Mathematics - 2015-01-11 (page 2 of 4)

Balarka Sen

18:01

LOLed so hard at this @user153330

Chris's sis

$$\lim_{\epsilon \to 0} \frac{1}{\epsilon}\int_{\large \epsilon}^{\large 2\epsilon}\frac{1}{2\pi+n(n+1)x}\left(\frac{1}{1+x^2}+\frac{1}{1+(2x)^2}+\cdots +\frac{1}{1+(nx)^2}\right) \ dx=\frac{n }{2\pi}$$

Here a mystery is about to appear ...

Talking about asymptotics, what would be the first three terms of $$\int_{\large \epsilon}^{\large 2\epsilon}\frac{1}{2\pi+n(n+1)x}\left(\frac{1}{1+x^2}+\frac{1}{1+(2x)^2}+\cdots +\frac{1}{1+(nx)^2}\right) \ dx$$ as $\epsilon \to 0$?

Daniel Fischer

18:23

@Chris'ssis If $f$ is sufficiently regular, then $$\int_\epsilon^{2\epsilon} f(x)\,dx = \int_\epsilon^{2\epsilon} f(0) + x\cdot f'(0) + \frac{x^2}{2} f''(0) + O(x^3)\,dx = \epsilon\cdot f(0) + \frac{3\epsilon^2}{2} f'(0) + \frac{7\epsilon^3}{6} f''(0) + O(\epsilon^4).$$

Chris's sis

@DanielFischer True :D

Daniel Fischer

@BalarkaSen What is Hatcher's 0.13?

Ted Shifrin

Hi @DanielF, @Chris'ssis

Chris's sis

@TedShifrin Hello Ted

Daniel Fischer

Hi @Ted.

Chris's sis

18:28

The dramma of my question is that I didn't manage to get a hard question as intended.

Ted Shifrin

Things seem peaceful, compared to Pedro's censorship squabble last night.

teadawg1337

Just pinned my location with the link on the right

Ted Shifrin

Hello, @teadawg !!

teadawg1337

??

Oh, hello @Ted

I'm a bit disappointed this weekend... I'm going to community college this semester, and I'm required to take Math 1720.

Which is basically the second half of my Pre-Calc course that I took junior year of high school

Daniel Fischer

@TedShifrin Oh, I missed that :( About which time?

Ted Shifrin

18:31

Oh no, @teadawg. Did you mess up the math placement exam?

teadawg1337

@Ted No, I was placed according to my Math ACT score. Which was a 34.

Ted Shifrin

Jasper said something that @Pedro removed, and Jasper was annoyed. I missed what it was. Some time around midnight my time, @DanielF.

user112495

I have two sets $X$ and $Y$ with $|X|=|Y|$. How can I prove that $Sym(X) \cong Sym(Y)$?

Ted Shifrin

That's a good score, @teadawg. I do not understand. Can you not talk to someone in the math department about getting at least into calculus?

user112495

I'm guessing I need to start off by letting $\psi : X \rightarrow Y$ be a bijection.

Ted Shifrin

18:33

I don't even know what Sym($X$) means when $X$ is a set.

teadawg1337

@TedShifrin I'm gonna try, but I'm not sure how to prove that I should be taking Math 1910

Apparently taking the course in high school isn't enough, which is baffling to me

user112495

@TedShifrin I'm using the notation that $Sym(X)$ is the group of all permutations of X.

Ted Shifrin

Well, @teadawg, when I was associate head of our department, I would give students a few problems to work to test whether they really knew what they thought they did. I don't know if someone there is willing to do that.

Ah, ok, @user112495. So are you wanting to show the groups are isomorphic, then?

user112495

@TedShifrin Yes.

Ted Shifrin

No, @teadawg, having taken high school calculus sadly does not mean one really knows precalculus and has algebra skills. We have seen this hundreds of times. :(

But we give a math placement exam to determine what someone's precalculus skills are :)

So, yes, you need to start with your $\psi$, @user112495. And use it to define what you hope will be the isomorphism.

Mary Star

18:36

Hello!! Could I ask someone about deleting nodes in a binary tree??

