Hi All,Are all submodules of Frac A of dedekind ring are fractional ideals?

@TedShifrin Ted.

@Ram no

e.g. Z[1/2] inside Q is a Z-submodule but not a fractional ideal, although it is a direct limit of the fractional ideals (2^-n)Z

Prof. @Ted it is generally an interesting questions. The use of indexes there isn't trivial as it feels as if the probability of each index willb e different, making it not as useful. But when you define the indexes well, and pay attention to the fact you don't care the $i$'th card sits as long as it sits before the 4 aces, it makes the solution much easier.

@anon thanks. But then is every submodule is a direct limit of fractional ideals in above case?

where the $i$'th card sits*

I get the 1/5, @Studentmath. I meant my summation formula with binomial coeffs.

yes @Pedro?

@TedShifrin Let $G=B(0,1)\cap B(1,1)$ in $\Bbb C$:

I have to see where $G$ is sent to by complex inversion.

Oh.

I know.

Inversion in the unit circle?

Inversion everywhere.

I mean $z\mapsto z^{-1}$.

Huh?

Oh ...

So, I think it is sent to the following region.

It is the plane $\Re z >1/2$, minus the intersection with $B(0,1)$.

@Ram every A-submodule M of Frac(A) is the union of the principal fractional ideals it contains (that is, for all m in M, we know M contains mA and mA contains m), and one checks the the collection of fractional ideals it contains is a directed system (if M contains (m) and (n) then it contains (m,n))

@Pedro: I need to ponder.

Winnie the Pooh ponders

Hi @nabla @anon

Hi @TedShifrin

Yes, @Pedro, you be right.

hi

@anon thanks. I did a mistake by assuming $\mathbb Z[1/2]$ as a fractional ideal. I just corrected my mistake.

You up to completing the square yet, @anon? ;)

we're doing more graphical things, solving/factoring quadratics is a bit later

Cool :) I approve of pictures.

I'm so disappointed in our real analysis course

We spent an entire 2 lectures (~2.5 hours total) proving equality of ordered pairs...which took me about a minute or so to do in my head

The professor kept talking about random stuff unrelated to the course

what was his answer to the fraction "contradiction"?

2/4 and 7/14

He said to ask him at the end of the course if we really want to know

I'm fairly certain he's implying well-defined concepts

That is such a lame response.

lol

@Gustavo, it's the former, but this is just sloppy classical notation.

Thank goodness. Thanks a lot @TedShifrin.

@nablablah It is simply the rule for simplifying fractions.

@Gustavo, if it helps, I have an undergrad diff geo text on my website you're welcome to look at.

Oh thanks! Please do share!

My profile has the link ...

Thanks

@Ted Oh I know, just meant that the question is interesting as is too - beyond the nice summation it leads to

Which I can't get how to solve, by the way.

But I've gotta catch some sleep, good night!

later

Hi EveryBody

Somebody tell me, what does $[a,b] \times [c,d]$ mean?

Hey @FractalHand. What you are describing is a cartesian product. Information can be found here : en.wikipedia.org/wiki/Cartesian_product

@GustavoMontano Does it mean the same in the Topology books?

Basically, $[a,b]$ and $[c,d]$ are a set of points. When you take the cartesian product of both sets you get another set of points.

Hmmm, I would think so.

yo ya'll know pdes? ^

PDE's require too much memorisation :(. I'm afraid I have forgotten all that good stuff.

come on many at least try and help me out with the second part of this... I am just lost on it :/

plzzz

Oh don't ask to ask - just ask.

:/

Yeh it has been a year too long. I'm afraid I don't have a real idea.

ugh so hard to find good pders I swear

All I know, is that questions like these always follow some sort of method.

Given the way it has been structured.

Whether its change of variables, etc etc.

I'm at the second part which is find 2 solutions that satisfy the pde in the form of $-1+2e^x$

-

I have $u(x, 3x) = -1+ke^x$ and $u(x,3x) = 1+2e^x$ I was reading over the notes... something to do with setting them equal and solving for the constant but I got as far as whatever I wrote so I'm stuck

As in equate the expressions you have noted and solve for $k$?

hmmmmmmmmmm

$\rightarrow k =\frac{-2}{-e^{x}} -\frac{-2e^x}{-e^{x}} \rightarrow k = \frac{2}{e^{x}}-2 \rightarrow k =2e^{-x}-2$

$ k = 2e^{-x}-2$

So $k$ is not a constant.

?

Well, the value of $k$ is dependent upon the value of $x$, correct?

Therefore it is not a constant.

then wtheck is it?

problem 4 was the question I posed and got math.stackexchange.com/questions/930314/…

but then I'm at now what land

Two different solutions? Oh ok.

Ugh, I'm just wasting your time.

