Mathematics - 2015-10-18 (page 1 of 2)

Lucian

00:51

Two questions: Do any of the two sites hosting the Inverse Symbolic Calculator work for anyone ? Does anyone know of a downloadable version of this very helpful tool ?

Random Variable

01:32

@r9m Cody asked me about that integral about a week ago. But I haven't thought much about it. Did you see those two integrals I posted 5 days ago? They're really not that difficult to evaluate (especially using contour integration), but I thought they looked cool.

Galois in the Field

01:57

@BalarkaSen $\bar{\Bbb Q}$ is notation for the algebraic closure of $\Bbb Q$?

Will Jagy

04:02

@Lucian, you can do a little yourself with the PSLQ algorithm and later variants, gp-pari has it. However, the value of ISC is the extremely large list of comparison numbers, maintained by Carma at Newcastle. Not something we could do ourselves.

lurker

04:28

Isn't Euler's identity based off the assumption that Cos (pi) is being calculated in radians and not degrees?

beinng that e^(i)(pi) = cos(pi) + isin(pi)

Remember me

And we know what $i\sin(\pi)$ is.

lurker

yeah its also calculated in radians

yielding 0

so -1 + 0 = -1
then e^(i)(pi) + 1 = 0

wouldn't it be radically different in degrees?

its kind of confined to the unit circle

not very beautiful if you ask me

Remember me

Your eulers identity is true for any real number

lurker

theres nothing to plugin

so i dont know how you can say that

its e^(i)(pi) = -1

im saying that the only reason it works is because pi is calculated in radians

Remember me

04:51

$\pi=180^\circ$ . We can also find out $\sin(180^\circ)$. It is just that it looks beautiful with your pi involved .@lurker

lurker

your notation doesnt work in chat. have no idea what you're typing.
but my point stands, for degrees, wolfram alpha yields approximately .9984 for cos(pi) and approximately i0.05477 for sin(x). Adding these two yields 1.053275

i1.053275

which destroys the identity

Remember me

05:08

@lurker pi =180 degrees .

lurker

thats a conversion based on radians

im talking about pi degrees

its never implicitly stated that pi indicates degrees or radians

cos (A) is an angle
apparently it can also represent radians, thus leading cos(pi) = 0

but thats a conversion thats assumed based on the unit circle

Remember me

For latex look at the starboard . By that I mean to say that you will see that at the right hand corner of the screen there are starred messages. Click on the first one. It says Chat guidelines....

pi is in radians
Pi radians = 180 degrees.
Obviously it won't be true for pi degrees . pi is approximately 3.14 . so you are just saying 3.14 degrees which will not give you the desired result.

lurker

right. so there should be a notation to indicate this difference

not just to assume one or the other for convinience

Remember me

When I did Euler's identity we used to write the angle as x^c where that c denoted it was in radians.

PVAL

05:26

There's no units in Euler's identity.

or any angles

lurker

right. so it all depends on the unit circle

cos(pi) must equal -1

no unit circle, no identity

PVAL

Eulers identity is that $\sum_{n=0}^{infty} (\pi i)^n/n!=-1$. I don't see the dependence on the unit circle.

lurker

according to the illustrations given online, one series of e^x when x = (i)(pi) is equal to a series of cos(x) + isin (x)

or rather cos (pi) + isin (pi)

Remember me

Though it is a very weird choice of notation

lurker

so e^(i)(pi) = cos (pi) + isin (pi)

leading to cos(pi) = -1 and isin(pi)= 0

therefore the only way cos(pi) is -1 is in reference to radians

no radians, no identity

Remember me

05:36

Euler's identity is true for any real number. @lurker

lurker

Euler's identity is e^(i)(pi) + 1 = 0

So when you say that this is true for any real number, when there is no variable available to use, makes no sense

Remember me

This is what I told you at first

lurker

where would you input the real number?

