Mathematics - 2014-05-10 (page 1 of 5)

Studentmath

00:01

Same here, unless they are in charge of that department of course..

Mike Miller

@TedShifrin I need to learn nonetheless.

skullpatrol

@TedShifrin See, I found an exception in your universe:

2 mins ago, by Studentmath

Same here, unless they are in charge of that department of course..

Ted Shifrin

That's not right, I bet @Studentmath. Department heads have all sorts of powers, but not to alter drop date.

No, @skull. You haven't.

ಠ_ಠ

Yea, I don't think the math department head here has any say either

Studentmath

I've had a friend whose head of department allowed him to drop after last drop date from a certain course and switch to another (of course with extra costs).

skullpatrol

00:05

They can ask for leniency...

ಠ_ಠ

The administration run by CUNY (the public school system here) is pretty much set in stone I think

But for private schools they might be more lenient

I know at Harvard one of my friends asked the professor to let him in the class and the professor added him to the registry

Ted Shifrin

Yes, we can with letters do a switch from one class to another, but the Dean has to approve it.

No, the professor didn't. Not literally.

ಠ_ಠ

Oh

Studentmath

@Ted I guess that's the situation here too, of course.

Ted Shifrin

I've been at universities both private and public. There is bureaucracy everywhere. Faculty are not super-powers.

ಠ_ಠ

00:07

Anyway, I thought we were going to continue with groups/rings/fields but we stopped after groups and we aren't doing rings or fields at all so I kind of don't want credit for this course so I can re-take it

Studentmath

Of course, just head of departments, if they care enough, usually have the connections to overpass them.

Ted Shifrin

Not in my experience at 3 big universities, @Studentmath.

Jack Schmidt

Our Director of Undergrad Studies (there is one in each dept) can add people to courses during the add dates despite capacity. The dean of the college has to approve any late drops. Late adds are handled by the registrar's staff, and usually only for typos.

Studentmath

Maybe because they're big. In my current math courses (getting 'advanced') we have about 13 students per course (not per teacher, per course..)

Ted Shifrin

I was Associate Dept Head here for 8 years and have advised thousands of students. I know the ropes :)

ಠ_ಠ

00:09

Everyone ignores our math class capacities

Mike Miller

@TedShifrin American universities... I recalled Studentmath as being in Israel.

ಠ_ಠ

There's always like 35/25 students enrolled for the course

Studentmath

That's right @MikeM

Ted Shifrin

Well, I'm tired of arguing. It's not helping poor mr eyeglasses learn his math, which really is the point.

Studentmath

Then again my university is also an exception in Israel, even..

skullpatrol

00:11

14 mins ago, by Ted Shifrin

Yes, @skull. But it has nothing to do with individuals. Colleges have universal policies.

Jack Schmidt

@TedShifrin Some colleges are more bureaucratic than others. College of Ag here handled a few late drops from my course -- I got an email informing me it happened. College of Arts and Sciences - I filled out a two page form requiring more info than my gradebook has.

ಠ_ಠ

If I pass algebra I'll have to study it on my own because we haven't covered rings or fields

Ted Shifrin

Nevertheless, @Jack, it requires deans' intervention.

Studentmath

What else have you taken, Mr. Glasses?

ಠ_ಠ

Calculus and Probability

Studentmath

00:12

How're you doing in probability?

ಠ_ಠ

Not that good

Studentmath

And single variable or multi?

ಠ_ಠ

Multi

Studentmath

Hah, almost the same courses I've taken this semester

Jack Schmidt

@TedShifrin Yes, definitely. I just meant that in some colleges you go ask the dean's secretary and you are dropped within the week, and in some colleges you wait a month for the committee to meet to review the binders of forms that have been turned in :-)

Studentmath

00:13

Just instead of Algebra, Graph Theory...

Where are you at with probability?

ಠ_ಠ

I'm going to have to study real analysis over the summer because I am doing so poorly in calculus and real analysis is much harder than this

Ted Shifrin

Mr eyeglasses, you can teach me probability so I can teach it in the fall. :)

ಠ_ಠ

We finished the textbook (that our professor wrote)

Jack Schmidt

business school is most reasonable - they ask me current grade, grade before the "event", and whether I think the drop fits uni policy. takes like 2 minutes to fill out.

