I found complex analysis nice but definitely not easy to grasp.

Much of it is just plain magic

@tb The edit? Surely a minor edit within 2.5 hours of the OP is within reason :)

Ah, the good old holomorphic functional calculus.

@robjohn Maybe Victor will get on soon so you can have your 500th

@JonasTeuwen one of my all-time favorite answers... even if I didn't contribute much

Would anyone care to consider this as a duplicate?

@TheChaz I cover my head with ashes and hang my head in shame.

@TheChaz that would be great. Wildberger certainly has a few more exercises in his videos.

@TheChaz Victor? I am working on an answer, but only because I know the answer, not because it is a really good question.

@tb I'm more into hard estimates 8-).

@tb Sackcloth. You're missing the sackcloth.

These students just skip the estimates and replace them with wordy thingies, which makes me unhappy 8-).

@tb Wildberger?

@TheChaz Wildberger. The prophet of rational trigonometry.

Oooh. I was getting lost in his personal webpage.

Why does a math Youtube movie have a sensual commercial about online dating?!

For obvious reasons :)

Oh, rational trigonometry. I remember hearing about that. Did it ever catch on with... anyone?

i saw it mostly dismissed

It's a nice idea. It would be taken more seriously if NJW wouldn't have this ... er ... 'tude

@TheChaz speaking of edits

You are too generous!

I didn't star that!

I only did the typesetting which was a complete mess (copy paste job from a LaTeX document)

Kunjan shah is really, really clever

@TheChaz drat! GEdgar posted almost the same answer, but earlier and got accepted.

@tb what's his "tude?

I was looking at his videos and in one he makes this comment: "The notion of metric space in modern mathematics is over-rated, as well as being improperly defined."

and apparently in another video makes an under-handed remark about the real numbers

the audacity!

@DavidWheeler irrationals are bad, mkay?

(what Tyler said)

well, there's a certain inherent tension between finitary methods, and non-finitary ones

Improperly defined? What?

oh yeah, I remember the comments about the reals. he says they are suspect.

He objects to the notion of infinite sets, at least as evidenced in Why infinite sets don't exist.

i'm no expert, but it seems to me there's whole swathes of mathematics where you have "fuzzy" notions (as in topology) replaced by more rigid ones (various homotopy and homology groups)

@tb "tude=attitude?

yes

@JonasTeuwen, I don't know, he didn't elaborate in his comment and I wasn't about to watch a bunch of his videos to find out :)

and the real numbers ARE suspect.

@JasperLoy It's all the same--I will visit the doctor the day after.

infinity is like a pandora's box, you can't just bite off a little bit of it...it keeps getting "out-of-hand"

well, i mean i think about this a lot....the real number system has been undeniably useful in the kinds of problems it lets us solve...but that doesn't mean the views that underpin it are correct

@TheChaz bye

i'm not trying to be pedantic, here. i'm not, in general, a firm adherent of "absolute truth" of ANY kind...i see things as "contextually true"

the views that underpin it? it's funny when mathematical constructs are relegated to opinions.

@robjohn: Q is a point in the interior of the triangle. The points R and S are defined to be the orthogonal projections of Q to two of the sides.

any idea is something that has a life of its own...in a sense, everything we've ever done (by "we" i mean the human race) is limited by our "human" point of view

@tb where is this? Is this the problem I just commented on?

1 more to go--then I should sleep.

mathematics has many useful models....i think it's important to recognize they are just models...the "ultimate nature" of reality is and most likely will remain, unknown to us

@robjohn yes. I had a tab open and saw your comment appear.

(that's a welcome new feature)

@tb how did you get what R and S are?

I drew a diagram :)

I just refreshed and don't see that.

if wildberger has reservations about certain kinds of infinities, he's neither right, nor wrong, but what is unarguable is that he is imposing some self-limits on the kinds of mathematics he's willing to do

@DavidWheeler My issue with Wildberger is that, in the little I've seen, he is not clearly distinguishing philosophical concerns (completed infinite? what?) with pragmatic concerns (what happens if we use the real numbers to model things in the world?).

@robjohn oh, just a crude one for myself.

We have had a victim...

@tb Using Geogebra?

yes.

:)

my take on him is that he is espousing a philosophical position while refusing to call it a philosophical position

Modes of convergence of RV seems to be an interesting area of math.

