Mathematics - 2012-05-01 (page 5 of 5)

David Wheeler

21:00

so we don't have to go case-by-case

AbstractionOfMe

what was the purpose of having me check that e does not appear twice in any row or column in the table just created?

David Wheeler

now, remember how i said you could think of groups as "functions"?

AbstractionOfMe

yeah

David Wheeler

if e only occurs once in each row and column, that product must be unique, right?

i mean if ab = e, and ac = e, then e would occur in the row a*_ twice.

AbstractionOfMe

yeah

David Wheeler

21:02

and if ab = e and db = e, then e would occur in the COLUMN _*b twice.

AbstractionOfMe

i feel like my knowledge that all groups have unique identities is like my knowledge that if i type 1+1 into my calculator, it will say 2

i'm pretty damn sure, but still

i feel like im missing some internal mechanism

David Wheeler

right. the binary operation is a FUNCTION. so it has a unique value for each pair of inputs

AbstractionOfMe

yes

David Wheeler

a*b is always just ONE result

AbstractionOfMe

well, ciao for now

David Wheeler

21:05

now writing ax = xa = e = ay = ya implies x = xe = x(ay) = (xa)y = ey = y

AbstractionOfMe

thanks a lot

wait

David Wheeler

is just a FORMAL way of expressing what happens in the cayley tables

AbstractionOfMe

maybe i'll be late for class

doubt my teacher will care

right, but you only showed me that identities are unique for a specific group

David Wheeler

but we DO need the associative part...and functions are a good example of "generally associative" things

the exact same reasoning applies to inverses as to identities

ok, suppose we had a vector space...can there be 2 0-vectors?

AbstractionOfMe

wait are you saying to think of each element in a group as a function?

David Wheeler

21:08

suppose so. then 0 = 0+0' = 0', and well, it looks like they are the same.

lots of different things might be groups.

but, in general, these can be thought of as functions "on some set", yes.

but not just ANY function will do. we can only consider INVERTIBLE functions (bijective ones), because we need inverses, yes?

AbstractionOfMe

hmm i'm thinking my mind might be a bit blown, i need to slow down

David Wheeler

well, let's consider an easy example. suppose we have a 2-element set {x,y}

how many bijective functions are possible?

AbstractionOfMe

so, given an arbitrary cayley table (which uniquely defines a group)

my teacher is kicking me out

David Wheeler

lol

AbstractionOfMe

thanks a lot though i will think abotu this

David Wheeler

21:13

i come here often, you'll probably see me again

Gigili

Okay, this is annoying

David Wheeler

sorry

Gigili

> @Gigili You need to add what you have done to avoid the users repeating it to you and, you saying "I have done that and no solution comes out of it" comes across to me as just saying and not having done anything at all. – Kannappan Sampath 1 min ago

@DavidWheeler Not you, you're perfect.

AbstractionOfMe

Testing does cht work on iphone

Sweet

Gigili

It does?

David Wheeler

21:24

we're just cool like that. we work on a variety of modern electronic devices.

@Gigili sounds like "edit wars"

one of the people now running for moderator edited one of my posts.

i had written a post where i used the variable $u$. i then added the parenthetical comment: (why $u$? because i like it. don't u?)

he changed it to "don't you?" totally obliterating the pun.

AbstractionOfMe

Is every group G isomorphic to another group H whose elements are functions and composition is the operation?

David Wheeler

yes! there is, in fact a theorem to that effect called (oh, the irony) "cayley's theorem".

AbstractionOfMe

omg I love it

David Wheeler

but to understand cayley's theorem, you have to get a handle on a special kind of group, called a "permutation group"

AbstractionOfMe

All bijections of a set

David Wheeler

21:31

and there are "nicer" groups to play around with, before that

and bijections are functions. yes!

in fact, every group is a subgroup of some "bijection on a set group"

but these groups can be VERY big, and so sometimes we'd like a "smaller set" to use.

AbstractionOfMe

Im looking it up on wikipedia

David Wheeler

in some ways, group theory is like chemistry...in order to understand molecules, we have to work with "atoms" first

and some of the "nicer" atoms we work with are "cyclic groups", which behave very nicely. they have good manners. they're abelian, even.

