Mathematics - 2012-04-12 (page 2 of 3)

Benjamin Lim

14:00

@tb I don't understand this:

2

A: Given a commutative ring $R$ and an epimorphism $R^m \to R^n$ is then $m \geq n$?

As mentioned in the comments to the question, it’s the first part of Exercise 2.11 in Atiyah-MacDonald, and I refer to the comments for an answer. The second part of Exercise 2.11 (which is perhaps more interesting) has been the subject of this MO question. I especially like Balazs Strenner’s...

The proof is like this:

Suppose that $m > n$

then there is an injective endomorphism of $A^n$ to $A^n$

(Actually it is an A-module homomorphism between an isomorphic copy of $A^n$ sitting inside $A^m$ and $A^n$).

Now how can we assume that such an injective endomorphism, say $b$ is such that $b(e_n) \in A^{n-1}$?

t.b.

sorry to interrupt: can you please star my undeletion request so that it doesn't get lost?

Benjamin Lim

ok

t.b.

thanks

Benjamin Lim

Basically I could figure everything out by myself, setting up cayley hamilton etc

Well for such an endomorphism $b$, we know that the characteristic equation of $b$ cannot have non-zero constant term.

For if it did, then $b$ would no longer be injective

t.b.

wait a second I can't follow this quickly I haven't even seen what the notation means, etc.

Benjamin Lim

14:05

the link is pierre's answer

Gigili

Hi @robjohn, do you think I need to add the quotient rule to my answer?

Benjamin Lim

Even though I have not studied integral extensions I know enough about algebraic extensions to understand what he's talking about there.

@tb And I don't have enough rep to comment on the MO thread: mathoverflow.net/questions/136/atiyah-macdonald-exercise-2-11/…

@tb are you there?

t.b.

Sorry, I dozed off because this is CA.

(why do you insist on asking me this stuff?)

Benjamin Lim

sorry man

I don't understand how applying say the characteristic equation of $b$ to $e_n$ gives zero.....

no worries

it's a bit late for me now

I should go to bed

12.15

@tb no problems, bye man!!

t.b.

Okay, do that :) Good night

@Gigili It looks like OP computed the derivative by differentiating both numerator and denominator so in as wrong a way as can be. Maybe it would be good to clarify this.

robjohn

14:35

@Gigili Ah, I posted an answer to that thread before I saw your question here. I guess we have covered it all :-)

@tb I agree :-)

@tb I think I covered that :-)

t.b.

@robjohn now this eejit has deleted another one.

robjohn

@tb He is losing a lot of rep in the process.

t.b.

Well, if he's afraid of losing real world rep, he might think about concocting a moniker...

I find this seriously annoying. You're inducing complete strangers to invest 20 minutes of their time and then you're grabbing the results, kill any trace and go away?

robjohn

@tb I understand and agree completely.

t.b.

Could you please vote on the second one, too?

robjohn

14:49

@tb done

t.b.

Thanks.

Nimza

good evening. Does anybody have reference to something about quadratically constrained linear programming?

Particularly, I have to solve linear programming problem in ellipsoid

Jonas Teuwen

15:15

Yuri Manin is speaking as we speak!

Tyler Bailey

Howdy

t.b.

15:36

hey there

Could somebody please be so kind and vote Philipp's (very nice) answer here up so that the thread is protected from deletion by the OP?

Tyler Bailey

Sure did. Is that what happened with the earlier question?

t.b.

Thanks. Yes, the OP deleted both his questions after he received rather complete answers.

Tyler Bailey

Weird.

How many votes are needed before it can't be deleted?

t.b.

I'm not sure how many you need exactly but I seem to remember that two answers with positive vote total each or one answer with two positive votes is enough.

According to this thread one vote on an answer is already enough. Much more than what you'll ever need to know is here

user25470

15:54

Hey ppl!
Perhaps this is a silly question but can monics in anyway be interpeted as limits?

t.b.

No, not in general.

user25470

thought so

but in specific?

t.b.

Well, monics in an abelian category are the kernel of their cokernel, so they are limits.

For example.

robjohn

@tb how does that protect it? (I upvoted any way :-)

t.b.

Monics that are limits of a parallel pair of arrows are called regular monics. This generalizes the previous observation.

@robjohn OP can only delete the question if no answer has positive vote count.

robjohn

15:58

learn something every day :-)

t.b.

I gave two relevant links here (10 mins ago)

Rob

Thanks for the useful info and the links ;-)

robjohn

@tb so another answer would have protected it, too.

t.b.

@robjohn looks like it. But clicking the up-arrow is substantially less work-intensive :)

robjohn

@tb Indeed

Rob

16:46

@robjohn @Rofler Would you agree with Brian that my question about the high school math definition of a variable invokes "either very little to say or too much for this forum." ?

