calculus?

0celo7

no no

I'm asking what the relationship between $vol(B^n)$ and $vol(S^{n-1})$ is

I've forgotten

Huy

ah

can't you just generalize the integral

dV = A * dr basically

0celo7

uhh

user227867

Basically is an overused word.

0celo7

14:56

these are unit spheres and balls btw

Huy

that'd be the physicist proof

so dV/dr = A

0celo7

...

Huy

wat

that's the relationship

0celo7

that's not an answer

Huy

???

0celo7

14:57

ugh I once wrote down this proof

gotta find it now

Huy

I just wrote it down

???

0celo7

???

what does that mean

Manolis Lyviakis

math.stackexchange.com/questions/1906371/find-a-row-space-basis it is an easy one

Huy

that you're confusing me

0celo7

ah, here it is

I have a full derivation of the induced metric on a sphere

computation of the curvatures in all dimensions

Riemann, Ricci, scalar

pretty cool

uhh

what kind of crappy notation is this

Huy

15:05

lol

you didn't say you wanted that formula

you said the relationship between volume of B and S

...

0celo7

yeah

hmm

what do

I really have no clue what these symbols mean

have to decode them

Huy

wat

which one

ö_ö

0celo7

I think $\mathcal V(n,r)$ is the volume of the $(n+1)$-ball

and $\mathcal A(n,r)$ is the volume of the $n$-sphere

user 1618033

@JasperLoy JASPER!!! I wanna listen to more songs from you!!!

0celo7

So what I need for this proof is the ratio of the volume of $S^{n-1}$ to $B^n$.

which is...

n?

user 1618033

15:08

@JasperLoy I saw the message you let in Tex chat, please don't call me genius anymore. If I were a genius I would have probably taught mathematics at MIT, Princeton, Harvard from an early age as a professor.

Huy

why n

0celo7

wait, no

hmm

yes, that's right

Agawa001

u taught in harvard ?

0celo7

compute $\mathcal A(n-1,1)/\mathcal V(n-1,1)$.

Huy

no

0celo7

15:09

it's $n$.

Huy

ok

user 1618033

@Agawa001 No. But a genius does it.

Agawa001

many geniuses exist in this world as grigory (he refused a nobel price and never taught in a world wide recongnised institute)

0celo7

@Huy I'm trying to show that $$\operatorname{tr}_gT=\frac{1}{\operatorname{vol}S^{n-1}}\int_{S^{n-1}}T(V,V)\,‌d\sigma(V)$$

Agawa001

Grigory Perelman

Huy

15:12

ok?

0celo7

that factor of vol S is killing me

Balarka Sen

What is $T(V, V)$

0celo7

tensor $T$ acting on $V$ twice

user 1618033

@Agawa001 I think so. Thinking freely without the bounds imposed by any insitution is one of the most powerful way to produce mathematics.

Without being pushed to publish because it is required so.

The inner manifestation of a free spirit in mathematics is what you want if you wanna obtain the results Ramanujan or other important figures of mathematics obtained. You have no bounds.

Agawa001

manifesto

keyword

user227867

15:17

@user1618033 I now have 4 videos on my channel. I made 2 today.

user 1618033

@JasperLoy let me see!

user227867

@user1618033 OK. But those professors are not genius. You are. =)

0celo7

Oh that's only for a symmetric tensor

user227867

@user1618033 Click link, as usual. =) Don't post in this room, please.

user 1618033

@JasperLoy would you bet those professors aren't geniuses?

user227867

15:19

@user1618033 Yes, I bet they are not as genius as you. =)

user 1618033

@JasperLoy lol

@JasperLoy I'm listening to sole mio.