Remember me

e^ix= cos(x)+ isin(x) for any real x.

lurker

yeah but the identity is exclusive to pi

im not talking about other numbers

oh, and no, you cant just input any number and get -1

this isnt about whether it's "true" or "false", the result of pi for x is based off radians

thats my only point

euler's identity falls apart without radians

it no longer equals -1

e^(i)(pi) that is

Remember me

05:43

When you put x=pi you get your identity. And I don't understand your question

lurker

identity is x^(i)(pi) -1 = 0
therefore x^(i)(pi) = -1
x^(i)(pi) = cos(pi) + isin(pi)
cos(pi) = -1
isin(pi) = 0

The only way those two values for cos and sin are possible is if pi is interpreted in radians

if pi is degrees you get a different answer

Boni

08:28

If you prefer, substitute in the argument of -1 instead of pi or "180".

Balarka Sen

09:02

@GaloisintheField yes

Mats Granvik

Equal to 3 when n is a prime or a power of 2:
1, 3, 3, 3, 3, 5, 3, 3, 5, 5, 3, 5, 3, 5, 7, 3, 3, 7, 3, 5, 7, 5, 3, 5, 5, 5, 7, 5, 3, 9,...

Galois in the Field

09:13

Hello @BalarkaSen, in regard to yesterday, we got $x^6+1 \equiv ((x+1)(x^2+x+1))^2$ over $\Bbb F_2$ which has irreducible $x^2+x+1$ over $\Bbb F_2$.

Set $L= \Bbb F_2 / \langle x^2+x+1\rangle$ and set $\alpha= x+\langle x^2+x+1\rangle$ and we have $\Bbb F_2 \cong i(\Bbb F_2)\subset L$

How do I write $x^2+x+1$ as its factored terms?

Balarka Sen

$x^2 + x + 1$ is irreducible over $\Bbb F_2$. That is why $\Bbb F_2[x]/(x^2 + x + 1)$ is actually really $\Bbb F_4$

Galois in the Field

Sorry that last sentence should have been, how do I write $x^2+x+1$ as factored terms over $\Bbb F_2(\alpha)$

Balarka Sen

I don't know what you mean by that.

Whoops, I have to go. I'll be back later.

Chris's sis the artist

09:53

$$\int_0^1 \int_0^1 \int_0^1 \log^2 \left((x-y)^4+(z-x)^4+(y-z)^4\right) \ dx \ dy \ dz $$

@robjohn the one above really has an exceptionally good-looking. :-)

(it's newly created)

Galois in the Field

10:06

@BalarkaSen Well I guess my question is two-fold, what does $x^2+x+1$ look like as a product of linear factors over its splitting field as an extension of $\Bbb F_2$ and what does the splitting field as an extension of $\Bbb F_2$ look like as an $\Bbb F_2$-vectorspace. Hopefully that makes sense, I am significantly more fatigued than yesterday

r9m

10:37

Best complement I have ever received :P

You really have sharingan power!!! :p — Masacroso 15 mins ago

Galois in the Field

It's some sort of eye disease @r9m?

Chris's sis the artist

@r9m hehe, good! Put the sharingan power on the integral above too. :D

r9m

10:52

@GaloisintheField ya! The kind that plagues your entire life with nightmares :P

Galois in the Field

@r9m That doesn't sound fun :(

Although it looks pretty cool ;)

Remember me

@r9m what about rinnegan?

Mangekyo Sharingan

r9m

@Chris'ssistheartist looks fun! I'll give it a try :-) I am trying to figure out Cody's integral (the one I asked you about the other day) .. I have a tedious approach in mind, I'm looking for short/slick approaches :)

@Rememberme The ability to see beyond the world of the living ?! :P you can't call that a disease :P

@GaloisintheField :)

Chris's sis the artist

@r9m You asked me more integrals. Well, I'm sure with enough research one can find clever ways.

