@robjohn Your recovery proceeding apace? I meet with the neurosurgeon re my back in a month!

I meet with my lawyer in a week

Your lawyer?

@TedShifrin yes.

I'm bankrupt after selling all my hice remember

So you clearly cannot afford even the shoddiest lawyer.

@TedShifrin Don't worry if all else fails I have ChatGPT, it can be my lawyer. What could go wrong, right?

Perfect!.

Make sure to visit me in jail

@TedShifrin I am doing well. Hormone therapy is a pain, however.

Is this a checkup, or more surgery?

@KZ-Spectra it looks as if you have mismatched '$'s

@robjohn didn’t know about hormone therapy … hope it gets better. First back surgery :)

I hope it goes well and helps

Thanks. I’ll let you know when it’s imminent.

Is the operation local, or will you be traveling for the surgery?

@robjohn Looks good to me, but of course it's hard to be sure. ;)

@robjohn Nah, just UCSD.

I always think an arbitrarily long sequence consisting of the same number is random.

FWIW, Brian May is a stereo photographer. He has a nice stereo portrait on his site: brianmay.com He revived the London Stereoscopic Company, and developed a stereo viewer that works with old stereo cards & with phones. shop.londonstereo.com/OWL-B-ENV.html

If you ask ChatGPT for a random sequence of digits it makes the same mistake that most humans do: it avoids repeats & things that look like patterns.

On the old XKCD forum we did an experiment to find the most popular random integer between 10 and 20. We used the site's polling facilities, so people could only nominate one number and couldn't see anyone else's choice until the poll was closed.

"Of course", odd numbers are more random than evens, 11 repeats, 13 is the well-known unlucky number, 15 is in the middle of the range, so it's clearly not random, and 19 is the highest odd number in the range, which makes it special. That leaves 17 as the only truly random number in the range. :D

Sounds like a prime example to me.

This might be obvious, but in this answer (math.stackexchange.com/a/4696887/109355), what computation is used to get from the first line to the second line of the multivariate case?

@user726941 I See What You Did There. ;)

:D

That sir, is A Title Statement.

But some keyboards ⌨️ start every word with a capital.

I think my google home speaker does that.

Does there exists a holomorphic function $f:D \to D$ with $f\left(\frac{3}{4}\right)=\frac{3}{4}$ & $f'\left(\frac{2}{3}\right)=\frac{3}{4} $?

Schwartz-pick lemma isn't very helpful here.

@leslietownes Thanks 🙏

Does Rudin's PMA have Arsela Ascoli?

I like the exposition of Arzelà-Ascoli from those notes if you want another source

koro: you may need to say what you mean by 'arzela ascoli' to get a yes or no answer to that. different books give this name to differently phrased results about compactness or precompactness in C(X)

PMA's theorem 7.25 is "an" arzela-ascoli theorem

7.25 is generalization of the Arzela -Ascoli that I'm looking for.

Thanks.

Complete +totally bounded iff compact.

Closed subset of a complete metric space is complete.

if {f_n} is unif. bounded + equicontinuous then {f_n} has a uniformly cgt. subsequence.

Is there a word or phrase to describe functions that are not easy to integrate?

@user977780 source?

chapter 7 is one of the few places he actually uses his chapter 2's general theory of metric spaces, and not just the special case for R^n and functions defined on subsets of R^n :)

Pre compact in a complete metric space iff relatively compact.

in the linked answer by Leslie, 1) is not possible because by SL, $|f'(z)|\le 1$.

Arzela-Ascoli theorem :\mathcal {F} \subset \mathcal {C}(X) is relatively compact iff equicontinous and uniformly bdd.

2),3) and 4) look tricky.

Compactness in C(X) where (X, \tau) compact Hausdorff space.

@Koro Exactly , Schwartz lemma.

In my complex analysis exam, it was asked to prove that that a non polynomial entire function is not injective.

I left it.

and that an entire function is ratio of two entire functions... I could say that if f is an entire function then f= f/g, where g=1 is the constant function but I thought they were asking something else so I left that as well.

@Koro f(z) =z?

then there was a question about calculating some hyperbolic area given some sets. I had no idea so left that as well.

@user977780 it's a polynomial.

Ohh.Non polynomial entire function.

Then infinity is an essential singularity

Now take help from Casorati

why is infty essential singularity?