Sorry buddy.

what

NO COME BACK!

do you know odes

maybe you can help me integrate with respect to z on another problem

it's just that part the end... then I got the rest.. divide sub back blah blah blah

ugh

oh hey @TedShifrin do you know pdes?

Hahaha, I'm here. Sorry - Had to move rooms.

humph *throws pde at @GustavoMontano

Dodges

I know there are software that solves pdes

maple .. mathematica

but not for this

HELLO!

GOOD NIGHT! I have a question about integral of tan in the morning.

Hi @robjohn

I just returned home from a one day event in the army.

@WillHunting hey there

@WillHunting in the army?

@robjohn Yes. National service. I finished 2.5 years, and then there will be a few days each year when I still have to go back.

But I do only mostly clerical stuff.

In other news, Jonas Teuwen is well and alive! He should finish his PhD next year!

I ordered 4 items from amazon and they were all "in stock" but they shipped 3 first, weird. Maybe the last item has to come from another warehouse.

Greetings

lol, preparing for another job interview

I saw you leaving and entering chat just now.

What job is it this time?

@robjohn this morning I created other 2 mind-blowing series, absolutely crazy awesome.

@WillHunting project leader in a certain company

@Chris'ssis You should use another instead of other.

@WillHunting Well, it's the plural ... or?

another car or other cars, right?

Not about singular or plural.

Hmm, let me think how to explain.

@Chris'ssis I created another series. / I created another 2 series. / I created 1 other series. / I created 2 other series. These 4 options can be used.

Hello @Did welcome to this chat.

@WillHunting Ah, got it! Thanks! :-)

@WillHunting I've just found this one - english.stackexchange.com/questions/87759/…

pummeled an inequality and I'm happy about it :P lol

@r9m You should see the new generation of crazy harmonic series ... (unseen so far) :-)

@Chris'ssis the one with trig terms ?

@r9m no no no, that one is a baby series

@Chris'ssis then which ones ? :)

brb, preparing to leave ...

@r9m Is that you in the picture?

@Chris'ssis OH LORD !!! The Book -_-

@WillHunting hehe .. you think so ? :P

@r9m Are you a girl?

@WillHunting okay .. what if I say yes :P

@r9m Nothing happens.

:P LOL

@WillHunting its a pic from deviantart.com (Hammer Girl form the movie The Raid : 2) :P

The Gentleman Zombie

Back.

@robjohn I sent you something.

@Chris'ssis Nice. I'll have to look at it. I imagine using Harmonic numbers will be a bit simpler. It will remove the $\log(2)\gamma$

@robjohn Good point. :-)

@IceBoy Sometimes people are not boring because they want to be boring, but they are silent because life has brought them to silence. Besides that, it may be about a disease, a treatment that makes you behave in a certain way.

It's good to try to see beyond the appearance.

@r9m

@IceBoy I try to avoid boring people. It often has the effect of spilling the messy stuff inside.

Tunnel boring machines are boring

@BalarkaSen hey there

I find a lot of people boring.

They talk nonsense. I waste my time hearing silly things instead of enlightening concepts - like mathematics :).

@GustavoMontano Life is not only mathematics, it's much more than that.

I'm well aware of that. But its not interesting.

Not as interesting as mathematics. Gossip, conversations about societies norms and convention are extremely boring.

@Chris'ssis So you and @robjohn send each other stuff, but both of you refuse to share email with me. Never mind. =)

@WillHunting we don't send email.

I learnt that when amazon says an item is "in stock", it still can take many days to ship. On the other hand, if they say "only 1 left", it may ship immediately, lol.

@WillHunting I didn't refuse you at all, but at the moment I don't have an email without my name in it. I still prefer to remain anonymous. :-)

@Chris'ssis I see. Anyway my email is jasperloy at outlook dot com, in case you want to know.

@WillHunting Thank you.

@Chris'ssis OK, you can delete, lol.

@WillHunting :-)

@Chris'ssis is there a M in your name ? :)

@r9m lol :-) You spy! :-)

@Chris'ssis You can tell me your name when you want to in future. =)

@Chris'ssis if you are talking about that (removed) I honestly didn't see it .. I was afk .. I am making wild guesses here :| .. 'spy' ?! interesting reaction !!! was I right then ? :P

@WillHunting The only problem with the real name is this: some may spread on internet a lot of s**t about you. For instance, once I commented on a site that was about some conest, and I was supporting someone, but some that offered support to others began to use our real names and wrote a lot of things about us. That is a kind of cyberbullying.

@Chris'ssis For me, I share a lot of stuff openly. The whole world already knows I am mad, lol.

Since that moment I preferred not to use my real name anymore.

@WillHunting I don't think you're mad at all. Someone mad never has your behaviour.

(you're pretty balanced)

Could somebody have a look at the question pentagon geometry?

I posted that problem about an hour ago..

