Mathematics - 2015-06-15 (page 1 of 3)

Karim Mansour

00:40

ok yeah I found my error in my question

ok good

anybody has seen concave functions ?

I am reading some paper that says it will prove Minkowski’s Inequality by relying on concave functions

Semiclassical

sounds like an appeal to jensen's inequality?

Karim Mansour

what is jensens inequality?

Semiclassical

Jensen's inequality

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean after convex transformation; it is a simple corollary that the opposite is true of concave transformations. Jensen's inequality generalizes the statement...

Karim Mansour

yeah exactly

Semiclassical

that's in terms of con-vex functions, but as mentioned in that opening that's just a sign change

Karim Mansour

00:50

I don't know about con-vex functions good that you gave me the name

I can now read about them

I mean when you look it at it makes intuitive sense

Semiclassical

it's terminology, really. concave up = convex, concave down = concave

Karim Mansour

I see

oh ok

I didn't encounter this in my analysis class before

Semiclassical

it's a handy tool for inequalities

Karim Mansour

yeah

Semiclassical

if you want to see a lot more of that stuff, track down "The Cauchy-Schwarz Master Class"

Karim Mansour

00:55

is that a book ?

Semiclassical

yeah

Karim Mansour

cool

will download it :D

I am currently reading apostol during the summer to make my analysis background much better

maybe I will see those stuff aswell

Semiclassical

here's the MAA's review of it, which includes the table of contents

maa.org/publications/maa-reviews/…

Karim Mansour

very nice @Semiclassical

Semiclassical

it really is

Karim Mansour

00:58

I was wondering @Semiclassical did you ever take a class in QM mathematical foundations ?

Semiclassical

not in foundations, no. i've taken the usual physics ones (and TA'd for two of the undegrad ones while in grad school)

i've done some reading, but not in any systematic fashion

Karim Mansour

I see

that would be really cool class to take

Semiclassical

(for instance, don't ask me what C*-algebra's have to do with QM)

Karim Mansour

yeah I would like to take something like this

but ofcourse after mastering the math behind it

to make better connections

Semiclassical

sure

Calculus

01:03

@AlecTeal i put a question on the site if you want to see

0

Q: Showing $\int_0^\infty \frac{x\sin(2x)}{x^2+3}dx=\frac{\pi}{2}e^{-2\sqrt3}$

Things are about to get real, prepare your mind! $$f(z)=\frac{ze^{2iz}}{z^2+3}$$ Now this has two singular points, $\pm\sqrt3i$ $$f(z)=\frac{ze^{2iz}}{(z-(-\sqrt3i))(z-\sqrt3i)}$$ and hence both poles are of order $1$. We can find $\operatorname{Res}_{z=\sqrt3i}f(z)=\frac{(\sqrt3i)e^{2i(\sq...

also @Semiclassical whats an example of an analytic function mmapping the unit disk onto the right half plane $\{z\in \Bbb C|Re(z)\gt 0\}$?

Semiclassical

unit disk as $|z|<1$?

i.e. not including the boundary?

Calculus

Yep

should be a mobius transform?

Semiclassical

a mobius transformation should do the trick, since the unit disk and the right half-plane are just different hemispheres of the riemann sphere

Calculus

how do i find it in general?

Semiclassical

pick some easy points, see where they'd need to go

and use that to find a usable mobius transform

think of some specific example which seems simplest to compute

columbus8myhw

02:03

@KarimMansour There are plenty of free copies of that book on the internet

Karim Mansour

I see

that is how I get my books @columbus8myhw

columbus8myhw

1 2 3

Karim Mansour

:D

columbus8myhw

And, slightly more legal, three chapters (though it's not the whole book)

Karim Mansour

thank you @columbus8myhw

columbus8myhw

02:04

Welcome

Minjae

can anyone tell me how to type mathematical symbols?