Ted Shifrin

You have more bureaucracy than we do, @Jack.

Studentmath

00:15

Hah, not there just yet, will be in.. two months I think, maybe one.

Jack Schmidt

We spend millions and millions on it, so it better be big!

this semester, for each failing grade assigned, we have to explain why it was assigned

Studentmath

I am overloaded with work, I never studied so much as I do now, it feels odd. I am used to studying few days before the test, doing half of the works and it's all fine...

Ted Shifrin

I'm looking forward to teaching something I've never studied :)

Studentmath

It's exciting, isn't it?

ಠ_ಠ

@Ted Do you have any recommendations for introductory real analysis books for slower students

Ted Shifrin

00:17

Generally, cramming for exams doesn't work too well with serious college math.

Studentmath

Yes, Prof @Ted, hence I started studying seriously this semester.

Ted Shifrin

Have you taken linear alg, mr eyeglasses?

Jack Schmidt

@TedShifrin Do you use webwork? My former office-mate just finished a junior level prob class. He is leaving to be a fancy six figure actuary, but had time to put together a pretty interesting (but standard) course.

ಠ_ಠ

@Ted Yes, it's a pre-requisite for our calculus course

@Ted But I feel my linear algebra is EXTREMELY weak

Ted Shifrin

What?

Studentmath

00:19

Hah, the opposite way around here, Mr. Glasses

Ted Shifrin

Prereq for calc? With proofs?

ಠ_ಠ

Yes

Ted Shifrin

Yes, @Jack. I actually wrote 300 problems for my multivariable math course.

Studentmath

If anyone could take a look at this question, will greatly appreciate it: math.stackexchange.com/questions/788479/…

I feel extremely weak with this flow and I have no lectures to study from...

Ted Shifrin

I'd have to go relearn it, @Studentmath ... It's been 30 years or so.

Studentmath

00:21

Woha. I thought you haven't taught Graph Theory at all?

Ted Shifrin

Linear alg is the place to learn proofs, mr eyeglasses. Analysis is hard.

ಠ_ಠ

@Ted Should I use your book if I can find it at the library

Ted Shifrin

I haven't, @Studentmath, but this was in Strang's applied math course.

ಠ_ಠ

omg I remember not liking Strang's linear algebra book for some reason

Ted Shifrin

Which, mr eyeglasses? Linear alg?

Studentmath

00:23

Ah yes, it has many uses I heared..

ಠ_ಠ

Yes @Ted Your linear algebra book or your multivariable book

Mike Miller

Can't decide if I want to walk home or take the bus./

Ted Shifrin

Linear would be better for you, my eyeglasses. It focuses on proofs and we tried to help students learn how to attack proofs. The multi book is quite hard.

ಠ_ಠ

@Ted I tried using Velleman's Proof book but it was pretty dry

Ted Shifrin

Most if those proofs books are ....

ಠ_ಠ

00:28

Haha, do you dislike them

Mike Miller

Clearly Ted prefers Lay.

ಠ_ಠ

Nooo

Ted Shifrin

Yes, @Mike, I always prefer linear books that save dot product for chapter 8. :(

skullpatrol

...proofs need to be taught starting right at elementary algebra, imo

Studentmath

Prof @Ted, mine did!

Jack Schmidt

00:32

(chapter 6)

Mike Miller

I dislike any linear algebra book that doesn't at least spend a little time on the geometric interpretation of the determinant

Ted Shifrin

Section 2 of chapter 1 in both my books! :)

Jack Schmidt

Do you teach QR methods before/instead of LU methods?

Ted Shifrin

I agree strongly, @Mike.

skullpatrol

@TedShifrin do you agree with me?

4 mins ago, by skullpatrol

...proofs need to be taught starting right at elementary algebra, imo

Ted Shifrin

00:36

No, @Jack, in truth. I'm also not biased toward a numerical sort of course, either. But I loved learning years ago the reversed QR alg to get eigenvalues/eigenvectors.

Mike Miller

Under the geometric interpretation/definition, "A invertible iff det A nonzero" is trivialized and doesn't come out of left field

Ted Shifrin

Not really, @skull, but having teachers/students explain would be a good start.

