Mathematics - 2013-02-19 (page 1 of 5)

So I suppose the existence of a matrix of TD such that it is in diagonal form right

?

I am just a little confused with the M and N notations

Do you know what a matrix to change basis is?

12:29 AM

as in Pass(B,B') ?

the matrix of passage from B to B' ?

Yeah

I know but I can't use it here

because the problem is regarding the material before

@user43758 Well, I really think this is the best way to go.

Maybe your problem is ill posed.

Well it is in the Apostoll book

Else, to diagonalize, I'd just use eigenvectors and eigenvalues.

@user43758 Oh, I have it.

That is useful for you to say!

Which page?

12:31 AM

It is in the volumee 2

@user43758 No problem.

page 51

@user43758 Add it to the question.

problem 20 in the 2.12

You have it?

@PeterTamaroff

@user43758 Found it.

12:34 AM

@PeterTamaroff It is the last one of the 2.12 section and I had to do the 19

That is why I was confused on what to put or not

@Ethan The quotient of two dirichlet series will also be a dirichlet series. I feel it should be the case that if $q(s)$ is bounded as $s\to1^+$, then $\lim\limits_{s\to1^+}q(s)$ should exist. (Also, I think any dirichlet series will have unconditional convergence past a certain point, but I have nowhere to point to.)

@PeterTamaroff So to use the nilpotence argument in order to reach a contradiction

?

@user43758 I must be misreading something. How can Apostol be wrong?

Let me re-read a little.

ok

math101

Would you classify the average mathematician as egocentric?

12:39 AM

Yes but it's not a bad thing :)

math101

egocentric has a negative connotation

@math101 Bleh.

I think not.

To your question, I mean.

@user43758 Note that ${\rm Im}(V)$ is the span of $(x,x^2)$

The polynomials of the form $ax+bx^2$.

I'm thinking what would happen if he really menat $DT$

That is, $p'+xp''$.

Maybe it would work out.

ok

Nope.

It is still nilpotent.

@PeterTamaroff Yes

So I can use the nilpotence argument for this problem right ?

12:46 AM

I think so, but I feel like I'm missing something.

@JacobBlack I already own it. :) I was looking for a different perspective.

@PeterTamaroff I was wondering if you knew any good websites to practice problems?

@user43758 Not really. I prefer books.

You should find a lot of exercises in math.SE though.

I prefer books too but I assumed you would know more website

Do you recommend any?

I am taking Multivariable Calculus with Theory this term

I am a freshman

I know none.

12:52 AM

ok..

books on multivar calc?

Yes and perhaps Real Analysis

I used Vector Calculus by Susan Colley in high school, I liked it.

@Ethan removed

:)

I didn't read your last comment lol

Now I know

12:55 AM

@anon

I am not in high school

I am in university freshman

The calc III class I took in HS was twice as rigorous and enlightening than the one I took in university. Your point?

Then again maybe I only felt that way because I had already seen the material, who knows.

I wasn't making any point lol

@anon Where did you take a course like that in hs

my high school was blessed with a mathematical man who had delusions he was really a college professor...

(I think he would appreciate that epithet)

@anon you are lucky.

12:59 AM

@anon Lucky you man!

I know you profited from that!

@anon I still don't understand, did you take a course some where else?

What do you mean? I took it once in HS, and once in uni last semester.

@anon Well I assume your high school didn't offer calc 3 as a course to take

@PeterTamaroff Maybe we have to find a basis for both V and W ie not consider the given basis?

@Ethan It did. It even offered Advanced Applied for one year, which was differential equations at a graduate level (but going 2-3 times slower). Only two other students took it with me. Both classes due entirely to the teacher I mentioned.

1:02 AM

@PeterTamaroff I mean maybe we have to find a basis for V and W such that the matrix of TD is in diagonal form

@anon How? Was it a special technical school or somthing?

It was a private Jesuit school. It was not a special technical school, but it did emphasize academics. (Unfortunately as I left the direction seemed to be tipping in favor of sports - which I couldn't care less about.)

