I have, @Balarka.

not really, @Ted

@TedShifrin i think you posted both that question and the answer here once before, but i don't recall what it is and the pair doesn't seem very small.

I've forgotten :)

Well, it certainly won't be prime or a very small composite number :)

i don't think any of the two can even be abelian.

"any" = "either" (flunking @Balarka in English)

i am sleepy @Ted

gah can't keep my eyes open anymore

I think one of them can be abelian.

Well, I said g'night earlier and you got all mad at me.

ok, i should really really go to sleep now

byes

bubye

@Ted WIll your video lectures teach me what a fibre bundle is?

nope @Kevin

it's freshmen and sophomores for goodness sake :P

You need to use Bayes' Formula, @evinda.

@TedShifrin Is P(A) the probabily that the cient gets the second orange from the farmer A?

Don't use $A$ when you have farmers named A and B. You need to name every event that's relevant. (By the way, I assume the two oranges are chosen without replacement.)

@Ted, note that any automorphism of $\bf Q$ fixes $-1$ since this is the unique element of order two. This means that the automorphism is determined by where $i$ goes (and by the previous observation saying where $i$ goes determines where $-i$ goes) and where $j$ goes. Since $ij=k$; this determines the automorphism fully.

I answered that for you ages ago @user159870. I gave you an explicit example.

Indeed, @Pedro.

@TedShifrin Then note we have $6$ candidates for $i$, and once we choose this, we have four candidates for $j$. This means the automorphism group has order at most $24$.

I wasn't asking you for a proof, @Pedro, but if you're going to give me one, we should see how we see symmetries of the cube. I haven't thought about this in about 3 years.

Events are not probabilities. Events are events.

SIGH.

The most important events, given the question, are $S_1$: gets a sour orange the first time and $S_2$: gets a sour orange the second time. What is the question asking for?

Pourquoi le sigh, @Pedro?

Image is not showing.

Preumably it should be that choosing where $i$ and $j$ go is like where choosing two pairs of opposite faces of the cube go :P

You seem like you're feeling healthy again, @Pedro.

@TedShifrin Yes, I'm fine.

There, that's the image.

The $\blacktriangleleft$ is just there to tease. =)

Right, just what I said :P

@user159870: Suppose $f(x,y)=x$ and $g(x,y)=y$, $P=(0,0)$. What's $I(P,f\cap g)$?

@Pedro: I gather your travel plans are more serious. Good that you'll get to visit some more universities.

1. why? how do we use it? @TedShifrin

@TedShifrin This travel in particular is not planned by me. But I will visit you know who.

OK, @user159870. Now what if $f(x,y)=x(y-x)(y-2x)$ and $g$ is the same?

@Pedro: Will you visit more cities and get to see more universities?

@TedShifrin I don't think so.

I'll do SF, SD and LA.

Oh, well, SF is a fabulous city, fabulous food, fabulous cultures. But you should think about visiting Berkeley across the bridge and Stanford ... if you and your dad rent a car and can travel a bit. In LA there are a few obvious universities.

LA, USC, CalTech, I guess.

@TedShifrin What do you mean when you say "visit"?

I wish more people in MSE (and this chat room) would try to understand concrete examples instead of doing so much abstraction.

Yup, @Mike,

I mean go by the math department and sit in on a class or two.

what does this mean @TedShifrin ?

Right, @user159870. Now do you understand what you were asking? You can obviously generalize to the case where $g$ has several irreducible components, as well.

When $f$ is as above, there are three irreducible components: $x=0$, $y=x$, and $y=2x$.

@Pedro: Since you're several years away from applying to schools, there's probably not too much point talking to a graduate coordinator, but you-know-who might facilitate a conversation for you at you-know-where.

do we factorize $f$ and $g$ into irreducible polynomials to calculate $I(P, f \cap g)$ ? Is the intersection multiplicity always equal to the Number of irreducible components of $f$ and $g$ ? @TedShifrin

No, intersection multiplicity will be the sums of intersection multiplicities. They needn't be $1$. For example, take $f=y-x^2$ and $g=y$.

His timing's bad for UCLA, @Ted, since he'll be coming for the last day of classes.

ah, well, that's better than his timing for UGA, when it was break before classes even started.

What a clown, that Pedro!

still, the last day of a graduate course wouldn't be so stupid.

Of course, I still treated him to a birthday party, @Mike :P

As if we're not going to treat him well here, @Ted.

I expect you specialize in maltreatment.

Of our undergrads, not of foreign ones.

I meant you personally, of course.

My above comment applies.

I still reserve judgment :D

OK, I have to go make a marinade for my pork ribs. Chinese, here we go ...