I'm so sorry to ask this question here, but I'm so desparate, any kind of links or help would be appreciated

skill patrol

960

Q: MathJax basic tutorial and quick reference

To see how any formula was written in any question or answer, including this one, right-click on the expression it and choose "Show Math As > TeX Commands". For inline formulas, enclose the formula in $...$. For displayed formulas, use $$...$$. These render differently: $\sum_{i=0}^n i^2 = \fr...

discussion faq mathjax reference

user24142

02:41

hi, I'm looking for a sanity check math.stackexchange.com/questions/1322199/…

Basically, got a question involving a metric space, and someone claiming to be able to use an order on that space. That doesn't make sense, does it?

Semiclassical

@user24142 i'm inclined to agree with you. at the very least, the poster has hardly clarified the matter.

plus, to tell the person who wrote up the question that they shouldn't downvote for not understanding something 'trivial' strikes me as altogether unconstructive

it's their job to make their answer comprehensible, not yours

(warning: i'm not an analysis guy, so i can only read that answer shallowly myself)

user24142

@Semiclassical thanks. I was pretty sure that i was making sense, just needed someone else to not feel crazy.

Semiclassical

about the only interpretation that i could see myself of having inf/sup on a metric space without order is that you'd still be able to talk about what points in the space are furthest/closest from a given point, and what those distances are

user24142

02:57

yeah, then you're mapping to the reals, using that order. He has stuff like "f(x)<c" in a space with no order, and there's just no meaning to impute to that clause.

Shin Kim

04:28

Hi people, one question about Laplace transform.

As you know, Laplace transform is a kind of function. Then what is its domain and codomain?

$\mathcal{L}:?\times\Bbb{R}_{+}\to ??$

Okay, i've figured it out till this: $\mathcal{L}:X\times\Bbb{R}_{+}\to\Bbb{C}$. But how is $X$? collection of functions? collection of convergent functions?

wait,, is it just $\mathcal{L}:\Bbb{R}_{+}^2\to\Bbb{C}$??

$\mathcal{L}:\Bbb{R}_+\times S\to\Bbb{C}$. This one seems bettter

anon

05:09

@ShinKim there are two spaces of functions, A and B, for which the laplace transform is a linear map A->B

I won't pretend to know what A and B are. For A, you want to find necessary and sufficient conditions for taking the transform, and for B you want necessary and sufficient conditions for taking the inverse laplace transform.

Vibhav Pant

How would I evaluate $\sum_k^n k\cdot k!$?

Karl Kronenfeld

@VibhavPant Hm, don't know how to give a hint without spoiling it.

Vibhav Pant

hmm

Karl Kronenfeld

But, if you are lost, do the usual: check a few cases and look for a pattern

Shin Kim

@anon Umm,, is a Laplace transform gives a function? For example, the Laplace transform of a function $f:\Bbb{R}_+\to\Bbb{R}$ defined as $f(x)=1$ is $1/s$ where $s\in\Bbb{R}\setminus\{0\}$ and also $1/s\in\Bbb{R}\setminus\{0\}$. Then the codomain of this transform is a set of numbers, not a function. isn't it?

if it gives a function, then it should be like $\mathcal{L}(f)=g$ where $g$ is a function, which is also a set, not a number. but Laplace transform, in fact, gives number that represented with variable.

anon

05:26

@ShinKim 1/s is a function of s

@VibhavPant try some partial sums and see if you pick up on anything

Shin Kim

@anon $f=\{(s,t)\mid s\in S\subseteq\Bbb{R}\wedge t=1/s\}$ is the function. $f(s)=1/s$ is just a statement of algebraic equalness between members of domain and range.

anon

$f$ is a function of $x$, and $\int_0^\infty f(x)e^{-sx}dx$ is a function of $s$. deal with it.

the whole idea behind calling the laplace transform a transform is that it sends functions to functions

Shin Kim

@anon intuitively, it is. but formally speaking, a function sends a number to number, hence the transform itself also sends number to number. take a look at its expression : $\mathcal{L}\{1\}(s)=1/s$. its domain is $\Bbb{R}$ and the codomain is $\Bbb{R}\setminus\{0\}$.