Ted Shifrin

@teadawg: Especially because most high school calculus courses are super calculator-heavy and many students literally cannot do arithmetic (let alone basic algebra/trig).

If your skills are indeed strong, they don't really want you in a precalculus course, bored out of your gourd.

Modded Bear

Noooo! someone deleted a question 10 second before I finished writing an answer.

user112495

@TedShifrin So I want to define a map $f : Sym(X) \rightarrow Sym(Y)$ that involves $\psi$?

Ted Shifrin

Yup, @user112495.

Daniel Fischer

@TedShifrin Found it. What Jasper said wasn't suspension-worthy, but deleting it wasn't a bad choice. Not necessary to delete it, but it wasn't constructive, and would have been offensive to some. But all in all, meh.

Ted Shifrin

18:44

Many of us are offensive to some :) Speaking of which, have I just been fortunate in avoiding my former nemesis, or has he actually disappeared?

Balarka Sen

I want to prove that if X def. retracts onto x in X then for every nbhd U around x there is a V in U such that the inclusion V --> U is nullhomotopic.

Ugh.

I want a hint.

user153330

@BalarkaSen :-)

user112495

@TedShifrin Hmm. Would you be able to give me a hint on how to start this.

Ted Shifrin

So what could the problem conceivably be, @Balarka?

Mary Star

@DanielFischer Can I ask you something about deleting nodes in a binary tree ??

Balarka Sen

18:45

@TedShifrin What problem?

Ted Shifrin

heya @dustin

@Balarka: The problem with your problem!

Mark Fantini

Hello, dustin

Balarka Sen

Heya Fantini.

Nitin

I just started contributing to this site - can anyone please explain what the little orange up arrow means? For example, for me it says X Reputation [Orange Triange] #

Ted Shifrin

@user112495: So think of $\psi$ as giving you a dictionary that translates $X$ into $Y$, right? So if a permutation of $X$ takes $x_1$ to $x_2$, what should the corresponding permutation of $Y$ do, do you think?

Daniel Fischer

18:46

@TedShifrin " seen Dec 11 '14 at 6:37 "

user153330

@Nitin the orange triangle is a badge, there's also a grey and gold one

dustin

Hello @TedShifrin and @MarkFantini

Ted Shifrin

wow, @DanielF. Well, I'm not in tears just yet.

P.S. @Balarka: I presume you have some hypotheses on $X$?

Balarka Sen

@TedShifrin Oh. Well, I have no clue how to start with this. I have to prove that there exists a map f : V cross [0, 1] to U such that f(-, 0) is the inclusion map and f(-, 1) is the constant map.

Daniel Fischer

@TedShifrin I guess not too many are inconsolable.

Ted Shifrin

18:48

@DanielF! There were so many negations in that sentence I'll have to spend an hour deciding what you meant.

Mary Star

@DanielFischer When we have a binary tree, where each node contains a key, and we want to delete the nodes of which the key is smaller that x, do we do that as followed??

Let f be a pointer to the tree.

while(f != NULL){
if(x < f->key){
f=f->RC;
}
else{
next=f->LC;
free(f);
f = next;
}
}

Ted Shifrin

So what makes it non-obvious, @Balarka, if you have a deformation retraction of $X$ to the point $x$? Can't you just use it?

Balarka Sen

@TedShifrin None other than what I have already said.

user112495

@TedShifrin Take $x_2$ back to $x_1$? So ${\psi}^{-1}$?

Ted Shifrin

No local compactness or anything, @Balarka?

Daniel Fischer

18:49

@BalarkaSen You know that $X$ contracts to $\{x\}$. Use that.

Balarka Sen

No.

Ted Shifrin

No, no, @user112495. $x_1, x_2\in X$. $\psi(x_i) = y_i\in Y$.

So, @Balarka, what could go wrong if you just used $V=U$?

Balarka Sen

Well if X contracts to x any nbhd is bound to contract to x, right?

user153330

@Chris'ssis i never had to do l'hopital with a limit involving integrals, adding that to my toolbox ...