Danu

11:20

Hi guys, does anyone have a canonical example of a Hausdorff (topological) space that is non-metrizable?

Alec Teal

No

Why? BECAUSE THEY'RE EQUIVALENT

Galois in the Field

Typing in capital letters is considered rude

Danu

@AlecTeal That's the other response that I was secretly hoping for :D

(not in the way you typed it, of course, but the mathematical content)

Alec Teal

@Danu I would never ever give you misleading information.

Galois in the Field

@Danu I believe what he has said is false

Danu

11:23

I think so too.

Galois in the Field

@AlecTeal Please don't intentionally mislead...

Alec Teal

Okay

starts a timer.*

Galois in the Field

What does that mean?

Alec Teal

It means I started a timer

Chronometer.

@Danu would it help at all if I was like .... I dunno

"I read that in a book by Spivak" or something?

@Danu why did you delete that?

Galois in the Field

It was essentially spam, so why not?

Alec Teal

11:33

@GaloisintheField howso?

Galois in the Field

I'm done talking, you are trolling.

Alec Teal

@GaloisintheField but I totally read that in a book by Spivak.

chat.stackexchange.com/transcript/message/24782264#24782264 chat.stackexchange.com/transcript/message/24782262#24782262 FOR FUTURE REFERENCE

Galois in the Field

11:49

Aug 4 '12 at 0:49, by Peter Tamaroff

@HenryT.Horton Is there a non-metrizable Hausdorff space?

Aug 4 '12 at 0:55, by Henry T. Horton

@PeterTamaroff Yes, some examples are $\Bbb R$ with basis open sets $[a, b)$, the long line, and $\Bbb R$ with basis open sets of the form $U - \text{countable set}$, where $U$ is open in the standard topology

Aug 4 '12 at 0:57, by Henry T. Horton

@PeterTamaroff The Nagata-Smirnov metrization theorem gives necessary and sufficient conditions for a topology to be metrizable

@Danu Relevant discussion can be found in the transcript

Alec Teal

There's like 5 minutes left on the timer.

I'll wait

Galois in the Field

Why are you timing 30 minutes?

Alec Teal

Gives you time to think about the question.

Galois in the Field

I am working on a different problem...

Alec Teal

The "you" wasn't singular @GaloisintheField

2 minutes or so left.

@Danu I didn't lie

@GaloisintheField I didn't lie.

Galois in the Field

11:56

If you prove that metric spaces are Hausdorff I will facepalm

Alec Teal

In Spivak's Different Geometry volume 1 in the appendix we see that requiring a manifold be Hausdorff and metrisisable are equivalent conditions, There are other such conditions.

So yeah screw you @GaloisintheField - as you can see I didn't lie. Don't accuse me of doing that again.

I wouldn't mislead when it comes to something serious @GaloisintheField - you were really offensive.

Galois in the Field

I own this textbook and I have it open, it says here that a manifold will be defined to be Hausdorff

@AlecTeal You are 100% trolling and I have you on ignore now

Angelus Mortis

Can I ask a very very noob question ? related to volume

Alec Teal

@GaloisintheField I'm not, I can take a picture of the page in the book if you like.

@GaloisintheField no it doesn't, it says the "manifolds we'll deal with" satisfy that, notice it uses metric ones too, not topologies. If you look in the appendix at the back ....

@AngelusMortis you can yes.

user159870

Hi everyone!

Does someone have the time to help me at the exercise in differential geometry math.stackexchange.com/questions/1485639/… ?

Alec Teal

12:11

See my comment @user159870

I haven't got the book to hand, and I don't recall the exact definition of closed (other than it returns to where it starts) so if that isn't it. I'll need the definition and 3 minutes.

Martin Sleziak

12:35

I vaguely recall existence of something like "ExerciseWiki", where goal was to collect various mathematical exercises (and solutions). Do I remember it wrong? Or does anybody remembers seeing such thing (and, ideally, has the link)?