Karl Kronenfeld

05:43

"send to"..."number"

Shin Kim

considering the term in the brackets $\{$ and $\}$, it should be at least $\mathcal{L}:X\times\Bbb{R}\to\Bbb{R}\setminus\{0\}$.

the problem is, what is $X$.

Karl Kronenfeld

a function is a set of pairs that has a certain property that reflects our intuitive notion of sending stuff to stuff.

Shin Kim

@KarlKronenfeld Exactly.

Karl Kronenfeld

you're missing my point

Shin Kim

@KarlKronenfeld We are discussing about the domain and the codomain of a Laplace transform.

Karl Kronenfeld

05:45

if $f$ and $g$ are arbitrary functions, then $\{(f,g)\}$ is in fact a function

Shin Kim

and?

Karl Kronenfeld

so why can't the laplace transform have a bunch of functions for its domain and a bunch of functions for its codomain?

Shin Kim

take a look at its notation. for example, $\mathcal{L}\{1\}(s)=1/s$.

Karl Kronenfeld

I read it as $\left(\mathcal{L}\{1\}\right)(s)=1/s$

Shin Kim

obviously, $s$ here is not a function. it is a member of the real numbers. $1/s$ as well.

Karl Kronenfeld

05:49

right you're defining the function $\mathcal{L}\{1\}$ by saying that its value at an arbitrary $s$ is $1/s$

Shin Kim

and $1$ here, is a constant function which is defined as $1:\Bbb{R}\to \{1\}$.

and this one is the problem i'm struggling. until now, what i've got is the following : $\mathcal{L}:X\times S\to\Bbb{R}$ where $S\subseteq\Bbb{R}$.

struggling what $X$ is.

Karl Kronenfeld

I give up.

Actually, lemme look for something that explains this better than I can.

Shin Kim

alright. just FYI, Fourier transform is also defined like that. $\mathcal{F}:\Bbb{R}^n\to \Bbb{C}$.

whatever it is, i bet we have to consider this as numbers, not functions. am trying to review some algebraic structures, like rings, fields, module, etc. since Laplace transform follows linearity between 'functions - precisely, the algebraic form of the values of the function'

anon

06:04

transforms send functions to functions

if your formal definition is ${\cal L}:X\times S\to \Bbb R$, where $X$ is a function space and $S$ is a set of values for $s$ to plug in, that strikes me as a crappy definition. it is not standard and I see no use in it.

the accepted, morally correct definition is $\cal L$ sends a function $f(x)$ to a function ${\cal L}\{f\}(s)$

I mean, given any function $f(x)$, one could argue "oh that's not a function it's a number, because $x$ is a number so to is $f(x)$." that's just obtuse.

Shin Kim

@anon the definition of a function is a relation which last entry is uniquely defined. you know that this means that a function is also a set. when you say that $f(x)$ is a function, it is fine. we all call like that because $f(x)$ represents the algebraic property of its values. but more correctly speaking, a function with the domain $X$ and codomain $Y$ which members satisfies the equality $f(x)=x^2$ (for example) can be expressed as $f:X\to Y:x\mapsto x^2$.

to sum this up, my point is that saying $f(x)$ is a function is not 100% correct.

Balarka Sen

06:19

hey, @KarlKronenfeld.

Karl Kronenfeld

hi @Balarka

anon

@ShinKim you began by saying transforms do not send functions to functions, and that was your original point

If $f$ is a function, then ${\cal L}\{f\}$ is a function.

Shin Kim

@anon I mean, the point on that statement.

so, what is the domain and codomain of $\mathcal{L}\{f\}$?

anon

certain sets of numbers of course, you know that

Shin Kim

god,

Balarka Sen

06:22

@KarlKronenfeld what kind of math have you been thinking about, then?

Karl Kronenfeld

@BalarkaSen I'm toying with this problem.

Balarka Sen

eh. not the kind of problem I'd like to think about.

Remember me

06:52

Hey @Balarka

Balarka Sen

hi

Remember me

I am on page number 86 Munkres..
The product topology $X \cross Y$@Balarka

Balarka Sen

ok

Remember me

And thats behind the gluing lemma @Balarka

The Substitute

07:30

Is probability theory useful to know when studying dynamical systems?