Ted Shifrin

Sure, @Balarka, in a sloppy sense.

Balarka Sen

18:50

Oh @Ted. It says strict subset of U.

Chris's sis

@user153330 It can be done in many ways.The bad thing is that the question is terribly easy the way it is.

Daniel Fischer

@BalarkaSen But you must choose it small enough to be able to guarantee it doesn't leave $U$.

Ted Shifrin

@user153330: Make sure you also remember the Fundamental Theorem of Calculus for differentiating an integral.

user153330

@Chris'ssis yup, seems i need some sleep

Ted Shifrin

@DanielF: Drat you!!

Balarka Sen

18:51

@DanielFischer But U itself def rets onto x.

Ted Shifrin

no it doesn't, @Balarka.

Chris's sis

@user153330 At this hour? By the way, where do you live?

Balarka Sen

OK, confus.

Ted Shifrin

<--- waits to be censored for that "drat you" :D

user153330

@Chris'ssis i love sleeping, in fact i love sleeping so much that i can sleep at any hour

Ted Shifrin

18:52

@DanielF gave it away, @Balarka.

Daniel Fischer

@TedShifrin censors Ted

Balarka Sen

PS don't reveal the solution

dustin

@TedShifrin can you tell me why my answer isn't clear here. To me, it follows but from the comments it says otherwise.

Ted Shifrin

Don't worry, @balarka. You should know me well enough.

Balarka Sen

@TedShifrin i don't see it unfortunately.

user153330

18:52

g night @all

Ted Shifrin

Besides, I'd have to figure out how to do it to give it away :D

Balarka Sen

so it's not a "give aways" for me yet.

Ted Shifrin

night, @user153330

Chris's sis

@user153330 My mom often tells me I end up badly if I DON'T SLEEP ENOUGH. :-)

Ted Shifrin

When you deformation retract $X$ to $x$, $U$ may travel all over the place before it gets to $x$. I bet you know examples of this sort of thing.

Balarka Sen

18:53

@Ted @Daniel it's my homework problem, so don't reveal it.

user112495

@TedShifrin Oh, okay. So am I looking for something that would map $x_1$ onto $y_2$ for example?

Ted Shifrin

No, @user112495: You're confusing yourself. $x$'s map to $x$'s, $y$'s map to $y$'s.

@dustin: I haven't thought about such things since I TAed differential equations in 1974.

Daniel Fischer

@BalarkaSen If you read all of Ted's and my hints and think about them, you should find it. Maybe just something more, compactness is a nice property.

Ted Shifrin

@Balarka: chat.stackexchange.com/transcript/message/19485461#19485461 is all I'm going to say. You should have some examples where you can deformation retract to one point of a space but not to another point. Think about those.

user134177

hi

Ted Shifrin

18:57

@DanielF: That's why I asked about local compactness. I'm a bit puzzled.

Daniel Fischer

@MaryStar You leak all the right children of nodes with key $\leqslant x$.

user134177

if f:IR->IR is a continuous function, g(x)=x^2+f^2(x) has always a minimum value, right?

Mark Fantini

Hey, @BalarkaSen.

user134177

x^2>=0 for every x in IR

teadawg1337

@TedShifrin Ugh, I'm getting so stressed out with the deadlines... I really hope that I'll be able to start with 1910 instead of 1720, I'm keeping my fingers crossed

robjohn

18:59

@Chris'ssis in the line just above? did you want me to delete it or add the $\frac1\epsilon$?

user134177

can i say f^2 is always nonnegative?

Chris's sis

@robjohn add $1/\epsilon$ please :-) Initially I thought it might be a question good for my collection, but no, it's too easy.

user134177

i think it is correct

robjohn

@Chris'ssis Just looking at it, I can see it is $\frac{n}{2\pi}$

Ted Shifrin

@teadawg: Here are two questions I used to ask students who insisted they knew precalculus. (1) Give me the surface area of a closed circular cylinder with total volume $10\pi$ as a function of its height. (2) If an isosceles triangle has legs $2$ long and included angle $\theta$, give me its area as a function of $\theta$.