Rofler

@Rob Yes

Rob

@Rofler Which of the two do you side with?

Very little to say
or
too much for this forum?

Rofler

@Rob Both

robjohn

18:31

@hhh: so much denial :-)

arete

19:37

Consider each of the set. Find an open cover for which there is no finite subcover. $\mathbb{Q}\bigcap[0,1]$.

I am still trying to figure out an open subcover for this problem. One of my classmates used the open cover $(-1, \sqrt{2})\bigcap \mathbb{Q} \bigcup (\sqrt{2}-\frac{1}{n},2) \bigcap \mathbb{Q}$ with $n\in \mathbb{N}$.

Is there any reason why this open cover would be chosen over a much simpler cover?

Matt N.

@arete What would be a simpler example?

Can you tell me why the point "in the middle" is chosen to be $\sqrt{2}$?

arete

Because it's irrational and therefore not contained in Q.

Matt N.

Yes and why do we want that?

arete

Because we're trying to create an infinite subcover.

Matt N.

No.

Ok, let's look at a simpler example:

arete

19:46

What is the impetus?

Matt N.

Can you give me an open cover of $(0,1)$ that doesn't have a finite subcover?

arete

no

because the set is not compact

Matt N.

What does compact mean? (can you write out the definition)

A topological space $X$ is compact if ...

arete

Compact means that a set is both closed and bounded (Heine-Borel Theorem). So, choosing $\sqrt{2}$ would give you an interval whose endpoint is outside of the interval $(0,1)$.

Matt N.

No, that's not the definition, that's the Heine-Borel theorem!

arete

19:50

OH

Matt N.

I would like you to complete this sentence: A topological space $X$ is compact if...

arete

Sorry definition would be that a set K is compact if every sequence has a subsequence $a_{n_k}$ whose limit is in the set K.

Where $K\subseteq \mathbb{R}$

Matt N.

Well that is equivalent to the definition I am looking for if we are in a metric space.

But I want you to give me the definition for a general topological space $X$, not necessarily metric.

arete

Weird thing is for my analysis class we haven't covered metric spaces.

Matt N.

Exactly, you don't need metric spaces to give me the definition of compactness.

(although in your analysis course you'll be doing everything in $\mathbb R$ which is a metric space anyway : ))

But let's forget about that.

arete

19:54

Honestly, this is the definition that we have been using. I guess we haven't covered the material that you're discussing. Unless that is I'm just completely missing something haha.

Matt N.

It might be that you haven't but if you haven't I'm not quite sure how you're supposed to be answering the question you're supposed to be answering : )

Have a look at the definition here.

arete

My prof has been leading up to this content, but he hasn't gotten to it yet. We'll probably be proofing this tomorrow.

Matt N.

You need it to do your question.

Have you read it? (the definition)

arete

Reading it right now.

Matt N.

Good : )

arete

19:59

Thanks Matt

Matt N.

Np : )

Let me know when you think you grasp what the definition is saying.

There is a separate room for topology, would you mind moving our discussion there?

arete

Sure

Matt N.

See you there : )

robjohn

@billcarson: hey there

Rob

@robjohn Is it official that Asaf is not coming back to the chat room?

robjohn

20:10

@Jasper good day

@Rob who know?

@Rob There has been no notarized documentation to that effect.

user19161

@arete Well as you study more you will realize that different books or courses will use different definitions. They may or may not be equivalent. Just something for you to take note.

user19161

@robjohn Hey! I see you still have the italics which means Asaf has not returned!

robjohn

@JasperLoy would I not have the italics if he returned?

user19161

@robjohn Oh I would have thought that he would continue to be owner of the room but you can be an owner of the room too!

robjohn

@JasperLoy there were three owners before; Gigili, Marc Gravell, and myself.

Rob

20:13

@robjohn Would you not hand him back his ownership?

user19161

@robjohn Ah I am not in touch with these worldly affairs! :-)

robjohn

Gigili relinquished ownership.

user19161

So what happened to Jordan? Is everything OK now?

robjohn

@Rob If he came back and wanted it, I see no reason not to.

@JasperLoy I have not read the transcript. Was there a problem?

user19161

@robjohn They should not have starred some of his comments.

user19161

20:15

I was there when it happened. People, can I appeal for you to show more sensitivity?

robjohn

This is the last I see.

user19161

@robjohn Yeah it was around there, plus minus a few messages.

Rob

I admit it was my fault for being so insensitive ...

user19161

We may all unintentionally hurt people at times, but sometimes it gets to the point where I feel the hurting may be intentional...

user19161

Just talking in general about my observation of SE chat rooms.

robjohn

20:21

I have unstarred his comments that are current. I cannot do so with the earlier ones.

Had I been here, I could have unstarred the comments when they were bothering Jordan.

Ping me if something like this happens again.