Mike Miller

08:06

I haven't ever seen it used much but I only have knowledge of random bits of dynamics.

Chris's sis the artist

09:27

@robjohn In my book I think I'll have at least 100 very advanced series involving harmonic numbers. I just created 3 more, absolutely crazy awesome.

Hope to cover all series in Philippe Flajolet's paper, all elementarily done, using the classical analysis (without making use of any complex analysis).

I was thinking to propose in Monthly $$\sum_{n=1}^{\infty} \left(\frac{H_n}{n}\right)^3$$ but the odds are poor to be accepted. I mean, Prove by real analysis that bla bla.

Maybe the best thing si to make an article with it (but there is a problem then, I reveal too many of the critical tools).

I should wait to publish my book first though.

skill patrol

Is the target audience for your book grad students?

Chris's sis the artist

@skillpatrol to both undergraduated and graduated ones, and not only. I often have in mind the word passionate, it's beyond all for passionates of integrals, series and limits (some being understood easily even by the kids in high school).

My book will turn the worst integrals, series and limits in tamed cats anyone like to touch and play with.

All will be very easy to understand, cleary explained.

(not just another book, but an unique one)

skill patrol

09:43

Have you @Chris'ssistheartist signed any sort of publishing contract yet?

Chris's sis the artist

@skillpatrol These details I wouldn't like to discuss here, but in private I might respond to that. :-)

skill patrol

I see.

Be sure to get a lawyer to read and explain to you all the fine print details

:-)

Chris's sis the artist

@skillpatrol Yeah, definitely. Thanks! :-)

skill patrol

You're very welcome.

Chris's sis the artist

Maybe at some point I'll be able to publish with him a book (I mean another book).

skill patrol

09:48

Sounds good :-)

Chris's sis the artist

:D

robjohn

10:39

@Chris'ssistheartist How much has been finished?

Chris's sis the artist

@robjohn Pretty much. However things changed a bit since I made the decision to increase the number of problems up to 500.

Initially it was 300, but I want more.

skill patrol

What's the price range of your book @Chris'ssistheartist

Chris's sis the artist

@skillpatrol Publisher (it's not up to me to decide that).

skill patrol

High, medium?

Chris's sis the artist

@skillpatrol I'd like to be very cheap, anyone can afford it.

skill patrol

10:44

Ok

Shin Kim

guys, how do i manage the ignored users? i accidentally pressed ignore button

skill patrol

I've never used it, sorry.

Try clicking on their avatar again.

robjohn

@ShinKim go to your chat profile and click on prefs... delete the user from those being ignored

Shin Kim

Ah, Thanks @robjohn

Chris's sis the artist

@robjohn it would have been far nicer to have 1000 problems, but this is hard, and push a lot the publishing date of my book.

Maybe the publisher would accept such a thing if writing it as an e-book only.

Calculus

11:03

hi

skill patrol

Hi

Calculus

@skillpatrol how are you?

@skillpatrol what do you study?

Huy

@BalarkaSen: Are you here?

skill patrol

Fine thanks. How are you?

Calculus

does $\sup_{0\leq x\leq 1} |f(x)-g(x)|$ give me the $x$ value that maximises $|f(x)-g(x)|$ or the maximum value $|f(x)-g(x)|$ attains in this range?

skill patrol

11:05

@huy he was here about 10 minutes ago

Balarka Sen

I am now, @Huy.

Tobias Kildetoft

@Calculus The latter

Huy

The latter, @Calculus.

Calculus

thank you

Tobias Kildetoft

@Calculus The former is often called argsup

Calculus

11:06

oh thats cool, thanks for that also

Tobias Kildetoft

(for argument sup)

Calculus

I guessed :P

Huy

@BalarkaSen: Just some very basic questions: $G(2,3) \cong G(1,3)$, right? (Grassmanian)

Balarka Sen

right

Calculus

what is the grassmanian?

Balarka Sen

11:07

Every plane through origin is parametrized by the perp of the plane.