Chris's sis

19:01

@robjohn Yeah, that's the answer.

Daniel Fischer

@TedShifrin We must assume that $x$ is fixed during the deformation, otherwise, the assertion need not hold. Then we don't need local compactness (not even Hausdorffness).

Ted Shifrin

Right, @DanielF. I hadn't thought it through, but I know the tube lemma tends to show up quite, quite often in these sorts of problems. Continuity alone will do it.

Daniel Fischer

@bunny Yes, and therefore your $g$ indeed has a minimum value (possibly attained at many points).

user112495

@TedShifrin I'm a bit confused as to what I'm actually trying to get.

So an isomorphism $\phi : Sym(X) \rightarrow Sym(Y)$ is a bijection from $Sym(X)$ to $Sym(Y)$ such that $\phi ({\tau}_1{\tau}_2) = \phi({\tau}_1)\phi({\tau}_2) \forall \, {\tau}_1,\, {\tau}_2 \in Sym(X).$

And then any element of $Sym(X)$, say $\tau$ will map $x_1$ onto $x_2$ where $x_1,\, x_2 \in X.$

user134177

@Daniel thank you very much :)

Daniel Fischer

19:05

@Nitin The orange triangle symbolizes bronze badges. So far, you have three.

Ted Shifrin

OK, @user112495, so what should $\phi(\tau)$ do?

Laters

@user112495: just think about given a bijection from X to itself you can make up a bijection from Y to Y

Ted Shifrin

We're working on it, @Laters.

Laters

the more interesting question is whether you can make Sym(-) into a functor

but idk

Ted Shifrin

Not the appropriate question for someone just beginning to learn abstract algebra, @Laters. Hush.

Balarka Sen

19:08

OK so our space $X$ deformation retracts to a point, i.e., there is a continuous family of maps $f_t : X \to X$ such that $f_1 = id_X$ and $f_0$ maps everything to $x_0$

Consider $f_t |_V$

user112495

@TedShifrin So would it give you a different permutation, $\pi$, that maps $y_1$ onto $y_2$?

Ted Shifrin

Where $y_1$ and $y_2$ are who, @user112495?

Balarka Sen

Ahh! What we have to ensure is that $V$ is small enough so that image of $V$ is inside $U$!

user112495

@TedShifrin Elements of Y?

Balarka Sen

That explains Daniel's comment.

Ted Shifrin

19:09

@Balarka: $f_t$ does not map to $\{x_0\}$.

Sure, @user112495, but shouldn't they be related somehow to $x_1$ and $x_2$?

teadawg1337

@TedShifrin Just saw that you tagged me, I'll work those problems in a jiffy. Want me to give answers when I finish?

Balarka Sen

@TedShifrin There you go.

Ted Shifrin

@teadawg: I'm just trying to give you some sense of what I wanted a student in your position to be able to do with confidence. :)

Laters

@user112495: you should think in terms of function composition

Ted Shifrin

Thank you, @Balarka :)

Laters

19:11

this is basically the only thing what you can do in this generality

user134177

if lim x->a \frac{f(x)-f(a)}{x-a} exists, is the equation lim x->a \frac{f(x)-f(a)}{x-a}=lim x->a \frac{f(x)-f(2a-x)}{2(x-a)} true?

Ted Shifrin

@Laters: You and I have very different teaching styles.

user112495

@TedShifrin So would it be $\phi(\tau(x_1)) = \pi(y_1) = y_2?$

user134177

i would say yes but

Balarka Sen

So question is, how to choose a $V$ such that $f_t(V) \subset U$ for all $t \in [0, 1]$.

Ted Shifrin

19:13

@user112495: If $\tau$ maps $x_1$ to $x_2$, then $\phi(\tau)$ maps $Y$ to $Y$. What you're writing doesn't make sense. $\phi(\tau)(y_1) = y_2$ where $y_1$ and $y_2$ are who?