In that case we have : $I(P,f\cap g)=I(P,(y-x^2)\cap y)=I(P,x^2\cap y)=2I(P,x\cap y)=2$ correct? @TedShifrin

Hi

Wow I got that LaTeX working, that's really neat

Hi, I need someone to tell me if my question is more clear now. It was put on hold.

@user159870, well, if you translate by $(c_1,c_2)$, the point translates as well. Because of the nonsingular matrix, linear independence is preserved, so linearly independent directions map to linearly independent directions, so intersection number $1$ is preserved. Can you deduce the rest?

@QuaxtonHale That's a given because you have 90-theta as the angle OAB, and I believe this forces angle OBA to be equal to theta. Because that is marked as being similar to angle OBW_2, they are thus equivalent. So this is an easy question

@Quaxton: Well, let's see if we can figure it out. $\phi$ is irrelevant. What do you know about $\triangle OAB$?

Oh, I'm too late :P

It's an isosceles triangle.

I feel pretty dumb right now. >.>

@TedShifrin you had it though :)

That is is

and yeah, $\phi$ is irrelevant

But something's wrong. If the $\theta$ that is marked for $OBW_2$ is correct, then $ABW_2$ has to be a right angle, which is certainly is not. Something's wrong.

That was what I am confused about...

Question: have you ever had a professor knock off half a letter grade on an assignment because you did not get rid of radicals in the denominator of fractions? I looked in the textbook and they even did it...

Because somehow $\alpha$ would be zero.

@bd1251252: I want to slap that professor. I keep telling my students to STOP getting rid of radicals in the denominator. That's an old fashioned custom that serves no purpose other than to make things too complicated and students prone to make errors.

@Quaxton: I think it's just plain wrong.

@TedShifrin Here is the solution to the problem I am working on. It relies on the fact that $OBW_2$ is $\theta$.

@TedShifrin I've researched it...I was really mad. It doesn't make any sense to me. From now on I'm just going to do it because I literally lose so many points to stupid things like that. It's like professors don't want me to have a GPA above a 3

It's a historical relic from before the calculator days. What kind of course are you taking?

Multivariable calculus

How old is your professor?

Probably in his late 30s...at least he looks that way

(Note that I'm over 60 and I'm trying to stop people from doing it.)

(haha)

Ridiculous. Utterly ridiculous. People need to know how to rationalize denominators with complex numbers or things like $1+\sqrt5$ in more advanced contexts, but this is not relevant in multivariable calculus (which is also what I'm teaching).

I'm sorry to be rude, but they're both idiots.

They can say that they require you to put things in that form, but they're being arbitrary and you are totally NOT wrong.

Wow, thanks for your opinion. I mean, I still would have thanked you if you had taken their side, but I mean...gee, at least I'm not the only one

No, in fact, my students make mistakes when they do it. One of my highly intelligent students screwed up because it would have been much easier to do $(1/\sqrt2)^2$ rather than $(\sqrt2/2)^2$ and more complicated versions thereof.

What sort of school are you at, @bd1251252?

I can't believe any of my colleagues would behave in this manner at a research university math department.

Well, I wrote to him afterward and asked if it was merely a matter of being "proper," and he basically beat around the bush in his email

This makes me totally embarrassed that we have people acting like this. I expect it from high school teachers or from old codgers, but not from someone teaching university mathematics.

It's so sad when professors care about such things.

Then people thing mathematics is about being a pedantic ass.

And people end up hating math.

Well, there are important things to be pedantic about. This is not a valid one. :)

You might fight me on more substantive issues, but I would be able to justify why I demand what I demand :P

Well...to be honest, I had a bit of a life crisis and had to leave the Institute I was studying at and now I'm attending Monroe Community College. It's honestly sad...some of the students there, I mean. Like the same professor had to write to me and tell me not to use Greek letters or vector notations that weren't angle brackets

OK, yeah, community college teachers are often high school-type teachers (not to be rude again). And they are not themselves actual mathematicians, in general.

Multivariable calculus has many challenging and beautiful concepts. I fear you won't get to experience the beauty of the material. :(

(It's been one of my favorite courses to teach my whole career, so I've been doing it well over 40 years.)

@TedShifrin Oh no, you're fine. I certainly agree. It's a joke and I'm ashamed of my position in life. My hope at the moment is to just graduate by the end of the year with my associates in mathematics and apply to the University at Buffalo. Hopefully that will work out and they'll accept me

Suppose we have finit oriented multigraph.Then in my book it says between 2 edges there is 0 or 1 connections

@TedShifrin Congrats! I love multivariable calculus as well

this is wrong it is multi?so between 2 edges ther emight be more than 1 connections

Well, I'm giving you moral support. But if you can't figure out what angle a creek is flowing down a mountain if I give you the gradient of the height of the mountain at a point and the direction of the creek, then I'll smack you :P

@bd1251252: If you're ever so inclined, you can check out my book. It's a math-major type book with multivariable and linear algebra all mixed together, with hard computations and lots of proofs.