@KannappanSampath Yes, there are entire books on that. This is an excellent one.

@tb Oh, I see. Thank you for that. Is it still active area of research?

no one has ever convinced me that the axiom of infinity is justifiable. that hasn't stopped me from using induction as a method of proof...but it has made me aware that there is a certain impredicativity in our thinking about "totality"

Will someone help me with a duplicate for this? I have myself written answers that concern this...

But just could not find one now.

@KannappanSampath arturo will find one.

for example, proofs by contradiction are ubiquitous. but how large a set ("collection") really, is our bag of "known facts"?

@tb hah. Yes, he has not yet come to the town for today, I guess.

@DavidWheeler I like it - that's actually a very concise summary.

@David You have managed to put me to sleep--I have one more problem to go to complete the assignment--Please do this to me after that. :P

@KannappanSampath I don't know. I'm not a probabilist.

we can't PROVE that ZF set theory is contradiction-free...we suspect it is, and have re-cast large amounts of mathematics in its terms....perhaps one day this will seem like a bad idea

(MattN failed in putting me to sleep once.)

@tb OK. Sorry about that. I should also ask less CA or less of anything concerning algebra.

@KannappanSampath first and foremost you should stop apologizing :)

i'm happy to accept that IF it's "true" (or at least "consistent"), then so are a lot of other things...but i'm not willing to say much more than that

@tb But, certain things come to me as if I have put someone in an odd position. Sure, I will if it does not come that way to people.

@KannappanSampath oh, there's nothing wrong with asking questions to which I don't know the answer. That's much easier than asking questions to which I do...

Oh, well. :)

what i see happen a lot, is that many mathematicians have a lot invested in their chosen field of study...they tend to get defensive about its value

@DavidWheeler Harmonic analysis and mathematics in general are completely futile.

i think it was dedekind who first came up with a truly nifty workable definition of infinity...but as i understand it, it's kind of a deep question as to whether or not his definition is the "right" one

> Occasionally a proof that there are no measurable cardinals is announced; but the last real fright was in 1976, and most of these arguments have easily been shown not to reach the claimed conclusion. My best guess is that measurable cardinals are safe. But even if I am wrong, and they are irreconcilable with ZFC as now formulated, it does not follow that ZFC will be kept and measurable cardinals discarded.

> It could equally happen that one of the axioms of ZF will be modified; or, at least, that a modified form will become a recognized option. This is a partisan view from somebody who has a substantial investment to protect. But if you wish to prove me wrong, I do not see how you can do so without giving part of your own life to the topic.

@DavidWheeler It's just infuriating that instead of delineating the known drawbacks and advantages of stepping onto strange philosophical ground, Wildberger will occasionally characterize these things as "voodoo and distortions of mathematical logic."

(this is from Fremlin's book on measure theory.)

BTW, How do you get to write it like that? @tb

haha

well, my guess is that he has his cherished ideas that he is unwilling to abandon...but so have many mathematicians

(With a line that pushes the content from the margin?)

K

@KannappanSampath It's a quote block. Add a > in front of the paragraph.

No, :/

>Now?

> like this

> like this

> Like this?

Hah! Yes.

not surprisingly, some of the deepest questions about measurablility lie in "how many sets we want to be able to have"

@lewist yes, he has this tendency of denigrating the work of others and overemphasizing the significance of his own gospel

Why debate about burgers? Ahem, bergers? We have got better things to do, no? :)

@KannappanSampath I swear, I'm almost done ;)

There was also this thread

(I see that André said about what I just said. I didn't parrot him consciously...)

well, i know finitism (and especially ultrafinitism) is out of fashion these days, but it seems unfair that they are given such short shrift.

Are Doron Zeilberger and Wildberger related?

Like the Conrad brothers...

as someone explained to me once: say we take the group Z4. it is a model of the group axioms, in that it satisfies all of them. there's lots of groups, they're not hard to find.

And dangit, David already said everything I would have said, but better. Wildberger's having a philosophical argument, but without calling it one, it feels disingenuous.

what would be the thing that satisfies all of the ZF axioms?