Correct, @Balarka.

user134177

i dont now how to reform the quotient \frac{f(x)-f(2a-x)}{2(x-a)} such that i can use that lim x->a \frac{f(x)-f(a)}{x-a} exists

Laters

@TedShifrin: it seems so. this perhaps explains why lots of people leave my classes (im TA for an algebra course) lol

Ted Shifrin

Well, I don't want to be a pompous ass, @Laters, but I have about 40 years of experience on ya :P

Chris's sis

brb

Ted Shifrin

Having said that, @Laters, the students who just want to be told the answers and memorize them leave my classes, too :P

user112495

19:15

@TedShifrin But doesn't $\phi$ map from $Sym(X)$ to $Sym(Y)$?

Ted Shifrin

Yes, @user112495. So, given $\tau\in\text{Sym}(X)$, $\phi(\tau)\in\text{Sym}(Y)$. We're trying to say what permutation $\phi(\tau)$ must be.

Laters

@TedShifrin: you certainly have. but to be honest when I talk about mathematics elegance and clarity is more important to me than trying to explain things in a "low-brow" way

Ted Shifrin

That's only appropriate for a very sophisticated, advanced audience, @Laters.

Well, not clarity.

But my goal is to get students to figure as much out for themselves and to learn how to think like a mathematician, not to give away answers for them to copy and not understand, elegant or not.

Balarka Sen

@Laters Mathematical mathematics looks interesting.

Ted Shifrin

The other thing you have to understand, @Laters, is that (a) most students will not think about mathematics exactly as you do, so you have to be prepared to explain things in different ways; (b) most students are not nearly as talented in mathematics as you are, either.

Laters

19:17

@BalarkaSen: it is better known as pure mathematics, so it certainly is

Ted Shifrin

For example, @Laters, when I teach undergraduate algebra, I draw pictures (with color coding) for LOTS of the course. Most algebra books/algebraists draw none.

P.S. @Laters: Teaching takes a lot of patience.

OK, I'm done.

user134177

oh, i solved this

Ted Shifrin

@LeGrandDodo !!

user134177

it wasnt difficult g

Laters

@TedShifrin: I just emphasize "structures" because that is the way I think about it and on the whole it seems to be a very conceptual viewpoint

Balarka Sen

19:20

Drawing pictures while teaching algebra is an excellent approach @Ted

Laters

@TedShifrin: pictures I do as well, if you consider commutative diagrams and exact sequences as pictures

I certainly do

LeGrandDODOM

@Ted Whoah, that was impressive. Never seen that many people (nowhere near after the 2012 elections)

Ted Shifrin

No, @Laters, that's not what I mean. I mean pictures for $G\to G/H$ (much as you might draw a fiber bundle in topology), color coding elements of $G/H$ and the corresponding cosets in $G$.

Yeah, I saw some pictures, @LeGrandDodo.

@Laters: Very few students approach math as formally as you. Indeed, even among professional mathematicians, many of us do not.

Laters

@TedShifrin: so do you this for general G/H or just for some example? e.g. the complex numbers modulo S^1

Balarka Sen

I always understood semidirect products via splitting of short exact sequences, and visualizing sections as topological sections.

Ted Shifrin

19:22

I do it in general and for examples, @Laters. Even color-coding $\Bbb Z$ when I first talk about $\Bbb Z/n\Bbb Z$.

And I do groups, rings, everything like this.

Laters

@TedShifrin: I don't approach all parts of mathematics like this. but to solve certain kinds of problems, which are termed of "general nonsense type" or "formal", I think it is very useful

user112495

@TedShifrin Sorry, I'm still confused. If $\tau$ can only act on elements in X, and $\phi(\tau)$ only acts on elements in Y, how are we getting between X and Y?

Ted Shifrin

I don't teach semidirect products in my undergraduate course, @Balarka. But when concrete examples came up, I never drew pictures for that, to be honest.

Right, @user112495. That was your $\psi$, remember?!! :) Use it.

Laters

Also in e.g. topology I think one only has to know about initial and final topologies, it is just a way of making things very concise

Balarka Sen

@TedShifrin Well, I kind of visualize them as "bundles" over Cayley graphs :P

Ted Shifrin

19:24

shakes head completely @Laters

Balarka Sen

I know, it's kind of lame, but works pretty well.