AbstractionOfMe

The proof relies on existence of inverses which I don't believe yet

Kannappan Sampath

Existence of inverses for elements of a group comes with definition, although I have no idea what proof you're talking about.

David Wheeler

what inverses do you doubt the existence of?

AbstractionOfMe

21:40

Sorry I mean unique Inverses

Gigili

@DavidWheeler It is it.

David Wheeler

@KannappanSampath of course an "axiomatic" group comes with inverses guaranteed...but with a generally defined structure, the existence of inverses must be shown

AbstractionOfMe

It says By the existence of inverses, this function has a two-sided inverse

Kannappan Sampath

@DavidWheeler what do you mean generally defined structure.....? Please define things.

David Wheeler

@KannappanSampath i think he's reading about Cayley's theorem

AbstractionOfMe

21:42

Yes

David Wheeler

@KannappanSampath for example, "symmetries of a square"

Kannappan Sampath

Have you shown that is a group by checking the axioms?

(in which case, you'll need to show the existence of inverses, yes.)

David Wheeler

@KannappanSampath yes, i know how to do that. But AbstractionOfMe is just learning things.

Kannappan Sampath

And, I thought you were talking about a "group" G. But, if you're talking about a set G which you need to show is a group. But, I wonder why call it a group before showing it is a group.

David Wheeler

@KannappanSampath what he is struggling with is WHY inverses are unique...why must there be ONLY ONE?

Gigili

21:45

And I explained what I've done so clearly and someone just jumped in ans asked what I have done.

Kannappan Sampath

Anyway, people making less precise definitions are apparently more decisive about users being rowdies on the site. I have nothing to offer them sigh.

What you have done is not: "I have done blah" You need to expand what blah is.

I can say I showed that $\zeta$ exists on the whole of $\Bbb C$ but it really does not work.

But how would you know what $\zeta$ is?

Anyway, I would be vocal about this; You HAVE NOT shown what you have done ang got your homework solved cleverly.

David Wheeler

wouldn't it be more polite to say "i would find it better if you did (insert something here)" instead of "you need to do (insert the same something here)"?

2

one is a request, for clarity's sake. the other is an imperative.

AbstractionOfMe

I'm assuming there is such a thing as a "general" group.

Kannappan Sampath

I don't like being polite to people who explicitly cannot be polite with other users.

David Wheeler

an imperative carries with it the tacit assumption of superiority

Kannappan Sampath

21:49

To you, but not necessarily to the rest of the world.

AbstractionOfMe

Like I picture a general triangle as 3 points moving around randomly in space connected by segments

David Wheeler

someone who is the judge of their own superiority can be hard to swallow

AbstractionOfMe

Given any specific triangle S, the general one will at some time assume The shape of S

Kannappan Sampath

Someone who never observes simple rules of Capitalisation in almost every messages of theirs can be even more.

David Wheeler

if someone is rude to you, being rude in return...it's a draw. if someone is rude to you, and you are civil in return...then THEY look bad.

2

Kannappan Sampath

21:52

What I said before still applies or does it? sigh.

David Wheeler

@AbstractionOfMe well, there are different "flavors" of groups

AbstractionOfMe

@david, what do you mean

Gigili

@DavidWheeler Good luck with being rude and polite.

AbstractionOfMe

I mean I don't yet know how to taste them

David Wheeler

people amuse me. "oh, you didn't latex that right". "oh dear, your grammar is bad."

Kannappan Sampath

21:54

I should stop talking to people who actually are unworthy of talking to.

David Wheeler

@AbstractionOfMe some groups are abelian, some are not. these display different kinds of behavior.

AbstractionOfMe

I picture abelian groups as tables symmetrical across the diagonal

Kannappan Sampath

Some people however amuse me even more -- In that, they think they are superior and can get away with abusing another NATION, a punishable offence.

David Wheeler

@KannappanSampath you shouldn't pick at scabs. it just causes more bleeding.

AbstractionOfMe

What does an abelian group taste like

David Wheeler

21:57

@AbstractionOfMe sort of sweet, with a chewy center.