ಠ_ಠ

This is sad. We only have the title "Geometry and Topology : Manifolds, Varieties, and Knots"

Mike Miller

what matters more than when we introduce proofs is how

the godawful geometry we subject high school students to ruins the excitement

Ted Shifrin

careful, @Mike, there might be some circular reasoning if you try to be rigorous. :)

skullpatrol

00:38

but elementary algebra is the perfect setting

Ted Shifrin

Well, mr eyeglasses, they have good taste!

ಠ_ಠ

Unfortunately this book won't even be useful to me until probably after I graduate

Jack Schmidt

@TedShifrin One way to view solving with QR is just orthogonal projection to pull out the coordinates. How much $\vec{i}$ in the solution? I don't know, let me use a dot product and find out! -- It immediately gives least squares too, since that is just what happens when you don't find all the coordinates.

Ted Shifrin

Mr eyeglasses, send me an email.

skullpatrol

@MikeMiller connecting arithmetic to algebra is important, imo

Ted Shifrin

00:39

No that book is a compilation of research articles. I merely edited that.

ಠ_ಠ

Okay, sent an e-mail

Ted Shifrin

But QR, with all the Gram-Schmidt, is computationally much more intensive than LU, @Jack.

Jack Schmidt

@TedShifrin nope, same asymptotic, more stable, only use 2 times as many ops

Ted Shifrin

Hmm, I'll have to work that out :) Thanks.

Studentmath

Gah, this depresses me. Think I will move to next work and leave it for now. Or go to sleep...

skullpatrol

00:45

@MikeMiller I do agree that the tradition set out by Euclid's Elements of geometric proofs given first is hard to go against.

Ted Shifrin

Mr eyeglasses, check your mail.

ಠ_ಠ

Ok

Victor

Hi, anyone could solve my problem? math.stackexchange.com/questions/787485/…

Mike Miller

Do you mean locally or globally, @Victor? Locally the answer is yes (c.f. Fourier series)

Victor

01:16

@MikeMiller - What is the difference between global and local as you say?

If you mean only the trigonometric integral could locally express by fourier series, then i probably mean globally

Mike Miller

I mean that, if you write $F(a) = \int_0^a f(t)dt$, then if one picks a point $p \in \mathbb R$, one can express $F$ as an infinite series of sines and cosines on $(p-\varepsilon,p+\varepsilon)$ (for sufficiently small $\varepsilon$).

Victor

@MikeMiller - Then what is globally mean?

Anthony

Hey everyone.

Mike Miller

That $F$ can be expressed that way on $(-\infty,\infty)$, @Victor

Anthony

Can anyone tell me what the actual definition of a differential is?

Not of a function, just something like $dx$.

Mike Miller

01:29

yikes

Anthony

@MikeMiller, what?

Pedro Tamaroff

@Anthony You mean differential forms?

Read Spivak's Calculus on Manifolds.

Much fun.

Anthony

@PedroTamaroff Well I've been reading about differential forms, and I'm running in circles.

Pedro Tamaroff

What does that mean?

Anthony

Well.

How is a differential form defined?

Victor

01:31

@MikeMiller - What about the integral of non- periodic function?

Then could it be express as fourier series?

Pedro Tamaroff

@Anthony It is something that takes a point in your space, and gives you an alternating multilinear form.

Anthony

But so a differential form is a map.

Pedro Tamaroff

A differential $k$-form is something that to each point in your space assigns an alternating $k$ form.

So it is a map $\Bbb R^n\to \bigwedge^k(\Bbb R^n)$.

Anthony

So, back to intro to analysis, what does the differential inside the integral mean?

$\int_a^b dx$

Mike Miller

integrating a $k$-form

Pedro Tamaroff

01:33

You're integrating the $1$-form $f(x) dx$.

Mike Miller

hence yikes

@PedroTamaroff Do you think "the map $\pi$ restricted to $V$" should be written $\pi|V$ or $\pi|_V$?

I prefer the latter.

Anthony

I thought integration was defined as a map acting on functions.