Sometimes I am gone, but sometimes I am simply in another window.

Tyler Bailey

hi

Rob

@robjohn BTW Kannappan had this last message.

@TylerBailey Hi

Tyler Bailey

Oh man, are people leaving math chat left and right?

robjohn

@Rob Ack! I had not seen that. Thanks for pointing it out.

Rob

@robjohn :-(

robjohn

20:29

@TylerBailey Some temperaments do not seem to be copacetic with each other these days.

Rob

copacetic

Tyler Bailey

I guess that's bound to happen...

it must be pretty bad if they in fact leave though

robjohn

@TylerBailey there are some who push others to the brink, and there are some whose brink is closer than others.

Rob

Copacetically put robjohn.

Tyler Bailey

haha

Agreed

Rob

20:36

;-)

Tyler Bailey

I did see Jordan post on the forum not too long ago, at least, I think, after this junk had happened.

So maybe he hasn't actually quit chat.

user19161

@robjohn I have unintentionally hurt others at times, but I feel others have intentionally hurt me at times (not in this room). :-)

robjohn

@JasperLoy I have to keep in mind that many people are a lot less restrained on the net and so I need to keep a thicker skin.

@TylerBailey Asaf has posted to MSE, but he has not been back to chat.

Tyler Bailey

Oh my...

user19161

@robjohn But I won't share with you here what happened. It's a bit personal. :-)

robjohn

20:49

@JasperLoy with regards to Kannappan or Asaf?

anon

I AM NOT WORTHY

user19161

@robjohn No, I mean with regard to myself. :-) Also it should be "with regard to" or "as regards". Note the absence or presence of the "s".

robjohn

@JasperLoy Ah, sure, I would not pry into matters that are too personal

user19161

@anon I don't understand why that is in the answer.

Jonas Teuwen

BACK.

Rob

20:53

Welcome

Jonas Teuwen

From 7:00 to 23:00 for the Dutch Mathematical Conference 8-).

Well, from home to home.

robjohn

@anon :-)

user19161

@robjohn Actually I think such comments ideally should not be in the answers as it is noise. But it is alright in the comments. But I ususally don't try to edit such things out from the answers.

robjohn

@JasperLoy I agree. I have accepted edits from others who delete such language, but I have not deleted any myself.

Jonas Teuwen

21:11

anon

I'm wondering about the discrete tomography of the audience's seating distribution.

user19161

@anon It is because the sample size is small.

Matt N.

@JonasTeuwen Welcome back.

Jonas Teuwen

8-).

Matt N.

So you were at the conference all day?

Jonas Teuwen

21:18

Yes.

Rob

Did you learn anything new?

Jonas Teuwen

Not much, I just looked at the cool images.

And was wondering why they didn't have any decent drinks.

Tyler Bailey

I don't care about new things, did you learn anything interesting? :p

user19161

@jonas I was thinking about your mentioning of reading short works yesterday. Well, sometimes reading a short work may cause you to waste more time as ultimately you don't learn anything really useful and still have to turn to the longer work.

Jonas Teuwen

Nah, that's not in my experience.

Stein's books are quite short for example.

Matt N.

21:20

Hey Jasper. You have done a good deal of algebraic topology, right?

user19161

@JonasTeuwen What would there be besides coffee, tea or fruit punch?

Jonas Teuwen

Well, there was beer.

But that hardly contains any alcohol.

Matt N.

I like beer : )

user19161

@MattN I have not studied much math and I have forgotten most of the things I learnt, so I am not the right person to ask. Sorry. :-)

Matt N.

@JonasTeuwen But it's tasty!

Jonas Teuwen

21:21

@MattN Not this beer.

Matt N.

What was it?

Jonas Teuwen

I drank it because there was nothing better, but I didn't notice anything except the high urge to pee :(.

Some Dutch beer.

user19161

@JonasTeuwen Wait, they have beer at math talks? Amazing!

Jonas Teuwen

Of course?

user19161

This is a surprise to me!

Rob

21:23

@JonasTeuwen Did you bring back any cool graphics?

Jonas Teuwen

No.

Matt N.

Is Amstel a Dutch beer? I can't think of anything else.

Sounds Dutch.

Tyler Bailey

They all are if you BYOB, Jasper :p

Matt N.

But then again I don't speak Dutch...

user19161

@TylerBailey I hate beer. But I like wine, though I won't buy it myself. My friends gave me a few bottles. :-)

Tyler Bailey

21:25

It's certainly an acquired taste. Wine is good too. I've almost completely stopped drinking though. Hangovers are awful.

Jonas Teuwen

@MattN Yes. And Bavaria, Heineken, Grolsch, Hertog Jan...

Matt N.

Lol, wat? "Bavaria" is a Dutch beer?

Jonas Teuwen

Yes.

Matt N.