If it is, then f(1/z) will have 0 as essential singularity.

@Koro Exercise: An entire function with infinity not an essential singularity is a poly.

Since f entire, infinity is isolated singularity

By Casorati- Weierstrass, f(1/z) will map deleted unit disk to all of C.

Infinitely often

too difficult for me... I'll think about it some other time.

the semester is over now. Some people are leaving the course. Algebraic topology results were declared. A few got 90+ out of 100. Then some got around 50. Many failed the exam. Almost no one got between 55 and 90. I passed this exam.

Throughout the semester, I studied AT the most and I barely passed it.

some didn't even write the exam thinking that it would be too difficult.

College/Ph.D are not for me. I belong to industry. I should stay in industry. I have clarity now.

that's one good thing that came out of this semester.

I got almost full marks in complex analysis midsem (almost because the teacher had given me 0 in a question which was correct. He said he'll fix that. But I don't know if he'll do that. Considering that full marks.) I expected my complex analysis endsem exam to be my best exam but it also went bad.

1 year is complete, now 1 more year left.

@Koro This a valid answer without further assumption on the hypothesis.

It'll be an amazing experience. I am happy that I atleast tried something new. from engineering to studying maths here :-).

@user977780 I didn't write it in the exam because it would be too easy.

:(

And my algebra exam also went very very bad.

A group of order p^n is isomorphic to a subgroup of upper triangular matrices in GL(F_{p^n}). The question was something like this.

I had absolutely no idea how to even start writing anything to that.

I also believe that if one has never seen it before, they can NOT prove it.

(think of it as algebraic brother of: If f:[0,\infty)-->R is continuous, f(nx) tends to 0 as n tends to infty, then f(x) tends to 0 as x tends to infty.)

I came here for doing Ph.D also but I have now decided to not do it. I feel like I became dumber after coming here. I DIDN'T learn anything at this college from any teacher here. I self-studied/took help from people here/referred to lot of online videos. That's how I learn. But here, classes disturbed me a lot, exams disturbed me a lot (in the sense, they wasted my time.)

Such disturbance was not there when I was in industry. I would come home from office and do maths. It was so fun.

The teachers at this college didn't aid me in my learning. Their presence is same as their absence to me. So no point of doing Ph.D here.

@Koro "I DIDN'T learn anything at this college from any teacher" 🐼

this is true.

You might think of the teachers here as 'one the best in the country' or so. I don't. To me, they are the worst I have ever seen.

Teachers are IITs are excellent. I miss some of them. They did their job amazingly and honestly. They encouraged students to ask them questions.

@Koro they definitely meant to ask that a meromorphic function on the complex plane is a ratio of two entire functions, because otherwise their question is trivial as you point out

One of my teachers there came to his office at around 9 pm to help me with some concepts.

Such dedication!

I'll never forget him my entire life.

on the contrary here, I don't even bother knowing names of the teachers here. They are all same to me -garbage/junk.

@porridgemathematics I'm sorry. You are right.

You see. I was thinking about so many things while writing those comments earlier so I made mistake in writing that question.

btw the gripes you're describing are honestly not uncommon in other countries too, its not totally surprising that when math becomes more of your 'job' than a hobby, it becomes 'harder' to do and you feel 'dumber'

i think this is just the case with virtually everything

@porridgemathematics I totally agree with the last part: when it becomes a 'job', it is different. You're absolutely correct.

I agree.

gotta just find ways to feel reinspired from time to time I think

easier said than done obviously

yeah. That's why I have decided to not do Ph.D.

thats fair

I don't like to keep crying here: Hatcher is too difficult. How to do exercise 2.1.1 etc.

I am 100% sure. If I do Hatcher while I'm in industry it would feel different.

It would feel better and I would actually learn :-).

Outside college, I would love even algebra.

2.1.1 isnt too bad

you can do it by just drawing

that was just for reference.

actually frankly most of his exercises can be done that way\

you just need to figure out what is permissible to do via cutting and pasting and most things are doable

I didn't mean to specifically point to that exercise.

yeah fair enough

honestly I think lees book on topological manifolds is kinda nicer than hatcher

maybe even for a first course in AT

its not about algebraic topology only of course

but since hatcher actually really basically only cares about things that are manifold like

its worth just reading that instead

wow, three modifiers.. actually, really, basically, ive outdone myself

Were you an industry person before? @Koro

But Hatcher is deep. I love his book actually. It's just sometimes being here, I say things about him.

lots of books are deep though

depends how deeply you read them too lol

like the first chapter now makes sense to me.

plus you have more examples at your disposable after you've seen something like the classification of surfaces, before reading hatchers

including hatchers use of appendix listed results @Koro ?