@DavidWheeler I guess most of us wouldn't admit it but we like to hide in the pack and follow the alpha wolves.

as i understand it, the best candidate is V, the class of all sets...which just has one small problem...it's, um, not a set

and even worse, if we did have a model for ZF, then there's a countable model for it...which i have a hard time reconciling with cantor's diagonal argument

people say: well, first-order logic is broken, so we'll just fix it with second-order logic

but then...well, i get confused: we state the axioms for the building blocks of math in some sort of "background system" like a formal language.

and so i look at formal languages, and they use "sets" in their definition

is that kosher?

Are people here are aware of a rumour(?) that there are atmost three counter examples to FLT?

(And that Annals is going to publish a short note on them, like a corrigenda, or whatever...?)

@DavidWheeler well, if I understand correctly, usually formal languages have finite words only, so they are small enough for sets

@KannappanSampath sounds like a belated April's fools joke.

@tb Is Doron into this(April-fool Pranks, I mean) as well then?

i agree that we could come up with a satisfactory concept of some finite thing

but calling it a set when we are using the apparatus defined by a set to define what a set is...that seems problemmatic

What the....

@KannappanSampath note the date of some of his most hilarious posts :)

Yes, you're right!!!!

An email attachment there says it all!!!

Why does he do this?

Fail.

it seems to me some people are still chasing hilbert's dream....i think mathematics should be (and is in practice) more robust than that. as much as i might admire some platonists for their idyllic view of things, i think there's mysteries we will never solve

@KannappanSampath One > per paragraph

> "In accordance with April Fool's Day tradition, all editors, referees, and their reports mentioned are entirely fictitious. Any resemblance to actual editors, referees, and their reports, living or dead, valid or inexcusably ignorant, is purely coincidental."

Yay, I get it right. Thank you @tb

i should look more into topoi....but i will never know enough about them to have an informed opinion...they do sound promising, though

@KannappanSampath this is hilarious, especially after his stab in the back of Hyman Bass...

Interesting Profile...

How do I see that $GL_n(k) / SL_n(k) = k^\ast$?

@MattN det

det is a homomorphism

I don't see.

what is ker(det)?

use the first isomorphism theorem

$\det: \operatorname{GL} \to k^\ast$ is an onto homomorphism and what David said.

@tb Ooh!! Obvious. Thank you : )

it's clearly onto since $a \neq 0$ has the pre-image diag(a,1,1...,1)

i forgot you get a factor for each row, duh!

bbl

reading some of Wildburger on his web-site...he really has an axe to grind, lol

he's not the only one...

Limit superior trouble me. Here is what I am facing trouble with:

$P(A_n \le b) \le P(B_n \le B+\varepsilon)+\delta$, holds for $n \ge N(\delta)$

Anyone disagrees that this is not an answer?

I flagged that as not an answer yesterday. Lazy mods!

@tb Flagged.

it's a comment, not an answer...off with his head!

same here.

@KannappanSampath What are $A_n$, $B_n$, $b, B$?

Now, how does this tell me that $\varlimsup_{n \to \infty}P(A_n \le b) \le \varliminf_{n \to \infty} P(B_n \le B+\varepsilon)$

seriously, find that guy, track down his address and defenestrate him

@KannappanSampath If $a_n \leq b_n$ for all $n$ then $\limsup a_n \leq \liminf b_n$. What is causing trouble?

@tb $b$ is a continuity point of the distribution function of $B$.

And $B_n \to B$ in distribution.

hi everybody

hi leo

@tb What is happening to delta? I fail to understand this stuff

can you prove: if $0 \leq c_n \leq \delta$ then $0 \leq \liminf c_n \leq \delta$?

Let me try: $\liminf \le \lim$ gives me that $\liminf \le \lim \le \delta$.

but you don't have a lim!

You mean to say, we don't know if $\lim c_n$ exists?

yes.

well, $c_n \in [0,\delta]$.

what is the definition of liminf?

I really need to review it, I am drawing blank, already.

No, it's $\sup_{k} \inf_{n\geq k} c_n$.

Right, I have it:

and for all $k$ you have $i_k = \inf_{n\geq k} c_n \in [0,\delta]$.

and the sup does the same.

I had from glancing wiki: $\liminf c_n=\lim_n \inf_{m \ge n} a_m$

Yes. The limit exists because the inf increases as $n$ increases

So we agree that if $c_n \in [0,\delta]$ then $\liminf c_n \in [0,\delta]$?