Ted Shifrin

Fine, @Balarka, and I think of it in terms of sheaf cohomology. shakes head

Balarka Sen

lolwut

Laters

haha @TedShifrin, I don't know why I am different than other students :(

lol

in any case it did explain a lot about Z/nZ and the corresponding unit group. it is sort of very classical stuff

Balarka Sen

I am partially influenced by the hyperbolic geometer I talk with, to be honest. Never thought I would :P

@Ted Are you being serious?

Ted Shifrin

19:27

@Balarka: It seems he and I have pushed you subtly, yes.

Serious about what? Have you finished your alg top question? I'm about to disappear.

Balarka Sen

Indeed.

@TedShifrin Serious about that sheaf cohomology thing.

Laters

In any case @user112495: lets try my approach. so you have your map $f:X\rightarrow Y$. If you have a bijection $g:X\rightarrow X$, how can we make a bijection $Y\rightarrow Y$? well you just form $f\circ g\circ f^{-1}$

the only kind of sheaf cohomology I know is derived functor cohomology. I have never learned about Cech cohomology, is it worthwhile?

Balarka Sen

Cech is good stuff

Ted Shifrin

Slightly. Splittings of exact sequences are computing Ext. For groups, for rings, for sheaves :P

Balarka Sen

i dunno about the cohomology part, but cech homotopy is very interesting

user112495

19:29

@TedShifrin So do we have $\phi(\tau(x_1)) = \phi({\psi}^{-1}({\phi}^{-1}))(x_1)$?

Ted Shifrin

@Laters. I just finished telling you NOT to just give the answer. That pisses me off. Seriously.

teadawg1337

@TedShifrin $$(1) SA=2\pi{h}\sqrt{\frac{10}{h}}+\frac{20\pi}{h}$$$$(2) A=2\sin\theta$$

Ted Shifrin

No, @user112495. You can't do $\phi(\tau(x_1))$.

Balarka Sen

@TedShifrin BTW, here is a formalization of thinking of Cayley graph of semidirect products as "bundles" : arxiv.org/abs/0912.2715

Ted Shifrin

So, in the interest of time, I will push you harder. Take $y_1 = \psi(x_1)$. If $\tau(x_1)=x_2$, what do you want $\phi(\tau)(y_1)$ to be?

Laters

19:32

@TedShifrin: it will take him a great deal of effort to translate this statement into his element notation

Balarka Sen

As you see, one of the author is him.

Ted Shifrin

@teadawg: First one, A+. Second one, not yet.

user112495

@TedShifrin $y_2$?

Ted Shifrin

But what is $y_2$, @user112495, if $y_1=\psi(x_1)$?

Here you really do need a picture, which I cannot draw here. :(

user112495

@TedShifrin $\psi(x_2)$

Ted Shifrin

19:34

OK, good, @user112495. $\psi(x_2) = \psi(\tau(x_1))$. Can you write the formula now?

Laters

it would be interesting how you would draw it @TedShifrin

teadawg1337

@TedShifrin Hmm... Is $\theta$ the angle between the two legs or one of the two congruent angles on the base?

Ted Shifrin

Arrows from one set to another, @Laters, of course. :)

Laters

would you draw X and Y as kidney shaped objects, like they do it for manifolds?

or just the arrows

Ted Shifrin

I meant $\theta$ to be the angle between the legs, @teadawg.

Laters

19:35

but how you would draw the sets

Ted Shifrin

Oh wait, maybe you had it.

Balarka Sen

Oh @Ted I think I have it.

Ted Shifrin

Sorry, @teadawg. I was too distracted. A+ on that, too.

@teadawg: You do not need to be in that class.

As blobs, @Laters :P

Balarka Sen

Eh, I could've applied the tube lemma if X was compact.

Darn it.

Ted Shifrin

@Balarka: I was thinking that. You only need a compact subset of $U$, not all of $X$.