I am taking topology again in my next sem.

:-)

Let him give me 0.

hatcher is a good book, i would just suggest reading it alongside another one

using it as a sole textbook imo doesnt yield the best results unless you've done a course in algebraic topology already

@porridgemathematics yeah. I am sure I'll agree more with this outside college.

Pierre Albin's lecture, I found them very helpful.

A great companion to Hatcher's book.

my gripe with him is how a lot of his appendix results can be more useful as tools than the main non appendix results, lol

but are just shoved into the appendix for some reason

(the first part- ch 1 and 2)

(Hatcher)

@porridgemathematics haha, like the topology of cw complexes.

yeah...

thats a BIG one

its what his whole friggen book is about

you would think that can be moved to Ch1

or Ch0

In my text sem. topology, the syllabus would be like ch 3 and 4 of Hatcher's.

so cohomology and higher homotopy groups.

And I actually didn't understand the calculation of 'boundary maps' in cellular homology.

I tried to know the answer to this via this question of mine.

But I didn't get the answer to my question.

@SoumikMukherjee yes

Teachers during my undergrad were so amazing. They met students regularly and would ask: are you facing any difficulties? etc. They ensured that everyone feels like home there. I miss them.

They even shared their contact nos. with students.

They cared for their students.

@PM2Ring There are primality tests much faster than trial division , but they give no nontrivial factor. PARI/GP or PFGW are tools to test numbers , one can also use factordb , if the numbers are small enough.

@Peter I used Sage, which often uses PARI for number theory stuff. However, I used my own modulus Fibonacci calculator (based on matrix squaring) to avoid dealing with huge numbers, but I later realised that my algorithm still used some big numbers.

I know Miller-Rabin, and often use a deterministic form of it, but for numbers that large I'd need to use a probabilistic test, eg probabilistic Miller-Rabin plus (maybe) Lucas-Selfridge. I was originally only going to test primes up to a million, but it was pretty fast, so I extended the test to 20 million.

@DLeftAdjointtoU hm, but what work is it doing, logically speaking?

since we're already assuming that a homomorphism exists, and then we're citing (1) prime ideal iff quotient ring integral theorem, and (2) the first isomorphism theorem, what role is that function playing, as opposed to just using the unexplicited homomorphism we're assuming to exist?

I want to solve using Laplace transform.

L(\phi) =\frac{s^2-2s-8}{3s^2+6s-8}

Ignore.

@user858770 maybe not, i have been traveling a bit, so waiting for an answer from me would be non optimal :-)

Welcome back @copper.hat

@copper Anywhere fun?

what do you think about x^9=u?

have you tried playing around with values for n?

When evaluating the exact value is difficult, try to approximate.

Try to find lower bound, upper bound, if possible use squeeze lemma etc.

Where the function increases , where decreases etc.

@user858770 THanks!

@TedShifrin my daughter's graduation at Oxford (same day as coronation) and then Ireland to see family & my Godmother for a few days. A bit busy, but enjoyable.

Had some great Indian, Greek & Bangladeshi food in Oxford. Very cute town (coming from a guy that doesn't really do cute unless it wears inappropriate clothing).

Indeed, Indian foods are great.

$$24\sqrt2-4-3\int_1^3 \sqrt{xf'(x)}dx$$

Is it possible to determine the integral?

Context: Trying to solve this:

@copper.hat Seems like just yesterday she started. Next you’ll be going to your son’s in a few months …,

@Wolgwang Looks suspiciously undoable. Are you sure?

You can try integration by parts — oh, that’s how you got to the first line.

What is the difference between the "Conditioned version" and the "Bayes' rules with 3 events" here (en.wikipedia.org/wiki/Bayes%27_theorem#Generalizations)? Is there a difference between the intersection symbol and the comma? I thought they were the same in probabilities.

I agree. The trouble with many authors and no one in charge.

Does Bayes' Theorem become non-unique in higher dimensions, based on all the different ways the multiplication rule can be applied to an intersection of many events, or is there a correct way to navigate that derivation?