Great song!!!

user227867

@user1618033 Some people may think I am sucking up to you by praising you. Not true. I truly admire your mathematical talent.

user 1618033

@JasperLoy Thank you. But don't overly appreciate me. ;)

user227867

Ethan is also another genius, but he doesn't come to chat these days. He is busy with school.

user 1618033

(I still have a lot of stuff to study)

I didn't see him either.

user227867

15:21

On the other hand, I am not a genius. So I need to work very hard just to write an article in future.

user 1618033

@JasperLoy You really have a great voice! Mayb you should do more than only having these posts on youtube.

user227867

Sometimes, I think, if I love mathematics so much, maybe I should not be a professor, maybe I should just read my books all day long.

user 1618033

@JasperLoy don't forget that I've always said I worked extremely hard for all I got.

user227867

But I think one needs a job, a career, so being a professor is not a bad thing.

user 1618033

@JasperLoy That's true.

Vrouvrou

15:23

hello, someone knows how to write this with latex

user227867

@user1618033 A mad person emailed me to scold me this morning.

user 1618033

@JasperLoy Why? What was it about?

user227867

@user1618033 This person probably thought I was ignoring him in chat, when all I did was go to sleep. Never mind, I will ignore him from now. He abused me years ago and did not apologise, and today he abused me again.

Huy

$\overline{L^\infty(\Omega)}^{\| . \|_{(\Phi)}}$

user 1618033

@JasperLoy Sorry to hear that. From this chat??

user227867

15:26

@user1618033 Yes, it is someone on Math SE.

user 1618033

@JasperLoy pfff. That's awful.

Vrouvrou

@Huy thank you

0celo7

good lord what is that

user 1618033

@JasperLoy Some dates lately?

:D

user227867

@user1618033 Sorry, what do you mean? I have no dates, no girlfriends. =)

user 1618033

15:36

@JasperLoy That's strange to me. If you go out in the street and sing sole mio as in the youtube clip, I can predit that a dozen of girls fall down at your knees in the shortest period of time.

:-)

user227867

@user1618033 LOL. If you go out on the street and solve your integral, the same will happen. =P

user 1618033

@JasperLoy hahahahaha :-))))

@JasperLoy NEVER THIS WAY. JUST TO SCARE THEM! ;)

user227867

@user1618033 When I was in high school, I had a genius schoolmate. For a few hard integrals, he could come up with several antiderivatives for each of them that nobody could think of. I copied them down and went home to study the solutions.

user 1618033

@JasperLoy do you have in mind such an example to see it now?

user227867

@user1618033 I have forgotten what they are. =) But it is not the same level as what you do. I think you can solve those when you were 5 years old. =)

user 1618033

15:43

@JasperLoy When I was 5 years old I think I was doing something else, playing some game? :D

user227867

@user1618033 Like having a date. =)

user 1618033

@JasperLoy :D

user227867

Have you heard of the Chinese mathematician Hua Loo Keng or Hua Luo Geng?

user227867

I think in China, there is a school named after him, and they train their students in mathematical olympiad problems from grade 1 on wards.

user 1618033

@JasperLoy It doesn't sound familiar to me although I might have heard about it before.

Hua Luogeng

Hua Luogeng, or Hua Loo-Keng (Chinese: 华罗庚; Wade–Giles: Hua Lo-keng; 12 November 1910 – 12 June 1985), was a Chinese mathematician famous for his important contributions to number theory and for his role as the leader of mathematics research and education in the People's Republic of China. He was largely responsible for identifying and nurturing the renowned mathematician Chen Jingrun who proved Chen's theorem, the best known result on the Goldbach conjecture. In addition, Hua's later work on mathematical optimization and operations research made an enormous impact on China's economy. Hua did not...

user227867

15:46

I don't have their textbooks with me now, but they were written in Chinese and are very very good. 12 books, one for each level.

user 1618033

@JasperLoy Are there English versions too?

user227867

@user1618033 Yeah, he has this beautiful Springer book 'Introduction to number theory'

user 1618033

@JasperLoy hmmm, thanks! Let me check that.

user227867

@user1618033 Unfortunately, no. I can't find a complete set of those books now either.

user 1618033

@JasperLoy OK. Let me know if you ever find English versions, I'm interested.

Transcript for

$Mathematics$ Mathematics