Alec Teal

@MartinSleziak I'm working on something like that (I'd put this on the page for closed curve as a caveat) there's proof wikit oo?

Martin Sleziak

Yes I know about proofwiki.

What is link to your wiki? (I am sure I saw it before.)

Alec Teal

maths.kisogo.com

Martin Sleziak

Thanks! That's not the one I saw before. But still it might be worth looking into.

Alec Teal

maths.kisogo.com/index.php?title=D-system I try and do stuff like this. I found 2 definitions, they're listed, with references and proved equiv. The search is now at the point where it is actually worth using though. Which I am quite proud of.

Martin Sleziak

12:48

@Danu I would either go with some space derived from $\omega_1$ or with some weak or weak* topology. Whatever you feel more comfortable working with. You can find some posts on the main with some examples. I listed a few of them here.

Alec Teal

Lastly, maths.kisogo.com/index.php?title=Topological_property_theorems I've given an index (as best as one can) to theorems, so say you know something about your space and you're looking to show something else, you can look up in these (user-sortable) tables given in dictionary order (dictionary as in on the keys, not name)

Georan

hey, can someone take a look at my question? math.stackexchange.com/questions/1485226/… thanks!

skill patrol

@user159870 $\gamma(t)=(\cos(t),\sin(t))$ is a closed curve on $[0,1]$ but not on $(0,1)$

r9m

@Chris'ssistheartist I gave the integral (from Cody's link) a shot! :-) Lemme know if the approach is correct! Thanks!

The inverse central-binomial harmonic sum seems to be highly resistant to series manipulation tricks from my arsenal though :(

Danu

13:05

@MartinSleziak Thank you

Chris's sis the artist

@r9m Using that splitting with the root unity is another option indeed. Are you sure the last form cannot be simplified further?

@r9m Cody puts on the last line in his question a question with a question mark that he probably knows since also a baby kid can do that.

r9m

@Chris'ssistheartist that last form can be simplified to $\displaystyle \frac{22}{9}\zeta(3)-\frac{4\pi\sqrt{3}}{27}\psi_{1}(1/3)+\frac{8\pi^{3}\sqrt{3}‌}{81}$

Chris's sis the artist

@r9m great

@r9m I didn't know the problem is posted on MSE.

r9m

@Chris'ssistheartist He replied to my post in I&S with the link to the question that was posted in M.Se before that

that's where I got it from :-)

Chris's sis the artist

@r9m Maybe he has a nice way he doesn't share, just test it.

Even affected by my condition I crawled myself anf got a lot of new identitites and other stuff, but I need more time to recover before being like an A-bomb for each integral, series, limit I meet. :-)

(especially some clever ones involving the root of unity)

r9m

13:14

@Chris'ssistheartist That won't bother me at all :P People rarely are willing to share their wives*( or *husbands) to others :P

Chris's sis the artist

@r9m hehe, I'm not bothered at all, it's not a bad thing to test your stuff. ;) Any opportunity to learn some more stuff is OK for me.

r9m

@Chris'ssistheartist Put it in your book please! (assuming that's where you might want to share it) :-)

Chris's sis the artist

@r9m What to put in my book? My book is already full! :-))))))

Georan

3

Q: Integration by substitution not working?

I've been working on a set of problems for the past 6 hours and I'm about to explode from frustration! Either my book has errors, wolfram alpha is broken or integration by substitution does not work! Help!! I'm trying to solve the following ODE: $$ \frac{dh}{dt}\; =\; \frac{1\; -\; 20kh\; +\; kh...

differential-equations

can someone help me out??

r9m

@Chris'ssistheartist OH!!! Then RELEASE THE KRAKEN :D so we might kneel before it's wrath ASAP :D

Chris's sis the artist

13:19

@r9m lolllllllllll :-))))))))))

@r9m I've been waiting a bit for some stuff to be published that is reference in my book too. It's awul how long you might expect for something to be published.

r9m

@Chris'ssistheartist Will it be published before end of November? :)

Chris's sis the artist

@r9m No, next year. With 300 problems all would have been already finished, just waited for the publishing moment. Even if you're done with your book, you need to wait until all your book is revised.