AbstractionOfMe

:)

Center like the core of the earth is in the center of the earth?

Kannappan Sampath

@DavidWheeler Why do you ping me?

David Wheeler

@KannappanSampath why bring personal issues to a math chat room?

Kannappan Sampath

@DavidWheeler Who has? and what have they brought?

David Wheeler

"Some people however amuse me even more -- In that, they think they are superior and can get away with abusing another NATION, a punishable offence." these are your words, yes?

Kannappan Sampath

22:01

Yes, how's that personal? And if it is and you agree it is, to keep things straight, you must have objected when those who did what is in question now did it.

David Wheeler

it might be just me, i'm prone to mis-interpret things at times, but that sounds kind of aggressive.

AbstractionOfMe

Can one prove cayley's them without assuming unique inverses

Kannappan Sampath

The person concerned leaving the room displays clearly that they are being a troll and I'd contend they would have to be banned from the network.

David Wheeler

@AbstractionOfMe unique inverses are part-and-parcel of what it means to have a group, so...no

@KannappanSampath i like talking with Gigili AND i like talking with you. so no matter which one of you forces the other to leave...I LOSE

Kannappan Sampath

(FWIW, it sounds more reasonable than your rowdy remark.)

You're being contradictory.

David Wheeler

22:06

@KannappanSampath you seem to be quick to take offense at things. the word i used was "rambunctious". it means "bristling with wild (or youthful) enthusiasm". it's not an insult.

Asher Rubin

lol

David Wheeler

i try to be a friendly person. i have no reason to bear anyone ill will.

Kannappan Sampath

But I differ. The word you used comes across to me in an entirely different context.

(I do read transcripts, so I try to make sense.)

David Wheeler

well, language can be subtle like that. words have NUANCES, the "dictionary meaning" of them isn't always faithful.

and cultural context can make a difference, too. perhaps where you live "rowdiness" would be seen as a greater social transgression than where i live, causing you to become even MORE upset.

so then you become "sensitized" looking for any perceived slight in anything i say. the trouble is, i'm more-or-less oblivious to that.

Kannappan Sampath

I am reasonably sure, rowdyism is not something people anywhere in the WORLD would welcome and your attributing that quality to me is more of an attempt to attribute falsehood to me.

(IOW: Do you know I am a rowdy? Then why attribute that to me?)

David Wheeler

22:14

but see, you're already reacting defensively, responding to a perceived attack. in your mind, a simple mis-reading of my INTENT leads you to equate what i said with slander.

AbstractionOfMe

@david this is how I currently taste the property of associativity in binary operations: I picture one after another all Cayley tables C_1, ..., C_n^(n^2) over a set N of elements. Then I see basically a little l.e.d. That flashes on when the current table is associative and off when the current table is not

David Wheeler

"a rowdy person" and "rambunctious" can mean two totally different things.

Kannappan Sampath

As I said, I am NOT misreading; please do not try to convince me that, your opinions of me are absolute truth, that for no positive epsilon, you'd find an element of false hood in the epsilon-neighbourhood of your statements.

AbstractionOfMe

I think I am bothered by not seeing any machinery in between the current table and the led

David Wheeler

you hear: loud, reckless, annoying, perhaps....and i MEAN: enthusiastic, outspoken, youthful.

AbstractionOfMe

22:18

I don't actually know which tables are associative, just that some are because eg integers under addition is

David Wheeler

@KannappanSampath i have only your behavior in this chat-room to judge by. in real life, you might be very different (and i would never know). i do not place so much importance on my own limited views of things as to call them "truth".

@AbstractionOfMe it can be difficult, just by looking at a table, to see if it is associative. that's why verifiying associativity of a group is often the hardest part.

@KannappanSampath what i AM saying is that you attach more malice to my words than what i type them with.

Kannappan Sampath

As I said, you're being contradictory. You say what you said were truth and I am perceiving them wrongly, but now you say, you don't know for sure if your opinions match the real life "me". This amply demonstrates, you quip words.