Balarka Sen

19:38

yeah. but there is no such assumption given there.

user112495

@TedShifrin Oh, so is it ${\psi}{\tau}{{\psi}^{-1}}?$

Ted Shifrin

Tube lemma shows up a lot working with homotopies ... but actually you need compactness of $I$, not of $X$, right?

Laters

@BalarkaSen: munkres student here?

Balarka Sen

Yes, @Laters

@TedShifrin Ahh

Ted Shifrin

Super, @user112495. I hope you would have gotten that without @Laters's answer. :)

Does that make sense to you, @user112495? $\psi^{-1}$ (the dictionary backward) takes an element of $Y$ back to $X$, then you permute it, then you map back by $\psi$ (the dictionary forward).

Laters

19:40

@TedShifrin he didn't even read my answer, don't worry about this. but @user112495 sometimes it is more convenient just to think in terms of function composition.

Balarka Sen

Conjugation is easy to see if you think of sets as points and maps as paths joining points :P

Ted Shifrin

Once you're sophisticated, it's obvious, @Laters. It's not obvious to a beginner.

I explain conjugation in terms of change of basis/coordinates (analogous to the change of basis theorem).

user112495

@TedShifrin Yeah, that makes sense to me. I think the main thing that was confusing me was what I was actually looking for. Thanks for your help though. Sorry it took so long :p

Ted Shifrin

You're most welcome, @user112495. Learn well :)

Balarka Sen

I think I'll turn out to be a homotopy theorist instead of a number theorist and work with Morse function on Lens spaces all day.

Laters

19:42

@TedShifrin: change of basis theorem for matrix representations?

Balarka Sen

sigh

Laters

a prof of mine told us that the only way he understands it is in terms of a commutative diagram

Balarka Sen

Just arrows and objects approach does the trick

So, @Laters, tell me : what do you work with?

Laters

in any case I learned the general steinitz exchange lemma in his class, which is sort of very rare

Ted Shifrin

Yes, @Laters. Thinking of a group of symmetries, for example, conjugation transports a symmetry with respect to one "position" or "orientation" to another "position" or "orientation," and normality of a subgroups indicates that all such transports are in there.

Of course, I even have that commutative diagram in both my linear algebra texts, @Laters :P

OK, I'm outta here for now.

teadawg1337

19:45

@Ted Hmm.... Now that I think about it, proper class placement should require more than just my ACT scores. I'll set up an appointment for a separate placement test, that should be enough to prove I can take 1910

@Ted cya

Laters

@BalarkaSen: usually algebras over a commutative ring

bye @TedShifrin may the force be with you

Balarka Sen

we'll all misbehave without you @TedShifrin

Laters

mostly commutative/homological algebra

Balarka Sen

@Laters Ah, I have only recently started studying comalg.

Laters

which book are you using @BalarkaSen ?

Balarka Sen

19:47

Atiyah-MacDonald

Laters

this is classic, amazing book isn't it?

Balarka Sen

yeah. great stuff.

Laters

so elegantly written

Balarka Sen

i don't care about the theory though. the exercises are excellent.

Laters

and the exercises, they contain some cool stuff like the grothendieck group

well it anyway only introduces what is the minimum

Balarka Sen

19:48

i especially liked their topological aspect of studying Spec.

i am starting to "see" spectrums a bit now, in fact, haha.

Laters

so can you draw spec(Z[x]) my fellow?

Balarka Sen

sure.

i use the treasure map technique.

Laters

yeah, so how would you draw x^4+1 in this map?

x^2+1 is the only curve which the only usual treasure map contains

x^4+1 is sort of special in the sense that it is irreducible over Z, but reducible modulo every prime

Balarka Sen

@Laters a big doodle with a string coming out, intersecting the lines V((p)) coming out of the prime ideals at different parts.

yeah i'd have to check the factors modulo different primes p.

Laters

my fav. part is the curve x^2+1 and its connection to classical theorems of number theory, so pretty

do you know about regular local rings?

Balarka Sen

19:52

i am familiar with local rings.

no idea what a regular local ring is

Laters

do you know some algebraic geomtry?