The process of publishing is not that fast, think that for some articles I waited for more than 6 months, and it was just an article. ;)

r9m

@Chris'ssistheartist oh! Okay! :)

@Chris'ssistheartist In case you care .. I was just stamping my feet like a spoilt kid 'Dad! I want that book! Right now!' :P

Chris's sis the artist

@r9m I have stuff to publish 150-200 books. Just to be entirely honest. ;)

2

r9m

@Chris'ssistheartist Mother of God!! :O

Chris's sis the artist

13:29

@r9m :D

@r9m lol, but who cares that? I still have no published book. ;) Well, patience, patience, patience ...

Why? For a simple reason: I work hard ceaselessly. While others talk about known problems in mathematics, or about philosophy related to mathematics and other stuff like that, I work extremely hard and get the results I also talk here once in while.

Galois in the Field

@BalarkaSen Were you willing to continue our discussion on the splitting field of $x^6+1$ for $\Bbb F_2$ whenever you are online? I'll be in the Linear & Abstract algebra room

Chris's sis the artist

talk here about

Galois in the Field

@Chris'ssistheartist Me?

Chris's sis the artist

Initially I wanted to write something like "While others talk about my mathematics ... and the rest ...", but I just don't wanna seem wrongly perceived. ;)

@GaloisintheField I was referring to my text.

Galois in the Field

13:45

@Chris'ssistheartist Okay sorry for confusion.

Chris's sis the artist

@GaloisintheField No problem. ;)

Galois in the Field

;)

Chris's sis the artist

Wait, did I post the cublic version?

$$\int_0^1 \int_0^1 \int_0^1 \log^3 \left((x-y)^4+(z-x)^4+(y-z)^4\right) \ dx \ dy \ dz$$

I don't think so.

Balarka Sen

@GaloisintheField Whoops, I totally forgot. I am actually a bit busy and somewhat ill. But let me see what the question was.

Chris's sis the artist

It was good to mention that since in my philosophy I'm only in a competition with myself, I'm not in a contest with anybody here or elsewhere.

Galois in the Field

13:55

@BalarkaSen The refined(?) question was put into the linear & abstract algebra room, no problem at all btw

r9m

@Chris'ssistheartist Is competing with self same as competing with one's shadow? :) Depends on the light source (if it's before or behind you) :)

Chris's sis the artist

@r9m lol, I mean to push your limits much, that to be a lifestyle, today to win the person you were yesterday in terms of performance. :-)

Balarka Sen

@GaloisintheField Well, if $\alpha$ is a root of $x^2 + x + 1$ in the splitting field of the polynomial, then the splitting field looks like $\Bbb F_2(\alpha)$. What else can be said? The polynomial factorizes as $(x + \alpha)(x + \alpha^2)$. You can provide an explicit isomorphism between the splitting field and $\Bbb F_4$, of course.

Therefore I'm confused what your question is

r9m

@Chris'ssistheartist :) I got that! :) I was trying to drop a PJ :P

Chris's sis the artist

@r9m :D:D:D

Balarka Sen

13:59

Trying to get LaTeX work in Inkscape is a pain in the neck.

Galois in the Field

@BalarkaSen I was just wondering what $\Bbb F_2(\alpha)$'s basis is as an $\Bbb F_2$-vectorspace

Balarka Sen

$\{1, \alpha\}$.

Er, I may have messed up signs in my factorization above. But you know what I mean.