Pedro Tamaroff

Usually, a $k$ form takes the form $$\omega=\sum_{i_1<\cdots <i_k} \alpha_{i_1\ldots i_k} dx_{i_1}\wedge\cdots \wedge x_{i_k}$$ where the $\alpha_{I}$ are functions, usually smooth or $C^2$.

Mike Miller

yikes

yikes as fuck i'm outta here

Pedro Tamaroff

@MikeMiller You scared bro?

It ain't gonna bite.

Anthony

01:35

Okay hold on. Let me see what I have.

Pedro Tamaroff

@MikeMiller Depends.

Mike Miller

@PedroTamaroff I ain't scared, I'm scared for Antonio.

Pedro Tamaroff

The first is more readable to me, sometimes.

For example if say $V$ has some ugly notation like $H_n^1(C;0)$.

Then I prefer the first.

For readability.

Anthony

Yeah so I thought an integral was a map from intervals to $\mathbb{R}$.

Pedro Tamaroff

@Anthony You mean from functions to $\Bbb R$, right?

Well, you have a "bilinear form" if you may!

$\langle f,I\rangle =\int_I f$.

Anthony

01:39

Yes.

That makes no reference to differentials though!

Pedro Tamaroff

Well, here you have something similar.

@Anthony Yes, you're integrating $f(x)dx$ along the curve $I$.

Anthony

Well I mean, that's what I've been doing since high school, but it seems like the $dx$ is almost just notation.

I mean, it isn't, but if a differential form is a map, then you're multiplying the product of functions?

Studentmath

Gah, I quit.. I won't manage it tonight, I just can't figure it out..

Can I set all the outgoing vertices from t to be 0 and the incomnig ones to have full capacity, and therefore cutting t would be the same as the net flow?

Anthony

@PedroTamaroff It always just seemed to me that the differential told you what variable you were integrating with respect to.

Ted Shifrin

Does @Anthony need a course on differential forms? I just blundered back in ....

Anthony

01:50

@TedShifrin it would seem that way. I've been shuffling through papers on them all day.

Pedro Tamaroff

@FernandoMartin HAI.

Atiyah is informal as f*** in some proofs. =O

Still awesome book.

Don't get mad at me.

=)

Ted Shifrin

Well, not to self promote, but you have my lectures on this stuff at your disposal on the web :D

Anthony

I'm just confused, again, because I thought that something like $dx$ was just a marker as to what variable you were integrating with respect to. In my analysis course we didn't really talk about differentials...

@TedShifrin where?

Ted Shifrin

See my profile, @Anthony.

Anthony

Aight.

Pedro Tamaroff

01:52

@Anthony Well, in that case $dx$ is that. Kinda...

Anthony

:(

Ted Shifrin

Yeah, we make it that in beginning calc and then we finagle when we do subs.

Which Atiyah? @Pedro (I know Sir Michael, so don't be a wise ass. :))

Mike Miller

@TedShifrin Instead of writing dir. lim., is it canonical nowadays to write $\varinjlim$?

Ted Shifrin

It was standard even when I was your age, @Mike :)

Mike Miller

I guess it was too hard to draw in the arrow in the 60s.

Ok, thanks.

Ted Shifrin

01:55

Yeah, typewriters sucked.

Studentmath

Hey, they had their class.

Pedro Tamaroff

@TedShifrin Commutative Algebra.

I really mean what I say Ted.

Ted Shifrin

Oh, ok ... You don't usually refer to CA by Atiyah :)

Pedro Tamaroff

I mean the book.

Fernando Martin

What do you mean @Pedro?

Ted Shifrin

01:57

Yeah, what do you mean?

Pedro Tamaroff

@FernandoMartin Some proofs are just "this will work", put in informally. But the proof doesn't really check the claim correctly.

Fernando Martin

for instance?

Pedro Tamaroff

Theorem 5.11.

Ted Shifrin

@Studentmath, without computers and LaTeX, I would never have started writing books. In many ways, I regret the technology :)

Studentmath

Haha

Pedro Tamaroff

01:58

They say $\overline A\subseteq\overline B$, when they mean $\overline A$ injects into $\overline B$.

Ted Shifrin

Incorrectness I am skeptical of, @Oedro. Not belaboring everything, sure.