Three. Why not? My current p-adic class is only me and one other student.

I was in such a class as well

I was only with another student

@anon I am not very familear with academia, also I suppose it depends where you go to school

The Substitute

Howdy Y'all. If I can find an upper bound on the Hausdorff dimension of $A_i$, can anything be said about the Hausdorff dimension of $\cup A_i$ ?

1:06 AM

It is the sup of the dimension of the A_i

@TheSubstitute

dimH(XxY)\geq dimH(X)+dimH(Y)

@PeterTamaroff Am I misreading something in the book ?

The Substitute

Thanks, this follows from the Def. Thanks @user43758

@anon, If I was to substitute a complex number z,with magnitude <1, into $$f(z)=-\ln(1-z)=\sum_{n=1}^\infty \frac{z^n}{n}$$, what exactly does $f(z)$ represent? In what sense is it still a logarithm? Would it satisfy $e^{f(z)}=z$?

user19161

@anon He is probably a banana.

@Ethan no, it would satisfy $e^{f(z)}=\frac{1}{1-z}$

@anon my bad, but why? I can add multiples of $2\pi i$ to an exponential with out changing the value, but that would change the value of the logarithm?

1:14 AM

@Ethan This would happen only on the disk $|z|<1$.

And as anon corrected.

keywords: "monodromy," "branch cut," "riemann surfaces"

@PeterTamaroff They are asking me to find bases for V and for W

@PeterTamaroff Therefore perhaps we can't use the one defined in the previous problem

We make an arbitrary choice of confining the logarithm function to take complex values with imaginary part in $[0,2\pi)$, similar to what we would do for say $\sin^{-1}$ or $\cos^{-1}$ - a noninjective function does not have an inverse, so we are forced to make a choice of where in the image we want to confine ourselves to in order to define an inverse map that is not multi-valued.

Any homomorphic image of $G$ is isomorphic to a factor group $G/K$ by a normal subgroup $K$.

@Ilya Yes.

1:19 AM

@JonasTeuwen Jonas. How's it chillin, brozzam?

@PeterTamaroff Yes. You want all of those isomorphism theorems to be engraved in your intuition eventually.

@anon My "intuition"?

saadtaame

What's a manifold?

@PeterTamaroff Am I reading something wrong?

@user43758 I don't know

I really gave up on Apostol this time.

1:21 AM

because they say find one for V

Not use the same one as the one before

Chokladkakan

It is late at night, but I felt I needed to have something in print here: http://math.stackexchange.com/a/307641/12880

Given the time I may very well be off. I would appreciate any and all scrutiny.

With that, I need to bed myself. Sleep well, MSE.

@PeterTamaroff

@user43758 What do you want to do?

Is it possible to proceed by constructing a base for V and for W

?

No, I don't think so.

@anon I think I have some nice exercise to do

1:25 AM

Ok well thank you for your help

@user43758 No problem.

Have a good evening

It's night here, hehe.

10:25 pm

But thanks.

it's 8:25 still

:)

@PeterTamaroff Good. I moved my blog.

1:29 AM

@JonasTeuwen Wiiiiiiiii!

Yes.

@JonasTeuwen During my vacations, I went to a brewery and tried a bunch of beers.

Some were pretty good.

That's great!!!

They served me a tray with their blends.

@JonasTeuwen I signed up for my math career today!

How so?

1:33 AM

I went to my uni's "mothership" and made a life-sucking time-consuming mind-killing errand, and now I have a auxiliary credential till I get the official one.

Excellent.

Good boy.

And I can sign up for courses in a few days.

Now treat yourself with a beer.

@JonasTeuwen Good idea

Orange Harvester

@JonasTeuwen If you install your wordpress instance on your own server, you can have MathJAX.

1:36 AM

@JonasTeuwen What do you think of Corona beer?

Like the joke with the cannoo.

@OrangeHarvester I have MathJax.

@JacobBlack Ah, this post is swimming in references.