@WillHunting I like to believe I'm genetically honest, I mean I don't make any effort to be like that, I'm just natural. You know, people are different, some simply don't care about ethics.

I posted that problem about an hour ago..

@r9m Why did you use that M? Just a random guess? ;-)

@Chris'ssis Maybe I will meet you one day. =)

@WillHunting Sure, we never know the surprises of the future! :-)

@Chris'ssis hmm .. here is only heuristics (no mathematics) .. so a wild guess :P but tell me if I am right :-)

@r9m You might be right. :-)

@Chris'ssis aha! okay next question .. have you been involved with teaching (as a professor/lecturer) ? (this is not a guess question .. I am eleminating from a list of potential Chris'ssiss :P lol)

(ps: I'm a creep)

Pentagon Geometry

@r9m Reading your question I realize you cannot accept what I told you in the beginning, the fact that I have no background in mathematics! :-) In a way I understand you, maybe if I were you I'd think the same way, but I really have no background in mathematics, so I couldn't be a professor/lecturer. :-)

This is simply the truth.

@Chris'ssis ^_^ I believe it if you tell me :) okay .. I have no experience with Romanian names but is Chris a Romanian name ? (I mean if you spell Chris like that or in a different way)

@r9m Chris like Cristi (in Romanian language), and some might simply call you Cris.

@Chris'ssis I see .. :) (at this rate I am only wasting your time with my childish questions .. I may be worse than the imp himself atm :P)

@robjohn let me know if you find a nice solution there.

@r9m hehe, no worry, it's OK. :-)

@r9m hmmm, have you seen this one?

$$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{H_n}{2n-1}$$ but done in the spirit of the art ...

@Chris'ssis hmm .. maybe but I'm not sure :o scratches head I have to try it ..

What's the point with solving questions only? This is not for me. I choose the art of solving things.

I notice that my enemy likes to downvote answers which have more votes than her answer.

@WillHunting Who is your enemy?

@Chris'ssis The user with the fifth highest rep now.

@r9m That one can be done in one line, honestly.

What happened to that guy who was amazing at topology?

I recall him being able to do every question in topology.

Had so much rep. I miss him.

Brian Scott

YES

Is he still around?

Seems to have stopped coming

@Chris'ssis okay :)

Seen :( "seen Jan 2 at 12:05"

@Chris'ssis I am surprised you forgot, lol.

@WillHunting Ah, I think I just remebered that case ... the old lady, isn't it? :-)

@Chris'ssis Yes. The more I know about her the more evil I think she is. =)

Is their a case when $(A \time B)-(C \times D)=(A-C) \times (B-D)$ ?

@WillHunting lol :-))))))

@robjohn do you believe me when I say that I have no background in mathematics?

I wonder who else believes me ...

@Chris'ssis Background meaning a degree?

@WillHunting Yeah.

@Chris'ssis I believe you.

@WillHunting I appreciate that.

brb

anyone ?

@Chris'ssis My grandpa even has no formal degree but he is one of the best problem solvers who I've ever known

@Anastasiya-Romanova Glad to hear that. So, it is possible ... :-)

@Chris'ssis "With man this is impossible, but with God all things are possible." Matthew 19.26

@Chris'ssis "Impossible is nothing." Adidas

@Anastasiya-Romanova True. I'm often told that my mind was touched by God.

(by some Christians I mean)

Only US astronauts have touched the face of God.

Quote by US President, I forgot his name.

I have touched my face, lol.

Last night I dreamt again that I was doing very hard integrals, and I knew almost instantly what to do for each one. I try hard to remember at least one of them.

@Chris'ssis I'm not a Christian. I am a Buddhist, but I almost go to church every Sunday, at least twice a mount.

I see.

Are you from US @Chris'ssis?

@Anastasiya-Romanova No

@Anastasiya-Romanova No

@WillHunting Where's she from?

@Anastasiya-Romanova Romania, the land of Dracula. :-)

@Anastasiya-Romanova Romania

Serious?

@Anastasiya-Romanova Yeap.

@Anastasiya-Romanova Yes

@Anastasiya-Romanova I know your location is a secret

@Chris'ssis Nicolae Ceausescu & Adrian Mutu

@Anastasiya-Romanova hahaha, right!!! :-)

My Romanian name is Jasperescu.

@WillHunting lollllllllll :-)))))))))))

@Chris'ssis My grandpa said that Nicolae Ceausescu is The Modern Count Dracula

@Anastasiya-Romanova lol, kind of. He was, he's dead now. :-)

I am the only dracula left.

:D

Okay, let me create something ...

@r9m did you see this one? :D $$\sum_{n=1}^{\infty} \left(\frac{H_n}{2n-1}\right)^2$$ a piece of art

My country had a president like Ceausescu, but he's dead

@Chris'ssis I have to try .. :)

@r9m First, let me know if you manage to do that alternating series in one line.