Mike Miller

that's pretty standard bro

Studentmath

I can't imagine doing works without the computer. Since I was in first grade it was already useable for pretty much everything I wanted..

Mike Miller

@TedShifrin Do you know when it became standard to have "sheaf" refer to the functor sending open sets to groups/modules whatever, instead of having it refer to the space of stalks?

Ted Shifrin

You're being way too pedantic, @Pedro. Grad level stuff expects an intelligent reader.

Pedro Tamaroff

02:00

The injection is given by the map $\overline{\pi i}:A/\mathfrak p_1\to B/\mathfrak q_1$ induced by $i:A\to B$ composed by $\pi:B\to B/\mathfrak q_1$ that has kernel $\mathfrak p_1=A\cap \mathfrak q_1$.

Mike Miller

I know the former is now standard and the latter is what Gunning calls a sheaf.

Fernando Martin

I don't mind filling in the details

The last line of the theorem is also not incredibly straightforward

Pedro Tamaroff

If $\pi':A\to A/\mathfrak p_1$, then we have $\overline{\pi i}\pi '=\pi i$

This is needed to prove the claim.

Fernando Martin

But that's ok with me

Pedro Tamaroff

They also write an intersection when again it is the pullback by $\overline{\pi i}$.

Mike Miller

02:01

oh no

oh no!!

Ted Shifrin

@Mike: with Grothendieck, maybe ....presheaf and sheaf are different. In alg land, there's still l'espace étalé ...

Pedro Tamaroff

Dunno, I'm just saying it is simple stuff that can be written down.

Fernando Martin

but then the book would be 800 pages long, and filled with boring details I can fill on my own

It's just a matter of taste, I suppose

Ted Shifrin

Right @Fernando. It's better to impart understanding and intuition than pedantry.

Mike Miller

@TedShifrin I know the difference. Gunning defines a sheaf as what would now be called l'espace étalé.

Pedro Tamaroff

02:03

@FernandoMartin Not true!

It's just three more lines, tops.

Mike Miller

You're right, @PedroTamaroff, it's not true that it's a matter of taste

Fernando Martin

three more lines for that line

Mike Miller

It would truly be a bad book if the authors filled in every detail

Fernando Martin

then five more lines for the last line

do that for every theorem in the book

Ted Shifrin

Right. But he does the associated presheaf, too. It's just a matter if the order, really.

20824

02:03

@TedShifrin: Could you answer my comments here math.stackexchange.com/questions/496906/…

Pedro Tamaroff

Well, I am just saying they could just give the hint and let you fill in.

Mike Miller

@TedShifrin I know; I'm not criticising it. I'm just curious as to the history.

Ted Shifrin

@Mike: you should also look at Godement in the library. Another classic.

Mike Miller

Which Godement? Topologie algébrique et théorie des faisceaux?

Fernando Martin

btw @Pedro, remember what we thought about today about rings where every $A$-module is free?

the result is true, but our proof works only in the commutative case

Ted Shifrin

02:06

Yes @Mike.

Mike Miller

@TedShifrin Unfortunately my library doesn't have it. I'm looking it up and it looks amazing, though.

I'll have to read this... both for the content and for my French.

Ted Shifrin

@20824: A partial derivative is not a number. It operates on functions. In general, a directional derivative operates on functions and we associate that operator with a geometric tangent vector.

Pedro Tamaroff

@FernandoMartin Why?

Also, I am just opposing bad notation @FernandoMartin @TedShifrin

Fernando Martin

I'm not sure :)

but this post suggests we overlooked some detail

(good) abuse of notation is good notation

Pedro Tamaroff

When things get more complicated, bad notation obscures thing.

At least to me.

Fernando Martin

02:09

I'm not sure where the argument goes wrong but they seem to be using a whole lot of machinery to prove it

Ted Shifrin

@Mike: Maybe another book for your moving van. I used to have a xerox copy from grad school before I bought the book in Paris. I'll see if I still have the xerox in my office.

20824

@TedShifrin But for example, f(x, y) = x + y and the partial derivative with respect to x is df/dx = 1 and it is a number, right? May be I should say partial derivative is a number when the function f is given. Right?