@tb Yes, this fact I recall: if $A \subseteq B$, we have that $\inf A \ge \inf B$

@tb Yes, we agree.

Now suppose that for all $\delta$ there exists $N(\delta)$ such that $0 \leq c_n \leq \delta$ for $n \geq N(\delta)$.

Then $0 \leq \liminf c_n \leq \delta$ for all $\delta$, so the liminf is actually zero, right?

Right. True.

Now. $c_n = a_n + (-b_n)$...

what does this tell you about the liminf?

$\liminf(a_n-b_n)=0 \implies \liminf a_n-\limsup b_n\le0$

yes, now I messed up $a_n$ and $b_n$. It should have been $b_n - a_n$. But then you're there.

Alright. I have never understood liminf and limsup. I don't know what should I do about that.

oh, have I messed up? Maybe I made this look too simple. Anyway it's no deeper than such a move.

I used the inequality: $\liminf(a_n+b_n) \ge \liminf a_n +\liminf b_n$

Is that right?

Yes, that's right.

but somehow I messed up the direction of the inequality.

May be , I should write all of that out and I probably will understand it better.

@tb Firstly my question had no $a_n$ and $b_n$. So, all we did was illustrative of what I should do. So, never mind about that.

Thank you very much for helping me with that.

(illustrative upto replacing $x$ and $y$ by $a$ and $b$...:))

Probably you should pick a subsequence such that $\lim b_{n_k} \to \liminf b_n$, take it on the other side and work with limsup. But I'm getting too tired to work this out right now.

Oh, I get it now. We wanted to have $\limsup a_n$ but ended up the other way round.

yes.

@tb No Problem. Thank you for that hint. I'll work through that now and let you know...

good luck!

But, firstly $0 \le c_n \le \delta \implies 0 \le \limsup c_n \le \delta$, right... The same argument works with $\inf$ and $\sup$ the other way around.

but then the inequality goes the wrong way around.

(the limsup gets bigger if you tear it apart)

Right, :/ Thinking of your technique now...

Right, let me strip all the details I don't need: I have $$a_n \le b_n +\delta$$ for all $n \ge N(\delta)$ and for all $\delta \gt 0$

Need to prove: $\limsup a_n \le \liminf b_n$

That is I need to have: $\limsup (a_n-b_n) \le 0$ or $\liminf (b_n -a_n) \ge 0$.

it is inappropriate to put a link to my own answer in order to get some reviews?

@leo into another answer of yours or as a comment or in a question, yes, it is appropriate to put in a link.

@KannappanSampath and in the chat?

@leo I have put it in a few times and I'd think it is appropriate.

I'm about to start a meta thread

about my doing that?

I have put it too, some with good results

@KannappanSampath no no

about advertise answers in the chat

own answers

That really is not advertisement--I mean you want them reviewed that's all right?

yep

I feel that I umcomfortable other people if I do that

I mean, I feel uncomfortable to others

if I do that

@JonasTeuwen are you there?

@leo I very much am.

@JonasTeuwen You are in analysis, right?

@leo I surely want to believe that I am in analysis, correct.

8-)

@JonasTeuwen :-)

@JonasTeuwen would you please see here and tell me if is there any grave sin

@leo I am grading exams! 8-).

I see

grading exams, no wonder you were concerned with your students work :p

@JonasTeuwen thanks any way (-:

@TomWijsman, pretty nice array of black and white squares :-)

Thanks @leo.

are they gliders?

Don't know exactly, think so.

Nice ^

it's definitely something, in any case!

haha

that's like the Conway's Game of Life

Yeah, it is from there. :)

can be the Cantor set obtained from a 1 dimensional version of The Game Of Life?

no

I am an illiterate when it comes to this game, well, this among several others... @leo Can you suggest some good mathematical article about this game?

(this=conway's game of life.)

related links: pentadecathlon.com/lifenews/engineered_objects

conwaylife.com

Well, I wanted something mathematical--I mean Wiki is too chatty. But, never mind, thank you @leo

@TylerBailey, some mathematical literature on Conway's Game of Life that you ca suggest

?

nope, sorry D:

I liked to watch the patterns when I was younger, that's about the extent of my knowledge, unfortunately

and this catb.org/hacker-emblem

I know that from a course of programming

apparently Riemann had an oversight