@Karlo: I don't understand. What is a connection between edges?

it is

Perhaps @Pedro understands multigraphs and can explain.

verticle

@TedShifrin What's the question?

here it is

@PedroTamaroff if we have finite oriented multigraph then we have 0 or 1 verticles between two edges.this is wrong rigth?

@bd1251252: If you're interested in serious math, check out my lectures from that course. There's a year's worth of lectures on YouTube. You might learn something :P (See my profile page.)

heya mr @Kaj

@TedShifrin Thanks!

@TedShifrin Yes I will take a look

@Karlo Multigraphs have repeated edges, but not "repeated" vertices.

Hey @TedShifrin

@PedroTamaroff ops i have written verticles instead of edges

@TedShifrin Would you mind taking a look at the solution I linked? Thanks.

@Karlo: vertex (vertices) is the English word you want, not verticle

@Quaxton: I've looked. I don't follow it yet.

@TedShifrin yes sorry ted translating is not my best side :D

But verticle is fun to say. Everyone in unison, now: verticle!

@TedShifrin Okay. Me neither .-.

@Mike: A vertical monocle?

@BalarkaSen I think most physicists are under the impression that she was 'one of us'

@Danu No biggie here, just a case of self promotion.

@PedroTamaroff So in finite oriented multigraph between 2 vertricles there are 0 1 or more edges

@TedShifrin Haha I liked your reaction when you asked the importance of a tangent line and the kid said "There's only one?"

@Karlo Yes.

@Quaxton: OK, so that $\theta$ is a typo. It should be $\theta-\phi$. Can you figure out why?

@PedroTamaroff Heh.

LOL, you have a long way to go, @bd1251252, but I hope you'll have fun :P

These students are very engaged and have a lot of fun, @bd1251252 :P

@Danu Hey, no stealing our celebrities.

@Danu Just saying. =)

@TedShifrin I'll watch all of the videos

@PedroTamaroff And in not oriented graph if between 2 edges we have 2 verticles we write them as {a,b} but we calculate them as 2 verticles right?

@MikeMiller What is she most famous for in mathematics? Algebra or something, right?

@bd1251252: If I can help with serious questions, let me know.

is @Danu referring to Emmy Noether?

Yes

@TedShifrin I've never taken a linear algebra course before actually, so those topics will probably be at the top of the list when it comes to questions

I'm seeing algebra on wiki

@Karlo I don't follow your question.

the Noether in physics is a different Noether :P

ah ... ok

There's also a Max Noether, and maybe there are others.

@TedShifrin Yes, that is what I computed. I wasn't confident and it confused me. Thank you.

@TedShifrin Oh... I'm talking about Emmy Noether in case there is any doubt

You're 1000% correct, @Quaxton. Books make mistakes. :(

She's the famous one in physics.

Oh, my fault, @Danu.

In particular for Noether's theorem

Emmy Noether is the one we always talk about

@TedShifrin :/

Right, I remember the theorem. I forgot it was she. Now I've forgotten what Max did.

Continous symmetries of the Lagrangian $\Leftrightarrow $ conserved quantities

i

@Quaxton: You should either remove your question or edit it and say you've resolved the issue.

@Danu Thats not quite the theorem in full generality

Right, @Danu. Somehow I didn't associate that with algebra :P

Oh oh, lurking @Kevin comes out of the shadows :P

No, I know, but that's the form that physicists care about

Careful, @Danu. Kevin is a physicist.

Btw, you're the Shifrin ?! Awesome

@TedShifrin I'm working on vectors in 3 dimensions right now in my class. We'll be done with them at the end of the week

LOL, what does that mean, @Quaxton?

Well in field theory it matters that it only implies a conserved CURRENT. Its only a conserved scalar quantity if the current vanishes at the boundary

Oh, you have a long way to go, @bd1251252, and I fear your course won't cover nearly all the stuff. But do well, please. :)

@KevinDriscoll Meh, okay.

But I dunno why I'm being pedantic this evening

By quantities I did not mean scalar

Because you want to make your students rationalize denominators and dock them half their points if they don't, @Kevin? :D

But I could see why you'd interpret it like that

@TedShifrin Yes...but I have taken a multivariable calc. course before at the Institute, last semester in fact. So most of what I am doing in this class is just review and at a much lower level

@TedShifrin HALF!? Pffffff $1/\sqrt{2} =$ ZERO CREDIT

So... what is Noether famous for in mathematics? I see she worked on fields/rings?