@Chris'ssis okay :-) atm I'm worried about a due assignments in a topic that I never study :P hard time for me

@WillHunting Luckily, your country never has a dictator as a president or prime minister

@r9m Take your time. No hurry (next days, weeks, months, when you have time).

@Anastasiya-Romanova Still, it is not a good place to live.

@Chris'ssis ;) okay :)

The Night Attack of Târgovişte by Theodor Aman is actually my coverpic in facebook :P lol I am gr8 fan of Vlad the Impaler :)

@r9m Really? :-)

@Chris'ssis ya I like characters with that kind of demeanour =}

hmmm, interesting. A good demeanour when tackling integrals, series and limits. :-)

@WillHunting have you ever heard about Lim Siong Guan?

@Anastasiya-Romanova Yes. Is he your uncle?

@BalarkaSen you might like this one $$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{H_n}{2n-1}$$

Do you notice that @user3290793 is so similar to @BalarkaSen avatar?

Is there a nicer way to do $\overset{\to}{PQ}$?

Haha someone downvoted my old question, might be the enemy, lol.

I am using \overset atm.

@WillHunting No, but he reminds me to grandpa

@Anastasiya-Romanova You mean reminds me of.

nothing

@WillHunting Oh, the correct collocation is of

@Anastasiya-Romanova Yes, and collocation is a difficult word, lol.

@WillHunting My English grammar is really bad.

Do y'all know of any books that go into complex numbers very deeply, starting from the basics?

I know of a really superb book!

Do tell!

Is it illegal to supply you with a link to the pdf?

Well, lets just say it's called: Complex Variables with Applications by James Ward Brown and Ruel V. Churchill. xD

@Khallil.

Thanks.

I'll try to look online for a hard copy.

;D.

wink

double wink

I'm on the PC way too much anyway. The strain can't be good for my eyes.

Seriously, @Gustavo!

$\mathbb{R}^{n}$ wink.

That makes no sense to me.

My winks are $n$ dimensional.

Everywhere.

Ah, I thought so!

wink ;D

I thought you meant that there were $|\mathbb{R}|^{n}$ winks!

Hmmm.............what does that mean?

Infinite number of ...

It's the cardinality of the real numbers (i.e. no. of elements in $\mathbb{R}$) to the power of $n$.

$\infty^{n}$?

$\mathfrak{c}^n$

(I just picked that up from here.)

Don't think I'm one of the smart guys on here!

Don't worry, we're in the same set of people on Math Stack ;D.

I am the reason why other people are smart (since smartness can be seen a measure of relativity).

Haha! ^_^

We can only do our best.

Exactly ^_^.

What is $\displaystyle \binom{n}{k}$ supposed to mean?

That is a notation for combination.

Have you seen the notation $^{n}C_{k}$ before?

They mean the same thing. That is, $$^{n}C_{k} = \binom{n}{k}$$

Yep, and I know they are identical and both equal to $\dfrac{n!}{k!(n-k)!}$, but I shamefully don't know what it represents, @Gustavo.

the number of k-element subsets of {1,...,n}

You mentioned combination. Could you elaborate on that?

Yes, it is a formula used to count a set of numbers under certain conditions.

the number of ways to choose k things out of n total, without regard for order

Oh, ok.

@GustavoMontano it doesn't count a set of numbers, it counts a collection of sets or a set of ways-of-choosing

Yes, that is correct. To understand what it means, you must first be aware of the multiplication principle and/or the axiom of choice.

axiom of choice is too advanced

They mean the same thing? Well, that's what someone told me "Refrain from using Mult.Prin - please say Axiom of Choice".

to understand the factorial formula for it you need to know the multiplication principle, but you shouldn't need it just to understand the definition

@anon. Thereby, counting a set of numbers.

@GustavoMontano wow, really?

Yes, apparently. That's what someone on stack told me.

And wiki it. I think it may be true.

pedagogically not a good idea methinks

I'm lost now.

Probably the simplest context to use Axiom of Choice is the multiplication principle.

@Khallil, do you know what the multiplication principle is?

and no. 3 choose 2 counts the number of elements of {{1,2},{2,3},{1,2}}, which is not a set of numbers

I've never heard of it before now, @Gustavo.

@anon. You are right then.

@Khallil. It is a very interesting principle. You should look at it when you have free time :).

Ugh, I'm only going to get 5 hours of sleep at this rate. Goodnight guys :( !

It looks interesting.

Good night, @Gustavo!

It is VERY interesting. Night! ^_^

So if there are $a$ ways of doing something, and $b$ ways of doing another thing, there are $a \cdot b$ ways of doing them both.

@GustavoMontano That does not go deeply at all.

@Chris'ssis (づ｡◕‿‿◕｡)づ