David Wheeler

@KannappanSampath there is nothing in your replies that convinces me otherwise, although i still hold out some hope.

lewist

@KannappanSampath I am not sure if you are confused as to the possible meanings of David Wheeler's statement, or if you don't trust David's explanation as being truthful.

Kannappan Sampath

@DavidWheeler So, you have now come to a point to agree with my interpretation of your remark. This alone suggests enough for me.

David Wheeler

22:22

@KannappanSampath life isn't like mathematics...verbal statements are rarely just "true" or "false". if you believe you are something other than what i said, that is your right.

Kannappan Sampath

And, my replies don't have to convince you otherwise. But, you simply are No ONE to form opinions and write them in a public domain.

David Wheeler

hmm...ok, in this country there is a difference between murder, and manslaughter. the difference is not "the result of the crime"(death), but rather, what intent there was. i don't know how many times i have to say i didn't intend to insult you.

Kannappan Sampath

My argument is this: You claimed I am a rowdy which I am claiming is simply not your right. You absolutely have no right whatsoever that lets you do that in a public domain.

David Wheeler

@KannappanSampath and who IS "someone to form opinions and write them in a public domain"? must we limit ourselves to purely factual statements at all times?

AbstractionOfMe

@david, I just had the thought that I could approach the question by considering a computer program that generates all groups of order n and show that such a program will never produce a group with non-unique identities

David Wheeler

22:27

@AbstractionOfMe such a program exists already, called GAP

AbstractionOfMe

It could just generate all Cayley tables and backtrack if they fail one of the properties

Benjamin Lim

@DavidWheeler Hey

@KannappanSampath Hey

Kannappan Sampath

You have done that and you quip now. Well, pointless arguing IMO. You simply will write things upside down every time and not agree on anything. Long live you and your opinions. And, support everybody else in their endeavours against other nations. Regards to you David Wheeler.

David Wheeler

@KannappanSampath the irony of it all is i was DEFENDING you and asking for people not to judge you so harshly.

Benjamin Lim

@DavidWheeler Can I ask you something

about galois theory

David Wheeler

22:29

@BenjaminLim you can ask me anything. but if it's for a date, i'll have to decline.

Benjamin Lim

I an assignment our lecturer asks to name an example of a non-normal, non-separable extension of degree 5

@DavidWheeler The thing is all the extensions I know that are not separable are normal anyway.

David Wheeler

non-separable extensions are kind of tricky

Benjamin Lim

well the one I know of is $\Bbb{F}_p(t)$

The problem is naming a non-normal extension of this that is not separable

Josh Gitlin

I see a number of chat flags popping up from this room, do I need to remind people to be nice? :-)

Alenanno

@JoshGitlin You stole my thunder. :(

Josh Gitlin

22:33

hahaha

AbstractionOfMe

Thus if the program generates a table with 2 e's xa and xb in a row and 2 e's in the corresponding column ax and b x, then it will check (ax)b and a(xb) and delete the table because they're not equal

David Wheeler

@JoshGitlin can you elucidate?

Kannappan Sampath

(FWIW, I did not raise any flag.)

Benjamin Lim

@DavidWheeler Do you any example of an extension like that?

Josh Gitlin

@DavidWheeler What is the question? Sorry, I am not up to speed here...

David Wheeler

22:35

who is flagging comments, and for what reason?

Alenanno

@DavidWheeler I think the "who" is private info there... :)

Benjamin Lim

@DavidWheeler I tried many things but I am out of ideas with this one.....

David Wheeler

well, in the absence of such information i can give you my understanding of the situation...but take it with a grain of salt.

Josh Gitlin

@DavidWheeler Yeah I can't tell you that

Benjamin Lim

@DavidWheeler Are you talking about extensions :D?

Josh Gitlin

22:37

I'm just here to keep the peace

AbstractionOfMe

@david: but how do I know the computer will behave as expected?

David Wheeler

at some point in the past, apparently Gigili made a comment which Kannappan took to be an insult to his country. i wasn't here for that.

lewist

(It is stricken from the transcript.)

David Wheeler

since then, things are "frosty" whenever they are both present. in a later discussion, people were discussing this incident, and i said something regarding Kannappan, to the effect of: "he's just rambunctious", to which he has taken some offence.