Huy

@BalarkaSen: So the two manifolds have the same dimension?

Balarka Sen

And that's a 1-dimensional subspace of R^3.

@Huy yes

Huy

@BalarkaSen: How can I go on to find the dimension of $G(n,k)$? Does this generalize somehow or is there a different reasoning involved?

Balarka Sen

There are both 2-manifolds. $\Bbb RP^2$, in fact.

Higher Grasmannians are a big mess, @Huy.

Calculus

How does one rigorously say that $\sup_{0\leq x\leq 1} |f(x)-g(x)|$ is non-negative? i mean we can obviously see that the sup of an absolute value argument is non neg

Huy

11:09

Is there a trick like stegeographic projection which works for arbitrary $S^n$?

Calculus

do i just say that |f(x)-g(x)| is non-neg by definition hence sup blah is

Balarka Sen

no, I don't think so.

Huy

@Calculus: Drop the "obviously" and you're done.

iwriteonbananas

hi balarka

Calculus

@Huy is saying 'obviously' 'clearly' and such bad?

Tobias Kildetoft

11:11

@Calculus Since it is non-negative even without the sup, it is indeed obvious

Balarka Sen

If I recall correctly, @Huy, there is an explicit homeomorphism type for Grassmannians out there. I am not the right person to ask about them.

Huy

@BalarkaSen: Hm. In my diffgeo notes from last semester I have this "exercise" finding the dimension of higher Grassmanians, that's why I was thinking there must be a trick. I'll go on then. Any nice Grassmanians apart from the two we've mentioned that are important?

Balarka Sen

hi @iwriteonbananas

Tobias Kildetoft

@Huy Hmm, I only recall seeing something about the number of elements in Grassmanians over finite fields, which end up being related to quantum numbers

Huy

@Calculus: It kind of is an insult for anyone who doesn't immediately sees it, and sometimes even very smart people don't see something "obvious" immediately, which is why I prefer to not use it.

Balarka Sen

11:13

@Huy My knowledge of Grassmanians are limited. I have only studied them when doing Lefschetz fixed point theorem.

Huy

@TobiasKildetoft: Ok, not very important then I guess.

Balarka Sen

You'd better ask Ted all this.

Calculus

@Huy that is a good point ill keep that in mind

Huy

@Calculus: If you find something very obvious and you think it should be obvious, it is better to write things such as "a short calculation leads to" or "by definition it follows" or so.

Balarka Sen

@iwriteonbananas doing anything interesting?

iwriteonbananas

11:15

@BalarkaSen the heat is getting to me, i've been slightly sick, hence not very productive

Huy

@Calculus: So in your case I would just write "by definition, the expression must be non-negative".

Calculus

@Huy thats a good idea even though i probably dont offend people yet

Balarka Sen

oh, i'm sorry, @iwriteonbananas. it's been raining loads in here, though, so i have quickly recovered.

Huy

@BalarkaSen: Do you know the configuration space of $3-4-5$ triangles?

Calculus

@skillpatrol I'm good thank you. Just doing some learning right now

Huy

11:16

(in R^2)

Balarka Sen

@Huy no, what's that?

iwriteonbananas

@BalarkaSen nice

the other day i asked you a question about the homology of the space $X$ which is the disk with two disjoint open subdisks removed and the boundaries of all three disks identified, preserving orientation. do you remember?

Huy

@BalarkaSen: I guess some parameters which completely "span" all 3-4-5 triangles. It's just mentioned in some notes online without further detail, and I hoped you knew. :D

Balarka Sen

nope. but you can "see" the space by taking a torus, removing a 2-disk and pasting a cylinder with on end circle to the bd of the removed disk and the other circle to one of the longitudal circles of the torsus, @iwriteonbananas

and putting a CW-structure on this is no big deal

Calculus

If I am proving things for a metric that is |f-g| do i still prove triangular inequality as $|a+b|\leq|a+c|+|b+c|$ or do I do it with the minus like:

$$|a-b|\leq|a-c|+|b-c|$$?