I saw that : )

user19161

Now Jonas has been infected with Matt's virus!

Matt N.

21:27

@JonasTeuwen I feel betrayed. (Not that I've ever seen "Bavaria" anywhere here before but the attempt counts)

user19161

I should try to use Jonas's smiley like this 8-)

anon

glasses-smiley

Jonas Teuwen

@MattN In Bavaria it is not sold...

Rob

How can a beer not be sold in the place that it is named after?

robjohn

@anon according to this page 8-) means "wearing contacts"

Matt N.

21:35

@JonasTeuwen Obviously not : )

anon

@robjohn Nonstandard smileys don't have standard meanings. ;)

Jonas Teuwen

8-)?

I once tried contact, it felt like my eyes were on fire.

robjohn

@JonasTeuwen That can happen when you insult someone who is smoking and they throw their drink on you.

Matt N.

They must be drinking a strong drink for that to happen.

robjohn

@MattN I know that 151 Rum can sustain a flame.

Matt N.

21:47

Not heard of that before.

robjohn

Bacardi 151

Bacardi 151 is an over-proof rum. The 151-proof liquor has an alcohol content of 75.5%, compared to the usual 35%-40%. Due to its high proof, it is typically used as a component in cocktails (whose final alcohol concentration may be greater than that of drinks made with conventional rum). The spirit is flammable and is used in flaming beverages such as flaming B-52s. Bacardi's 151-proof rum has been available in the US since at least 1981. The product, as shipped by Bacardi, is equipped with a flame arrester in the neck of the bottle to prevent large volumes of the flammable liquid f...

Matt N.

Ew.

robjohn

It is good with Coke or Dr Pepper

Although I have had Everclear 190

Matt N.

That doesn't sound nice...

robjohn

That article says it is illegal to sell the 190 in California. My friend must have gotten it out of state, or it must not have been illegal back then.

Jonas Teuwen

21:58

There was no harmonic analysis at the Dutch Mathematical Conference :-(.

This was an unbearable thought.

I will talk to the new chairman of the Royal Mathematical Society (of The Netherlands, of course).

It's a bearded guy.

Our chairman.

Matt N.

22:24

Is a measure continuous if it's continuous from above and below?

Jonas Teuwen

Yes.

Matt N.

Thanks : )

Heh but the definition of absolute continuity is much simpler. $\mu$ is absolutely continuous w.r.t. to $\lambda$ if $\mu (A) = 0$ whenever $\lambda (A) = 0$.

Doh : )

Are there any measures apart from the counting measure that only have zero measure for the empty set?

Jonas Teuwen

Hmm.

Probably.

Sums of Dirac measures?

With the counting measure.

Stuff like that.

Matt N.

For 1, can't he just take $\mu$ to be the counting measure? : )

What am I missing?

I should go to sleep.

Good night folks.

Jonas Teuwen

Good night, ding ding ding!

robjohn

22:37

@JonasTeuwen ding, ding, ding?

Jonas Teuwen

@robjohn Yes.

robjohn

@JonasTeuwen why?

anon

The chatroom is a barbershop or something.

Jonas Teuwen

I don't know 8-).

robjohn

okay

anon

22:38

And when someone walks out the bell above the door dings.

Rob

But who shaves the barber?

anon

The barber is a woman.

user19161

I know why there is ding ding ding.

user19161

Because good night symbolizes the end of something, and so does ding ding ding!

user19161

So our subconscious associates them and hence Jonas types that.

Jonas Teuwen

22:44

Hmmm, thanks Jasper.

Rob

@anon What if the barber was a half-man, half-woman like your avatar?

user19161

@Rob I just realized it was half-half!

anon

My avatar depicts twins, not an individual.

Rob

Question: How can two individuals born at the same time who biologically have the same mother and father not be twins?

Jonas Teuwen

Just put the egg in another woman 8-).

anon

22:50

Same mother Jonas.

Jonas Teuwen

That's the same mother.

The mother is the one that delivers the genetic code.

Rob

The biological mother gives birth to both.

Jonas Teuwen

That's a twin by definition holy monkey.

Rob

But in this case they are not twins.

anon

The woman becomes pregnant after her first child with the same father and then goes back in time, birthing the second child at the same moment as the first.

Jonas Teuwen

22:54

An interesting twist to the plot.

Rob

This case of the two individuals can and has actually happened in reality.

Jonas Teuwen

I fail to care about that. Sorry.

Rob

Np

anon

Happened in reality? If you change the meaning of words like "twin" or "reality" I guess.

Benjamin Lim

Where are the algebraists in this chatroom? Why is it now dominated by analysis people?

anon

22:57

It's our shift right now.

Benjamin Lim

oh crap.....

Rob

Want the answer?

Jonas Teuwen

No.

Benjamin Lim

No.

Transcript for

$Mathematics$ Mathematics