Mike Miller

@20824 Ah, no. "1" in this case means the function on R^2 that sends every point to the number 1.

Pedro Tamaroff

@FernandoMartin I have to go now. Let me know what you find out.

Fernando Martin

ok, see you

Mike Miller

02:12

@TedShifrin Hahaha, I like this comment in a review of that book.

""We have therefore tried to write a book which presupposes no
familiarity with algebraic topology . . . ." This statement is taken
from the preface of the book under review, and surprisingly enough,
the author may have succeeded in his attempt. Of course, until the
appearance of the second volume of this work, the reader may find
himself still unfamiliar with algebraic topology"

Ted Shifrin

No, it's only a number when you have a function AND a point. But you need n such numbers to reconstruct the first-order Taylor polynomial of $f$ at $p$, assuming $f(p)=0$.

LOL ... It presupposes the topology knowledge, but it does spectral sequences @Mike.

Mike Miller

I know, I saw that. I just found that bit funny. The review right after it is a review of Bourbaki's Algebra, calling the author "polycephalic".

What does Cohomologie à valeurs mean?

Ted Shifrin

Cohomology with values in ...

Mike Miller

Ah, okay.

Ted Shifrin

Did you take the bus home?

Fernando Martin

02:17

Anyone up for some algebra proofchecking? it's simple

I promise

Mike Miller

Yes, @TedShifrin. I decided it was too hot to walk.

Ted Shifrin

I don't trust you, @Fernando.

20824

@TedShifrin Oh, that is right. It made more sense now. But another question I have is whether the definition of tangent vector based on partial derivative depends on the function f. For example, given a chart at p of a manifold M, then we have a basis for the tangent vector space as d/dx_1, ..., d/dx_n. But it is clear that if we use different f, then the basis with change. Right?

Ted Shifrin

No, the vectors stay the same. They tell you in which directions to take directional derivatives of any $f$.

Fernando Martin

I'm trying to prove that if $R$ is a ring such that every $R$-module is free, then $R$ is a division ring

I think my proof is incorrect since all proofs I found use a lot of machinery I don't, but I can't find the mistake

Ted Shifrin

02:21

You need @anon or @Karl.

Fernando Martin

Ahhhh, I found it. Sneaky.

Ted Shifrin

Surely there must be a counterexample to test your proof out with?

Fernando Martin

I ended up proving that $A/(x)=0$ for any $x$

Ted Shifrin

Ah, good.

Mike Miller

@TedShifrin I forgot to say... I'm sick of Serre's notation (in reading his rep. theory book).

Fernando Martin

02:22

but that doesn't imply $x$ is a unit in the non-commutative case :)

Mike Miller

He identifies representations with the vector spaces they act on. Usually one can decipher his meaning. Sometimes it's pain.

Anthony

@TedShifrin So if a differential is a map, why doesn't it get written as such under the integral?

Jack Schmidt

@FernandoMartin Don't use two-sided ideals. if Ax=A, then 1 is in Ax...

20824

@TedShifrin So no matter what f I have, for a given chart, 2*d/dx_1 + 3*d/dx_2 + ... + 4*d/dx_n represents the same tangent vector at point p. However, for different f the result will be different. Right?

Fernando Martin

I can't quotient by an ideal if it's not two-sided

I didn't prove that $Ax=A$, I proved that $A/(x)=0$

Ted Shifrin

02:24

The integral is given by applying that map to the derivatives of parametrizations, and the chsnge of variables thm says you get something welk-defined.

Subtle @Fernando.

Jack Schmidt

You can quotient the regular module A with the left-submodule Ax :-)

Ted Shifrin

Right @20824

Fernando Martin

hmmm

Ted Shifrin

Ah, I didn't know @Jack was still here. Nice to have a resident algebra guru :)

20824

@TedShifrin Thank you very much! Could you shed some light on this question please? math.stackexchange.com/questions/788529/…

Fernando Martin

02:26

Then maybe my argument holds. What I thought of is the following:

Anthony

@TedShifrin Applying the map to the derivatives of parametrizations? I'm not following. Can you tell me what $\int_a^b xdx$ means, then? In light of viewing the differential as a linear map

Fernando Martin

Consider $A/Ax$, it's free by hypothesis, so $A/Ax \simeq A^Y$ for some set $Y$

Jack Schmidt

(but yes A/(x) = 0 is not sufficient to conclude A is a division ring or that x is invertible. the Weyl algebra A=k[x][d/dx] also satisfies A/(x) =0, but that is because d/dx * x = 1 + x * d/dx, so 1 is in the two-sided ideal containing x.)