Balarka Sen

11:18

@Huy I dunno, I think I have a vague idea of what you might be talking about, but I am not sure if that's what you want.

iwriteonbananas

@BalarkaSen yeah, i was gonna ask you if it has a homotopy type we can recognize. are the homotopy groups $H_2(X)\approx H_1(X)\approx H_0(X) \approx \Bbb{Z}$ and $0$ else?

skill patrol

@Calculus you can particularly sound offensive if you say "obviously" at the end of your sentence.

Huy

@BalarkaSen: I have as a remark "even the space $\{a-a-a \text{ triangles in } \mathbb{R}^2| \, a \geq 0\}$ is a manifold". How would I possibly find charts for that one?

Calculus

@skillpatrol yes my bad ill not do that

Balarka Sen

@Huy I don't even know what's the topology on them.

If you can define them, I am willing to have a look.

Huy

11:20

@BalarkaSen: Is that towards me or banana?

Calculus

did n-e-1-c my question above?

Huy

@BalarkaSen: Quotient?

Balarka Sen

@iwriteonbananas I don't think $H_2 \cong \Bbb Z$, but let me compute.

Calculus

Probably my last question for this hour at the least.

Balarka Sen

@Huy ? You have just given me a set over there. What's the topology you have in mind?

Huy

11:23

@BalarkaSen: Last semester we started just putting charts on arbitrary sets, then after having an atlas we would use the quotient topology to get a topology on the manifolds.

Balarka Sen

ok, @iwriteonbananas, $H_2 \cong 0$.

skill patrol

"Clearly" is ok at the beginning of a clear explanation :-)

Calculus

@skillpatrol Clearly you should only use clearly at the beginning of a clear explanation :-)

skill patrol

:D

iwriteonbananas

@BalarkaSen how did you compute it? what CW structure do you have for $X$ btw.?

Balarka Sen

11:31

sorry, I was away. @iwriteonbananas I used Mayer-Vietoris :P

iwriteonbananas

@BalarkaSen oh?

there is 1 2-cell right?

Balarka Sen

who needs cells to use Mayer-Vietoris?

I just told you above that it's torus minus disk $\cup$ cylinder.

iwriteonbananas

i wanna write the space as a pushout

Balarka Sen

noo

iwriteonbananas

lol

ok, i dont fully "see" why the torus minus disk $cup$ cylinder is the space $X$

Balarka Sen

11:34

OK. Take your disk, punch two holes in it.

iwriteonbananas

yea

Balarka Sen

Now identify the bd of your disk and bd of one of the holes.

This give your a punctured torus.

iwriteonbananas

yeah, true

Balarka Sen

Now take the bd of the remaining hole and identify that with bd of the original disk, which is now one of the longitudal circles.

iwriteonbananas

right, ok

Balarka Sen

11:35

So take a cylinder, attach appropriately, and voila.

iwriteonbananas

that's a cool way to see this space, how did you find it?

Balarka Sen

Practice :)

iwriteonbananas

cool. then you chose as open cover of that space (an $\epsilon$-fattening of) the cylinder and punctured torus?

Balarka Sen

yes

iwriteonbananas

and why exactly is $H_2(X)=0$? the MV sequence reads $0\to H_2(X) \to \Bbb{Z}\to \Bbb{Z^2}$, no?

Balarka Sen

11:41

The boundary map is most certainly zero.

i.e., the second map in your sequence.

@iwriteonbananas PS : it should be $0 \to H_2(X) \to \Bbb Z^2 \to \Bbb Z^2 \to \cdots$

Huy

@BalarkaSen: How do I see that $G(4,2) \cong S^2 \times S^2$?

iwriteonbananas

@BalarkaSen why is that?

and i think the $\Bbb{Z}$ is correct. the intersection is a circle, right?

Balarka Sen

@Huy You mean $G(2, 4)$?

@iwriteonbananas No, the cylinder intersects the torus minus disk on the longitudal circle and the boundary of the removed disk.

iwriteonbananas

true :/

Huy

@BalarkaSen: Yeah.