Пожар в левом двигателе

@冥王Hades problem with your port engine?

@user10478 That doesn’t make the theorem “non-unique.”

Hi @shintuku

what's up koro

Do you know LPP?

what's lpp

linear programming

none

I want to learn it so I want to know some references on this.

why are there two 'p's?

it's programming in linear programming

not to be confused with lppp

that's one 'p'

no, programming in linear programming is lpp

linear programming is lp

What's new?

It's been a bit.

@BalarkaSen 0 or 1?

where can I find sums of ultrafilters on $\mathbb{N}$ i.e. $\sum_y \{x_n\} = \{\bigcup_{x\in Y} M_n : Y\in y, M_n\in x_n\}$?

@robjohn $$\frac{1}{\sqrt{2}} |0\rangle + \frac{1}{\sqrt{2}} |1\rangle.$$

As the cool kids say nowadays

(in other words, a bit of both, I suppose)

it's been a q-bit

I found some in the book Ultrapower axiom

Please suggest me a book to learn differential geometry.

What kind? Riemannian geometry, or curves and surfaces?

curves and surfaces

hi @BalarkaSen

alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

for a beginner

That's a very beginner-friendly book

thanks.

Balarka ! Long time no see!

Hi @Ted!

How are you doing?

Writing my Master's thesis this semester as part of my graduate coursework. I will formally join a PhD afterwards.

Currently at home for a holiday, thinking about some research and editing a paper which got accepted in IMRN

@Jakobian what do you mean? Are you asking for a reference?

@BalarkaSen congrats!

@BalarkaSen What are you writing?

@TedShifrin I am writing an expository of sorts for this paper. It's actually pretty interesting, I can send you the final draft once I'm done.

Roughly, it is about the following: Let $(M^{2n+1}, \xi)$ be a contact manifold. Call an embedding $f : L^n \to M$ as Legendrian if $df(TL) \subset \xi$. What is the homotopy type of the space of such Legendrian embeddings from $L$ to $M$?

In particular, are there some nice algebraic topology obstructions which, once they vanish, say if two such embeddings $f, g : L \to M$ are isotopic through Legendrian embeddings?

This is famously difficult for $2n+1 = 3$, where this boils down to Legendrian knot theory. For $2n+1 \geq 5$, Murphy's work provides an answer for a large class of Legendrian embeddings called "loose".

@AlessandroCodenotti Thanks.

I'm a bit tired of editing though.

Still h-principle!

Yeah :P

Good to get practice at clear writing!

Agree, good exposition is hard. But I do find expository work a little bit more interesting than research.

I enjoyed joint research much more. Most of my papers are with a coauthor or two.

It's much more fun

I nudge my colleagues here and there to work on something with me. They're interested, but busy getting their own research done, which is very different from what I do.

Not enough people in India who do topology, sadly.

Oh, but on a hopeful note, the group who wrote the text on Freedman's disk embedding theorem is coming to my uni on Sept, @AlessandroCodenotti.

Some workshop on low-dim topology.

Plenty around the world, though. You should meet some people at conferences and do math by zoom :)

My advisor gives me a lot of flack regarding that, because I'm very lazy regarding going to conferences.

Hmm, too many "regarding"s in a sentence.

If only our research interests weren't disjoint!

@BalarkaSen oh that's cool

I will side with your adviser!

You have to be pro-active and get to meet good people.

Regardless and fond regards.

:)

BTW, Balarka, did you give a link to the current version of my notes? With alpha in there, not.

That is very old and a server we no longer have contact with.

Ah, I wasn't aware. It's linked in your MSE page, I think.

Should have gone there.

@Koro Here's the right version: math.franklin.uga.edu/sites/default/files/users/user317/…

Right. It’s also on the AMS Open Notes site.

Oh, wonderful.

My advisor also strongly encouraged me to go to conferences, but she also warned me that while they're fun I shouldn't go to too many and make sure I also have time do maths every now and then

But connections are important, and contacts with experts and potential collaborators are very helpful.

@Koro I am not a fan of Hatcher's book. It is a bit too talkative, and not very conceptual.

@robjohn No its a warning I was getting in the game DCS while flying a Russian Sukhoi aircraft

Turns out it means "Fire in the left engine". No wonder the jet crashed

@冥王Hades When will your funeral be scheduled?

@冥王Hades that’s why I asked about the port engine.