Kannappan Sampath

Done now.

Benjamin Lim

22:41

@KannappanSampath I have advised you before: "He abused me, he struck me, he overpowered me, he robbed me." Those who harbor such thoughts do not still their hatred. - Dhammapada verse 3

Alenanno

@DavidWheeler I'm translating that, lol.

David Wheeler

personally, i like both of them, which makes the situation rather awkward for me.

AbstractionOfMe

@david did you see my question?

Not to rush you

David Wheeler

apparently, they (Gigili and Kannappan) had a disagreement over an edit (i think), which brought all this to the surface again.

Kannappan Sampath

@BenjaminLim Yes, Ben. You did, but I am rather upset how some people behave.

Josh Gitlin

22:43

@DavidWheeler Well if there's anything I can do to help, I'll do my best!

David Wheeler

@AbstractionOfMe well for sets of small order, only a few tables will fit the bill. for a set of order 4, there are just two "different" tables.

Alenanno

Me too, hoping @JoshGitlin doesn't steal my thunder again.

:P

David Wheeler

again, without knowing who flagged (and why) it's hard to know if i am hitting the nail on the head...but that's my story, and i'm sticking to it.

Benjamin Lim

@DavidWheeler Can you help me?

David Wheeler

@BenjaminLim i am still thinking about it

Benjamin Lim

22:46

Ok

Alenanno

@DavidWheeler Well, I still learned a new word, thanks. :P

AbstractionOfMe

@david, but how do I know I can trust the computer to remove all tables with 2+ inverses for an element, especially if it's calculating groups over some huge set?

Kannappan Sampath

Anyway, I should sleep now. Let me see what I can do to feel good.

AbstractionOfMe

I know that it should

But I dont know all the details of the insides of computers

David Wheeler

what i am thinking about is $\Bbb{F}_5(t)(\sqrt[5]{t})$

AbstractionOfMe

22:49

Is there an axiom in math that says if person A says they will do X, then they will indeed do x?

X rather

David Wheeler

@AbstractionOfMe the trouble with your approach, is that to be fully confident...you would have to try "every" group, and there's an infinite number of them....

AbstractionOfMe

So how do you fix that approach?

Kannappan Sampath

This place, after some reflection, looks like exactly not the place to hang around atleast when an "user" (I would like abuser here.) is here. So, I'll best achieve this by not being here at all.

David Wheeler

remember where we started? if ax = xa = e AND ay = ya = e, then x = xe = x(ay) = (xa)y = ey = y?

AbstractionOfMe

I've thoroughly enjoyed this conversation

@david yes I temember

David Wheeler

22:53

think of that as a statement where a,x, y and e are functions.

so "ay = ya = e" means "y undoes what a does"

AbstractionOfMe

Ok

Then what

David Wheeler

or logically, sort of like this: imagine "a" means: turn light off", and "y" means "turn light on" is x means "something" (who knows what).

then something = something + do nothing = something + turn light off + turn light on

and we know that something + turn light off = does nothing, so

something = something + turn light off + turn light on = do nothing + turn light on, and thus something = turn light on

@Alenanno @JoshGitlin before, Kannappan has left, vowing never to return (or some such). what to do?

Alenanno

@DavidWheeler About what?

One thing: flags are for serious stuff. Don't flag posts at random. Every time you do, all moderators (from all sites) are alerted.

9

Josh Gitlin

@DavidWheeler I guess hope he reconsiders? I've come late to the party and felt ill equipped to help much unfortunately.

David Wheeler

i can't remember ever flagging anything.

Alenanno

23:02

@DavidWheeler I'm obviously talking to whom flagged, not you. :P Sorry if it was ambiguous.

David Wheeler

Kannappan is quite gifted at math...when he's talking about it here, he really shines

Alenanno

@DavidWheeler We cannot do anything about it. You can tell him that you hope he reconsiders.

David Wheeler

sigh i try...but not. getting. through.

i suppose its possible that someone flagged something, and then went afk, or just left. i won't lose too much sleep over it. maybe the last five minutes before bed. perhaps seven. or only three.