Orange Harvester

@JonasTeuwen Yes, I saw that later. :-)

Dawg. THis is my uni's bibliography for Measure and Prob and Real Analisis

J. Cerdà, Análisis Real. Universitat de Barcelona, 1996.
N. Fava y F. Zó, Medida e Integral de Lebesgue. Red Olímpica, 1996.
G. B. Folland, Real Analysis - Modern Techniques And Their Applications. John Wiley & Sons, 1984.
P. R. Halmos, Measure Theory. Van Nostrand, Princeton, 1950.
S. Igari, Real Analysis - With an Introduction to Wavelet Theory. American Mathematical Society, Volume 177, 1998.
H. L. Royden, Real Analysis. Mc Millan, 1968.

What do you think about it?

Looks good.

1:44 AM

@JonasTeuwen Thanks.

@anon

?

I am determining $\text{aut}G$ for $G$ ciclic, infinite and then finite.

I' starting with infinite.

I'm thinking it is "simply" determined by $\eta(a)$

I mean, you take a value $\eta(a)$ and then it is $\eta(a)^k$

For any $k$.

what are $a$ and $\eta$?

$\eta$ would be the automorphism.

$a$ would be $\langle a \rangle =G$

a cyclic infinite group is just $\bf Z$ up to isomorphism, so might as well work with $\bf Z$

1:50 AM

Yes, I know.

any homomorphism out of a cyclic group is determined by where the generator is sent, yes

Oh, OK.

an automorphism is a homomorphism

just like a square is a rectangle

Sure.

Thus there are $|G|$ automorphisms?

there are |G| endomorphisms. the automorphisms are the invertible ones.

1:53 AM

Oh, silly me.

Wait.

But if $G$ is infinite, $\eta(1)=1$ only for the identity.

rephrase

So any endomorphism is injective.

@anon It is cyclic of infinite order.

yes, endomorphisms on Z are injective

not necessarily surjective though

Well, so there are $|G|$ endomorphisms.

@anon No?

pick some number a and consider the endomorphism $x\mapsto ax$

with $a\ne\pm1$ of course. $a=0$ would be the most extreme example. (indeed, the trivial endomorphism is never an automorphism, unless we are working with the trivial group.)

1:56 AM

Oh, OK.

So the only homomorphism is the identity, yes?

cough -1 cough

But that is just the same.

Since we go through negatives.

Oh, well, OK.

no, $x\mapsto x$ and $x\mapsto-x$ are not the same map

Sure, sure.

I was being an ass to $x\mapsto -x$

the inverse map is an automorphism in any abelian group. it is only identical to the identity map if the group is of exponent 2, which is necessarily abelian, and hence necessarily a direct sum of cyclic groups of order 2 (perhaps infinitely many - no conditions on cardinality are made)

2:00 AM

Wait, isn't any infinite cyclic group isomorphic to $(\Bbb Z,+)$?

...

SHUSH ME!

Both $x\mapsto x$ (the identity map) and $x\mapsto-x$ are automorphisms of $({\bf Z},+)$.

Yes, yes,.

Moving on to finite cyclic groups.

Case $|G|=6$.

Any finite cyclic group is isomorphic to a factor group $\Bbb Z_n$, yes?

@anon

yes

So for this case it suffices that $\mod 6, \eta(1)=\pm 1$.

2:20 AM

Alright. Good night guys.

@anon So $x\mapsto 5x$ would be an automorphism, yes?

And so will $x\mapsto x$

in Z6 yes

OK, and those are them.

Wiat.

Wait.

Yes, I think that is it.

show that if (a,n)>1 then the map x->ax in Zn will have nontrivial kernel. hence you can identify Aut(Cn) with multiplicative by units in the ring Zn.

My question is: why do we talk about $5\times x$ when our operation is $+$?

@anon Yes, indeed. I was thinking about $\varphi(n)$

2:25 AM

it is $x\mapsto x+x+x+x+x=-x$. IOW any abelian group is a Z-module.

$\eta(n)=\sum^n \eta(1)=n\eta(1)$.

@peoplepower Yes, I know.