Chris's sis the artist

14:41

@r9m You're still very young but one day you'll understand how is to get hit only for being smart (you're smart enough to get in trouble). ;)

Trust me, I know what I say. However your humour wellllll developed might alleviate all to some extent. :-)

Angelus Mortis

14:55

Q : A man sold an article at a gain of 5%. Had he sold it for Rs 40 more, he would have gained 8%. Then the cost price of the article is?

my answer coming is 4K / 3

but actual answer is 10k

please someone help )

lurker

are all non-algebraic numbers transcendental?

or can there be a non-algebraic and non-trascendental

Balarka Sen

@lurker By definition, yes.

lurker

15:14

If the square root of non-perfect squares are all irrational, then does this definition continue to the cube root of all non-perfect cubes are irrational?

and so on to infinity?

r9m

@Chris'ssistheartist I am not sure I understand what happened .. did someone hurt you for being smart? :O

Chris's sis the artist

@r9m I only say if you're smart enough you probably get in trouble. ;)

r9m

@Chris'ssistheartist Better to be smart and get into trouble than being dumb and do nothing! :P I like things when they are in motion! :P

Chris's sis the artist

@r9m :D

Galois in the Field

@r9m The happiest people I know are some of the dumbest people I know.

Chris's sis the artist

15:23

@r9m I'm more a retarded these days, so I'm not worried too much :-)

r9m

that didn't come out right :P

Chris's sis the artist

(especially during the allergy episodes)

lurker

found the answer

The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two nth powers) are irrational.

Galois in the Field

@r9m My point is, if you have really thought about what life is about, you will have found that there isn't really a point, and you are left with assigning your own point, which the common choice is enjoyment. If enjoyment is your life-value-metric, then from my observation, dumb people are usually winning.

lurker

We are but blades of grass tossed to an uncaring wind. All effort is vain. All order an illusion.

Galois in the Field

15:29

I don't think of it as depressing anymore though @lurker :P

Chris's sis the artist

@GaloisintheField In terms of enjoyment I'm hard to beat when a attend some integrals, series and limits and make some discoveries ... :-) It's about music, dancing, jumping around, shouting, screaming, drinking champaigne ...

lurker

that was a quote from an onion video about a football team's pre game coin toss that makes them realize "the randomness of life and the triviality of their own existence"

its satire, quite hilarious

Galois in the Field

@Chris'ssistheartist Well if you have found what you love and you never tire doing it, then you have won.

@lurker I've gotta see this haha

barznjy

Hi

lurker

ill send u a link

Galois in the Field

15:31

Pre-Game Coin Toss Makes Jaguars Realize Randomness Of Life

►

lurker

Pre-Game Coin Toss Makes Jaguars Realize Randomness Of Life

►

lol

r9m

@GaloisintheField can't argue over that on a full stomach! Excuse me ..

Chris's sis the artist

@GaloisintheField I think the little secret is to do your best and find the enjoyment in everything you do. Sometimes we cannot do what we like to do.

Galois in the Field

@lurker This is seriously awesome

lurker

glad you like it lol

user159870

16:03

Why is the chain of rules called like that?

Dominic Michaelis

16:16

if i have some balanced convex and absorbing set, what may I say about it, if I know that it is unbounded in every Norm?

Tien-Cheng Huang

17:10

Could someone recommend some courses outside mathematics for one with some mathematical maturity?

Gigili

Hi

What is efficiency in mathematical context?

Pedro Tamaroff

17:55

@Tien-ChengHuang Gardening?

skill patrol

Sanic Hodgeheg

@Tien-ChengHuang amazon.com/Introduction-Botany-Murray-Nabors/dp/0805344160/…

Chris's sis the artist

Sweeeeetttt God what a result I got!!!!!!!!!!!!!

morphic

18:14

@skillpatrol lol

skill patrol

:D

Ted Shifrin

18:32

hi @morphic @skull

morphic

Hi @TedShifrin

skill patrol

Hi Professor

Ted Shifrin

anything good goin' on?

Balarka Sen

hi @TedShifrin

Ted Shifrin

heya @Balarka

I guess not ...