Balarka Sen

nope

hope to study it at some point of time

you sound suspiciously like an arithmetic geometer @Laters

Laters

in any case it is all about zero sets of polynomials ("varieties"), and these have a natural ring of functions on them. if you localize at prime ideals, you get some local rings

saying that the local ring is regular local amounts to saying that the variety is regular/nonsingular at the point

basically looks like a manifold

Balarka Sen

i get what you're saying

Laters

@BalarkaSen: not quite, but I do find it interesting

have you studied dimension theory in Atiyah macdonald?

it tells you a bit about it

chapter 11 I guess

Balarka Sen

19:56

no, i am just a beginner :)

barely going through chapter 2.

Laters

ah ok. in any case you should try to appreciate the geometric meaning, lot is hidden but it is quite worthwhile thinking about the geometric meaning

user134177

@DanielFischer where i need that f is continuous, that x^2+f^2(x) has a minimum value?

Laters

so can you prove A^n isomorphic to A^m implies n=m for any commutative ring A?

Balarka Sen

@Laters yeah i kind of try to do every exercise by thinking about it geometrically

Laters

this is sort of a classical question, like R^n homeomorphic to R^m

Balarka Sen

19:57

but the later takes a lot of stuff like homology @Laters

user134177

@Laters yes, the proof using homology is very nice

Balarka Sen

@bunny you know homology?

:P cool.

Laters

indeed. do you know whether R^n can be isomorphic to R^m as an abelian group?

user134177

@BalarkaSen yes:D

Laters

so you are learning about tensor products @BalarkaSen? one of the best things about tensor products is extension of scalars

Balarka Sen

20:00

@Laters i haven't thought about it.

Daniel Fischer

@bunny It can have a minimum value also if $f$ is not continuous. But without a specific $f$, you need continuity - well, lower semicontinuity suffices - to guarantee that the infimum is attained and hence a minimum.

Balarka Sen

@Laters yeah.

Laters

actually they are isomorphic, in fact they are isomorphic as Q vector spaces because dim_Q R is infinite

it is a cool problem, is sort of surprising

Balarka Sen

oh ok as "abelian groups"

Laters

the problem with A^n isomorphic to A^m appears in the chapter 2 I guess

Balarka Sen

20:02

i was somehow assuming you meant rings.

then we just have to show if they are iso as Z-modules

and it looks pretty obvious

Laters

nope, I said as abelian groups

why do you think obvious?

user134177

@DanielFischer ok, thanks, i wasn't sure if i forget something. but now everything is fine, thx

Balarka Sen

ack, what i was thinking about won't work.

nvm, it's dead of a night in here.

Laters

are you talking about A^n isomorphic to A^m?

there the point is to do extension of scalars

to reduce it tho the case of a field

in any case I am off laters

Balarka Sen

i need to think about it :P

i can't think super-fast like you

user134177

20:06

@Daniel Fischer but i have to consider first the function x^2+f^2(x) on a compact interval?

Daniel Fischer

@bunny You don't need to do that (at least not explicitly), but that's a convenient way. You know that $g(x) \to +\infty$ for $\lvert x\rvert \to +\infty$, so the minimum will be attained in the interval $[-(1+g(0)), 1+g(0)]$.

(And you need lower semicontinuity of $f^2$ of course, not of $f$, to know the infimum is attained.)

shcolf

hello people, does anybody have idea how to compute lim(x->0, (1+sinx-cosx)/(1+sin(px)-cos(px)) = 1/p
I'm having difficulties for the first step. I have tried to divide each of fraction terms by cos(x) but got 0 in result

Daniel Fischer

@shcolf Taylor expansion of $\sin$ and $\cos$, you don't need to go far (the first non-constant term of each suffices).

Mike Miller

Morning

shcolf

well, Taylor expansion is yet beyond of my course

user134177

20:21

@DanielFischer oh ok thank you

shcolf

@DanielFischer thank you, google and wiki helped me, I'm done with this :)

Mary Star

What does this mean @DanielFischer ??

user134177

@DanielFischer you are good in math

Daniel Fischer

@bunny Only in some parts.

user134177

@DanielFischer :)

Transcript for

$Mathematics$ Mathematics