Fernando Martin

If $Y$ is non-empty, then there's an element $y\in A^Y$ such that $yx\neq 0$

(namely, pick $1$ in some coordinate of $y$)

Ted Shifrin

@Anthony: the meaning doesn't change in regions in $\Bbb R^n$. It just becomes the usual Riemann integral, because the parametrization in that case is the identity.

Fernando Martin

02:29

This is a contradiction, since $x$ right-annihilates $A/Ax$, so $Y$ must be empty.

Have I lied somewhere? :P

Ted Shifrin

@20824: I hope you've taken abstract algebra or a good linear algebra class. You shouldn't be doing this level of diff geo course without that background.

Jack Schmidt

@FernandoMartin That looks good. I didn't check your left-right, so there might be small problems, but easy to fix if they are there.

Fernando Martin

Thanks for your help @Jack, I'll think about it

Jack Schmidt

i'm headed out :-) good luck!

Ted Shifrin

Night @Jack

20824

02:33

@TedShifrin It is near the the end of my diff geo class actually. We mostly studied curves and surfaces before. But suddenly the teacher started talking things like this.

Anthony

@TedShifrin So then with something like $\int_a^b udu$ where $u=x^2$. What is the du telling me to do, in terms of this map? because the normal Riemann integral I was taught made no reference to differentials!

Studentmath

I think I got the solution!

I feel like a genius!

I should be ashamed it took me so long though

20824

contratulations

Mike Miller

@Anthony there are multiple ways of interpreting the integral

in the cases we're talking about, differentials and the riemann integral you were taught are equivalent

Studentmath

Thank you @20824

Anthony

02:48

Are you saying differentials together with the Riemann integral? I just want to know if the differential next to the integral always carries more information than just which variable you're integrating with respect to, and if it does, what's the meaning? In @TedShifrin 's lecture he mentions a map. I don't I'm terribly misunderstanding things, it's just in the (only) rigorous definition of an integral I've seen it doesn't make reference to differentials.

Mike Miller

I can

t comment here, I'm just saying that the standard Riemann integral and the integral in terms of differentials can be defined differently - and in the full generalities they lead one to consider be different things - while still giving the same results in this case

Anthony

That makes sense. And I don't doubt it. I'm just fishing for what the definition in terms of differentials is.

Mike Miller

Oh, okay.

Anthony

Thanks for your comments though.

Mike Miller

No prob, for what they were worth.

Studentmath

03:11

NNah, still haven't got it right..

Mike Miller

03:25

$\exp$ test

$\text{exp}$ test

Enjoys Math

03:46

0

Q: How do you graph a fractal on a line, as a function of time and position?

I'm writing ANSI C code for work and just got bug-free my LED Light Show Library. The first project it's going on is a sound bar with 10 volume indicator (1-color (white)) leds that are lined-up. I've already done a lot of the UI indicator light shows (such as dimming down to zero brightness), ...

Fernando Martin

@Mike: Here's an elementary argument: given $x\in R$, the left $R$-module $R/Rx$ is free, and so it's isomorphic to $R^X$. If $X$ were non-empty, we'd have an element $y\in R^X$ such that $xy\neq 0$, which contradicts the fact that $x$ left-annihilates $R/Rx$. This proves that $R/Rx=0$, and so every element is left-invertible; a symmetric argument over the right $R$-module $R/xR$ proves right-invertibility.

Jack Schmidt

04:05

@FernandoMartin Let $r + Rx$ be in $R/Rx$. Why is $xr + Rx = 0 \in R/Rx$? -- I think you need one more step, you need to choose $r$ specially.

Mike Miller

@JackSchmidt I just had this conversation with him - he doesn't think it's salvageable, though you seem to imply it is

Fernando Martin

Yes, what I wrote there is not correct.