Balarka Sen

18:33

I revised a bit of differentiation plus some topology today. Nothing interesting though.

Ted Shifrin

Well, keep me posted.

Balarka Sen

Sure. How's your day?

skill patrol

How's your bridge games going?

Ted Shifrin

Going to another in an hour, @skull. :) We're doing decently, but still making stupid mistakes.

Huy

is it common to write $\bar{A}$ instead of $A^c$ for the complement of an event $A$ in probability theory or is that just the high school way?

Ted Shifrin

18:40

I hate the bar, because bar is used for so many other things in mathematics (e.g., closure). But some intro to higher math texts use it for complement, for example. I certainly would never, never use it.

Balarka Sen

Me neither.

Huy

that's what I thought

I'm just preparing some prob/stat for my high schoolers but I hate that notation

I'll just write $^c$

damn I never studied these tests properly and now I have to teach them soon =_=

Ted Shifrin

tests?

You sound testy.

skill patrol

and testimonial :)

Huy

you know, this $\chi^2$ stuff for example

I really never cared about it at all

to me the most boring thing in maths ever

but maybe that's just me

Ted Shifrin

18:44

oh the statistical tests ...

Huy

@Ted BTW I talked to anon the other day because of identifying $H^2$ with upper triangular matrices

@Ted I looked up QR. Now if I want to identify SL(2,R)/SO(2,R) with upper triangular matrices with positive diagonal entries, wouldn't I have to show that if I take a matrix A in SL(2,R) with A = QR, and multiply it by a matrix B in SO(2,R), then AB = Q'R for the same R?

anon

In the QR decomposition, the orthogonal matrix goes on the left, so you are actually identifying SO(2)\SL(2,R) with upper triangulars with positive entries. Anyway, if G=HK for trivially intersecting subgroups H and K, to show H\G can be identified with K do you think you have to show for all g=hk that multiplying on the right by some h' yields h'k for the same k? Why would you think that?

To show SO(2)\SL(2,R) can be identified with upper triangular matrices, it suffices to show that SL(2,R)=SO(2)*B (where B=upper triangular matrices with positive diagonals) and SO(2) intersects B trivially. This is true more generally: whenever a (lie) group G acts (smoothly) on a set (space) X, if H is a point-stabilizer and K is a subgroup that acts transitively, you can prove G=HK (or G=KH, they're equivalent) and H,K intersect trivially.

Huy

19:03

I'm thinking of a map $SL2/SO2 \to U2$. The version $SL2 \to U2$ is already well defined by the QR decomposition and requiring positive diagonal. now if I look at an equivalence class, the map shouldn't depend on the representative and quotienting groups is done by right translation, no?

anon

correct

Huy

so what's wrong with my thought process?

anon

It looks like the only problem is you're multiplying by B on the right instead of the left, whereas like I said QR puts the orthogonal matrix on the left so we're talking about the right coset space SO(2)\SL(2,R).

Huy

ok

anon

Say X and Y are in the same right coset of SO(2). Then X=AY for some A in SO(2). Write the QR decompositions X=QR and Y=Q'R'. Then QR=(AQ')R'. Since QR decompositions are unique, Q=AQ' and R=R'.

Try to prove what I said about group actions in general though: if G acts on X, H is a point-stabilizer and K a transitive subgroup, then G=HK and H,K intersect trivially. For example, let E(2) be the group of rigid motions acting on R^2. Then H=SO(2) is the point-stabilizer of 0, and K=(R^2,+) acts by translations. Then the decomposition E(2)=HK means that every rigid motion is uniquely a rotation around 0 followed by a translation.

err, actually a translation follows by a rotation around 0 sorry

err, I want more than transitive

a regular group action

so that K can be identified with X

another example: $G=S_n$ acting on $\{1,\cdots,n\}$. We can regard $H=S_{n-1}$ as the point-stabilizer of $n$, and $K=\langle(12\cdots n)\rangle$ a subgroup which acts regularly on $\{1,\cdots,n\}$. Then $G=HK$, so $K$ is a right transversal for $H$.