@BalarkaSen: I've seen both notation.

iwriteonbananas

11:47

mhh i guess the map $\Bbb{Z^2}\to \Bbb{Z}^2$ is an iso

Balarka Sen

it is :P

but that's not the way I did it.

iwriteonbananas

how did you do it?

Balarka Sen

using the geometric interpretation of snake maps.

iwriteonbananas

interesting, what is the geometric interpretation?

i assume you're saying snake map because of how the boundary map in the MV sequence was defined

Balarka Sen

@Huy I don't know.

Huy

11:52

@BalarkaSen: :(

Balarka Sen

Interesting, though. I didn't knew it was homeomorphic to that.

I'll think about it and get back to you if I find some visualization.

@iwriteonbananas OK, do you know about the geometric intepret. of the snake map in the homology LES?

iwriteonbananas

what is the snake map of the LES?

Balarka Sen

$\partial : H_n(X) \to H_{n-1}(A)$

iwriteonbananas

oh, ehh, im not sure. dont think so

Balarka Sen

I nicknamed it snake map because it comes from the snake lemma :P boundary map is confusing, because you also have those in the chain complex.

iwriteonbananas

11:55

fair enough

Balarka Sen

@iwriteonbananas take a relative $n$-cycle in $X$

iwriteonbananas

yea

Balarka Sen

by definition, it has it's boundary lying inside $C_{n-1}(A)$

iwriteonbananas

sure

Balarka Sen

as $C_\bullet$'s a chain complex, the boundary is not only a chain, but a cycle in $C_{n-1}(A)$

iwriteonbananas

11:57

yep

Balarka Sen

thus, define the map $\partial : H_n(X, A) \to H_{n-1}(A)$ by $\xi \mapsto \partial \xi$, i.e., grab the relative cycle and map it to the boundary.

that's all there is to it

iwriteonbananas

alright

Balarka Sen

this is an immensely powerful visualization. exercise : use this on the LES for $(\Sigma_g, A)$ where $A$ is one of the punctured torus in your connected sum to compute $H_\bullet(\Sigma_g)$

iwriteonbananas

hmm ok

Balarka Sen

anyway, a similar interpretation works with M-V LES. you take a cycle in $H_n(X)$, split it up into a chain in $A$ and a chain in $B$, and then define $H_n(X) \to H_{n-1}(A) \oplus H_{n-1}(B)$ by $\xi \to (\partial x, -\partial y)$ where $x + y = \xi$ is your splitting.

i gotta go.

iwriteonbananas

12:01

so i should compute $H_\bullet(\Sigma_g)$ w/ LES and the visualization you gave me?

Balarka Sen

yes

iwriteonbananas

ok

Balarka Sen

i remember being so excited when i learned about that interpretation that i almost fell off my chair.

:P

it's pretty cool, though, as you'll see when you do this exercise.

iwriteonbananas

it sure sounds nice, but i didnt fully grasp it yet...im super groggy right now lol. i'm gonna do the exercise now

@BalarkaSen was this a typo btw.? did you mean to write $\partial: H_n(X,A) \to H_{n-1}(A)$?

Balarka Sen

12:37

yeag, @iwriteonbananas

Ted Shifrin

14:38

@Huy: FYI, oriented $G(2,4)$ is diffeo to $S^2\times S^2$.

Balarka Sen

hi @Ted

Ted Shifrin

hi @Balarka

You healthy again?

Balarka Sen

loads better, yeah. the heatwaves have finally died out, and it's raining.

Ted Shifrin

good

Balarka Sen

@TedShifrin I don't know how to visualize that, though.

Ted Shifrin

14:45

It's linear algebra any way you do it.

One is to look at eigenvectors of the Hodge star operator on $\Lambda^2 \Bbb R^4$.

Balarka Sen

blank stares

Ted Shifrin

Another is to write down the Plücker embedding in $S^5$.

Balarka Sen

blanker stare

Ted Shifrin

I'll shut up now.