Alenanno

Disputes happen, but we're grown-ups. I think we can solve things by talking in a polite, constructive way. Misunderstanding in chat is easy. :P

David Wheeler

one disadvantage of chat-rooms (versus something like skype, for example) is all sense of intonation is lost...simple kidding can erupt into something ugly

Josh Gitlin

23:07

That is true

Alenanno

I add the :P after it or over-exaggerate the tone to show I'm joking. :D

Josh Gitlin

Same here

Alenanno

Like I did with the thunder, about @JoshGitlin... Well, he did stole my thunder, but I'm not offended by that. Maybe. :P

David Wheeler

well i'm deadly serious, mister. that's right, "mister". with a capital M.

:P

Alenanno

This is my first chat on Math by the way.

Josh Gitlin

23:09

Likewise I think

jrg

By the way, in addition to what @Alenanno said about chat flags:

in Ask Ubuntu General Room, Mar 1 at 14:44, by Arda Xi

Guys, guys. That flag button is a great responsibility. I realise it's a lot of fun to hate on/love Justin Bieber, but that doesn't mean you have to pull guys like me off of my cozy star ship and to here just to make sure nobody gets banned. :)

David Wheeler

well welcome to the loony bin. just throw words like "measurable" and "uniformly continuous" around from time to time, you'll be fine.

jrg

in Ask Ubuntu General Room, Mar 1 at 14:49, by Arda Xi

So, Flagging 101: Flagging is srs bsns. If a flag passes, it suspends the flagged for at least 30 minutes. Not only that, but it pulls every active 10ker/mod into the room. Try to talk to the person before flagging, and only use that flag button if something needs to be removed NOW and the user can't be reasoned with. With that, I am off. Enjoy yourselves. :)

(I'm done. And yes, I have those bookmarked. :P)

Josh Gitlin

Thanks @jrg, also LOL

jrg

@JoshGitlin howdy! :)

Josh Gitlin

23:10

Hey there

You're here for the flagfest too? :-)

David Wheeler

wow, 3 blues. must be defcon 2.

3

Benjamin Lim

@DavidWheeler hhahahaaha

jrg

I was here, just didn't find the chat messages until now. :)

Benjamin Lim

FPCON DELTA

David Wheeler

@AbstractionOfMe are you still here?

Alenanno

23:13

What's this machine-gun starring?

David Wheeler

i guess if you can't flag, star!!!!1111

Josh Gitlin

oh boy, LOL

David Wheeler

gotta love those interwebs..they're really happening

@BenjaminLim did you check out that field?

Alenanno

Ok guys, I'll let you talk about numbers, letters, equations and so on. Goodnight!

t.b.

Gee. Looks like I missed some fun...

David Wheeler

23:18

sorry. i'll save you some next time.

David Wheeler

23:45

oh well, looks like the party's over. a bientot, gendermes....

Chris

Hello all!

Can the rank of a matrix affect the solution set of the equivalent non-homogeneous linear system of equations? Thank you!

Mariano Suárez-Alvarez

what do you mean by "affect"?

Antonio Vargas

@Chris, yes, the rank can tell you whether the system has one, none, or infinitely-many solutions.

Chris

Well, I am thinking maybe, the solution set has one and only element or be infinite.

Well, I've been asked this question and I am trying to figure out the answer..
I am thinking that it does affect the solution set. We know that, if rank(matrix) == rank(matrix_) the solution set is nonempty...where matrix_ is the matrix containing as a last column the b1, b2 ,b3 .. etc (i don't know how this matrix is called in engilsh)

@AntonioVargas: Thank you! Is there any theorem?

Antonio Vargas

@Chris Yes, here.

I didn't know it had a name until just now :)

Chris

23:56

@AntonioVargas: Thank you very much! I didn't even know its existence (obviously :P)

Also, I have this question here: http://math.stackexchange.com/questions/138969/can-a-solution-set-be-an-image-of-a-linear-transformation and I've got two answers which are difficult for me to understand. :-/
Could somebody help?

Transcript for

$Mathematics$ Mathematics