Studentmath

How is it even possible.. wolframalpha.com/input/…

Well, I am off to bed, I gave this question way more than enough, 7 am here already.

If anyone could take a look and see if he can aid me in my struggles, I will appreciate it greatly:

1

Q: Maximal flow and minimal cut in complete graphs

The question is as follows: We define on the complete graph $K_n$ with the vertices {$v_1, v_2, ... , v_n$} the following directions: for every j>i, the edge $v_i v_j$ is directed from $v_i$ to $v_j$ if |i-j| is odd, and from $v_j$ to $v_i$ if |i-j| is even. We define the capacity function c(...

homework graph-theory summation

Mike Miller

04:23

@Studentmath $\sum_{x=1}^{n-1} n = n(n-1)$

$\sum_{x=1}^{n-1}x = n(n-1)/2$

Studentmath

And 2x.. nice, yes.

Precisely 0.

Was just playing with possible routes and flows, nothing works.. at least I practice some long lost summation skills/knowledge

Good night..

juniper

08:03

hi

can anyone help me with math.stackexchange.com/questions/785353/… ?

Matt N.

10:24

I'm getting sick of book recommendation questions.

Daniel Fischer

> However the hint says I should use uniform convergence of integrals to show analyticity.

What does "uniform convergence of integrals" mean?

Karl Kronenfeld

@MattN. Looks like they have a new tag for it. Add book-recommendation to my list of thirty or so ignored tags.

Matt N.

@KarlKronenfeld Yeah. : ) And thanks for the information, I hadn't noticed the new tag.

@DanielFischer The sup norm applied to the integrals?

I'm not sure I should have posted that answer.

Daniel Fischer

10:40

@MattN. Context. (The obvious answer would be Morera, but for some reason, the OP doesn't want that.)

Matt N.

@DanielFischer I'm confused. Does he mean to use the sup norm in the definition of the derivative?

ALJI Mohamed

Hello :)

Daniel Fischer

@MattN. I have no idea what the OP wants to do.

ALJI Mohamed

I have a special question, can someone help me with ?

Daniel Fischer

@ALJIMohamed Without seeing your question, we don't know. Ask it - or link to it, if it is a question on the site - and we shall see.

ALJI Mohamed

10:48

0

Q: Can you please suggest how do they get equation (2.9) on this book?

I have a problem of $minimization$. When $E(w)$ is minimized, that means $w$ verify (2.8). My question is I don't know how do they get (2.9) ?? here is an extract of math book ... ?

integration euler-lagrange-equation

here it is

it is simple

I am sure It is obvious

I simply do'nt get it that's all

but It is definitely simple

Matt N.

I'm also sick of people posting a question on SE and then additionally posting it on chat.

ALJI Mohamed

sorry @MattN. , I tought it is a special question

Matt N.

@DanielFischer I don't understand the idea of posting a question on SE and then posting a link to it on chat.

ALJI Mohamed

it is not like others, and It may be out of subject

Matt N.

I mean, within a couple of minutes. It's ok if the question is more than a week old and didn't get enough attention.

ALJI Mohamed

10:54

okey @MattN. I won't do that again ...

Matt N.

That's of course just my personal grumpy opinion.

ALJI Mohamed

:)

Karl Kronenfeld

@MattN [How](http://math.stackexchange.com/questions/379713/why-is-n2-fracn22-fracn22/379956#379956)|
[about](http://math.stackexchange.com/questions/411875/what-are-two-continuous-maps-from-s1-to-s1-which-are-not-homotopic/411884#411884)
[people](http://math.stackexchange.com/questions/450703/injection-mathbbn-times-mathbbn-to-mathbbn/450707#450707)
[who](http://math.stackexchange.com/questions/627969/induction-without-a-base-case/627993#627993)
[post](http://math.stackexchange.com/questions/484765/is-p-to-q-to-r-logically-equivalent-to-p-to-q-to-r/484773#484773)

Matt N.

@KarlKronenfeld Hahaha : D

I don't think I need to say anything.

Karl Kronenfeld

can't figure out why that doesn't work

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$Mathematics$ Mathematics