Australia

Does exist $\{a_{n}\}$,such $\lim_{n\to\infty}\dfrac{a_{1}+a_{2}+\cdots+a_{n}}{n}=0,\lim_{n\to\infty}(a_{n+1‌}-a_{n})=0$,but $\lim_{n\to\infty}a_{n}\neq 0$

Ted Shifrin

14:46

I'm just throwing out all my notes from math classes in college and grad school. Just got to my algebraic topology class.

Balarka Sen

Send me those notes, @Ted.

Ted Shifrin

@Australia Good question

They're not in sendable form, @Balarka.

Australia

Thank you

Balarka Sen

ahh. the shame.

Ted Shifrin

I would have to scan them, and the pencil might not scan well, @Balarka.

So, @Australia, if $\lim$ exists, it must be $0$.

Australia

14:48

Balarka Sen

nah, don't bother about it too much, @Ted. who cares about your notes if you yourself is available? :P

Ted Shifrin

Oh, so there is a counterexample, @Australia.

I don't know everything in those notes, @Balarka, far from it. John Stallings, a very famous topologist at Berkeley, taught a good course.

Australia

Now,I can't take it. can you post your counterexample

Ted Shifrin

No :) What's the easiest example of a series you know of where $\lim (a_{n+1}-a_n)=0$ but the series doesn't converge?

Balarka Sen

@TedShifrin oh well.

user3491648

14:53

math.stackexchange.com/questions/1326283/… - compelling enough for big list?

0

Q: counting the number of solutions to $x^2 + y^2 = 1$ over $\Bbb Z/p$

List proofs of the fact that the number of solutions to $x^2 + y^2 = 1$ over $\Bbb Z/p$ where $p$ is a prime $\neq 2$, is $p-(-1)^{\frac{p-1}2}$. I thought of two. I write one below.

abstract-algebra finite-fields big-list

Australia

such $a_{n}=1/n$

but $lim_{n\to \infty}a_{n}=0$

Balarka Sen

Hint : $a_n$ are the partial sums of a series you know, @Australia

Ted Shifrin

@Australia: Yes, so consider the series now, i.e., look at what Balarka said.

Oh yeah, what I said was misleading.

Antonio Vargas

I'm interested in the answer as well. Are you suggesting to take $a_n = H_n$, the harmonic number?

Balarka Sen

that doesn't work.

Antonio Vargas

15:05

right

Balarka Sen

Harmonic series isn't [big-word-I-don't-want-to-spell-out].

Antonio Vargas

let me put my telepathy hat on

Australia

$a_{n}=H_{n}$ not such fisrt condition

Ted Shifrin

$a_n = 1+\dfrac12+\dfrac13+\dots+\dfrac1n$.

Balarka Sen

No, it doesn't work.

:P

Ted Shifrin

15:08

Oh, rats, right, I confuzled myself.

Antonio Vargas

The numbers $a_n$, if they exist, must change sign infinitely often and can't be bounded away from $0$.

Ted Shifrin

I'm too busy doing too many things.

Balarka Sen

$a_n = (-1)^n$

Ted Shifrin

No, the second condition fails.

Balarka Sen

swears hard

oh, sorry, I meant to say $\sum_k (-1)^k$, but surely that doesn't work!

Antonio Vargas

15:10

It's hard to guess what kind of formula would give it, but my gut tells me it would be something like $a_n = \sin \sqrt{n}$

Balarka Sen

big-word-I-didn't-want-to-spell = Cesaro summable, btw

OK, the second condition messes up everything.

Antonio Vargas

@Australia, considering it's so hard to guess an example sequence $a_n$, try to prove the statement is false instead :)

Balarka Sen

Surely $a_k$ must be very oscillating.

Ted Shifrin

but, although not Cauchy, almost so, since $\lim a_{n+1}-a_n = 0$.

Balarka Sen

@AntonioVargas Are you sure $1/n\sum^n a_k \to 0$?

hi @SohamChowdhury

Soham Chowdhury

15:40

hey

project is a mess.

just got an emergency haircut, because I had to.

still ~15 pages of writing and a bunch of bar graphs to go.

:(

should go.

good night, B.

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$Mathematics$ Mathematics