Huy

19:20

what's a regular group action?

anon

you can look it up, but the way I think about it is: $K\curvearrowright X$ regularly if $K\cong X$ as $K$-sets, ore equivalently if for all $x\in X$ the map $K\to X$ defined by $k\mapsto kx$ is a bijection. thus, $K$ can be "identified" with $X$. For instance, the cyclic group $C_n$ acting on $\{1,\cdots,n\}$, or a vector space acting on itself by translation.

Krijn

(Or just that for $x,y \in X$ there is a unique element that sends $x \mapsto y$)

anon

yes, that too

that's probably best for the purposes at hand

Krijn

Althought I never thought about how you could actually "identify" $K$ with $X$, which is smart

anon

it's like affine spaces versus vector spaces. affine spaces are just regular group actions of vector spaces.

by picking an origin in the affine space, you can identify it with the vector space

then all those "displacement vectors" are the unique group elements sending one point to another

if K acts regularly on X, then picking an "origin" in X allows us to identify K and X

Krijn

19:30

How do you set yourself to doing something?

anon

eh?

Krijn

I have been procrastinating my work for the last day but I can't get myself to start

anon

realize it's no big deal and start off doing it in between other things until you get momentum I guess

or just wait for the adrenaline rush of last minute work to give you the boost you need

whatevs

Krijn

I've been doing that adrenaline rush thing for years and its getting tiresome

But somehow I can only do the harder exercises on these adrenaline rush moments

Why does my own brain work against me so much :(

skull petrol

It works that way for everybody pal :-)

Krijn

19:36

So how do people working on a Ph.D do this?

They obviously cannot write a doctoral thesis in a matter of months I'd say.

Balarka Sen

@Krijn keep some problems at the back of the mind, start off doing it in between other works, as anon said. most of the time, subconsciousness helps you to do mathematics. this is a well-known theory proposed by Poincare.

skull petrol

^

et al

Balarka Sen

some people make serious progress by sleeping with a problem on the mind, letting the subconsciousness do the thing. this might not work universally. e.g., i have never done mathematics in my sleep. but i can make nontrivial progress by walking with a problem in the mind and never thinking about the problem at all

Krijn

I do not do mathematics in my sleep, but somehow I managed to solve most problems in my bachelor thesis at around 3 am while trying to sleep

But then of course you have to get up to write it down, because I was too excited before I would have written it up.

I have also resolved to taking showers when I cannot solve a problem. There is something about a shower that just solves problems for me

Clarinetist

0

Q: Teaching about absolute values not centered at $0$

One of my peers is studying for the GRE and ran into the following problem: When is $|x-4|$ equal to $4-x$? I tried to attempt teaching this by developing the intuition for why $$|x| = \begin{cases} x, & x > 0 \\ -x, & x < 0 \\ 0, & x = 0 \end{cases}$$ by taking positive numbers, negative n...

algebra-precalculus education absolute-value

Krijn

19:47

Do not say'take $x-4$ instead of $x$'

Show how it works for $|y|$ and then ask her what happens if $y = x-4$

Also, the question is probably better suited at matheducators.stackexchange.com

r9m

20:33

@robjohn as you mentioned ,, that is exactly the Hlawka Inequality :-) and as for the resemblance to IE formula ,. the inequality is not true for $4$ or more variables in general :)

robjohn

@r9m Ah... I was about to check out a few cases in 4 variables

Chris's sis the artist

20:50

@r9m What did you use for drawing that picture?

r9m

@Chris'ssistheartist Geogebra online :)

Chris's sis the artist

@r9m Nice picture, btw. Thanks. ;)

r9m

@Chris'ssistheartist -_-'

Transcript for

$Mathematics$ Mathematics