@DanielFischer: Is "Completely continuous" often used instead of "compact"?

@MikeMiller encyclopediaofmath.org/index.php/Completely-continuous_operator

Kaaaaaaaaaaaaaaaaaaaaaaaaaaaaarl.

@Pedro: See the last line of the first paragraph. I'm aware of the history. I was asking about its prevalence.

@MikeMiller Ah, OK.

I would presume the use was dropped and "compact operator" is used now.

@MikeMiller Not in my experience. Sometimes, but not often.

Just sort of bizarre, since the book I'm using is not old. It's on a Hilbert space, so it's the same thing, but all the author actually cares about is that it's a compact operator.

@PedroTamaroff What'd I do this time?

@KarlKronenfeld Mud stains in the hallway.

@KarlKronenfeld Wanna read some homological algebra stuff?

@PedroTamaroff That's just the tip of the iceberg :D (mudstains)

@MikeMiller Well, people also say "bounded operator" when what they're interested in are continuous operators. Often, they use the terminology they learned when they were young.

Yes, I cannot get into the grimmy details. This chat is PG-13.

@PedroTamaroff sure

@DanielFischer Fair enough.

Next post should be about Künneth formulas and Koszul resolutions, primarily in polynomial rings and perhaps a bit about quadratic algebras.

ah

Silly question, since I never learned Fourier theory: $\hat f$ is the Fourier transform of an $L^2$ function. Why does the inverse transform of $\hat f$ exist iff $f$ is continuous?

Perhaps something about gibb's phenomenon @mikem

@MikeMiller ? It doesn't. You have problems with pointwise formulae without continuity, but ...

@DanielFischer That's what I thought. I must be misreading here.

@MikeMiller The Fourier(-Plancherel) transform is an isometric isomorphism of $L^2$. When $f \in L^1\cap L^2$, you can write it as an integral directly. When $f \in L^2$ is continuous [hah-hah, elements of $L^2$ are not functions] and $\hat{f} \in L^1$, then you have the inversion formula as a pointwise equality.

OK, they're invoking that, then. The claim is that if the integral exists, $f$ is continuous. [Sure they are, they're just lots of functions.]

Oh, this is clear. If I write down the integral formula I get something that's obviously continuous, and one can show that if this defines a function, it agrees with $f$ a.e.

Thanks.

@MikeMiller Obviously continuous if the thing is in $L^1$.

@DanielFischer: It's not defined otherwise.

So fair enough, I suppose.

@MikeMiller Well, we can use sloppy notation and mean something other than a literal Lebesgue integral.

Right, but that's not what I meant :)

Watch out, you might be joining our pedant ranks ;)

@NormalHuman higher algebra useful in quantum theory?

Interesting, @MikeMiller. Thanks.

@Moses Yes. But it's a slash, $X/F$.

@Chris'ssistheartist nothing comes to mind immediately.

@pedro Differential Graded Algebras: Is $\partial\circ\partial=0$ a consequence of the condition on $\partial(ab)$ or is it assumed?

you need to assume it. it doesn't follow from the Leibniz rule

thanks

@pedro I've lost track of your notation by the paragraph starting "extending scalars we obtain $K'$.."

What's $A$ supposed to be?

Oh, $B/(f)$?

@robjohn OK

@pedro Alright, done. Good post. It looks like the Kunneth Theorems are the key ingredient here, so I will be interested in reading your next post.

Are primitive integrals supposed to be antiderivatives without the constants of integration?

From Paul Nahin's book on integrals

The question from the beginning is pretty cute. I just noticed that there is a very very nice way to do that that randomly I found in my work.

It's a pretty condescending introduction.

Who in their right mind would start a book like that?

@Khallil I would do it.

It's as though the author believes he's better than you if you don't find a question like that interesting.

I don't get that vibe at all

the author is acutely aware that arcane interests of his may not interest many others

I don't feel it diminishing readers who aren't interested

It's all I can see from the final sentence.

do you feel mystery novels, historical biographies or vegetarian cookbooks are the stuff of pejoratives and condescension?

I don't see them that way

@anon Show me a passionate of integrals (a true one) that says the questions posted by the author is not interesting. Calculating integrals don't turn you into a passionate in integrals, neither in an expert. In case you don't know, this question, in its generalized form, is related to some very nice and interesting infinite series.

As opposed to putting the book down for another book that addresses the same stuff but in a different way, the author suggests that we pick up a book that has almost nothing to do with what the book we chose to read concerns.

@Chris'ssistheartist passionate is a noun, not an adjective. in any case, I don't understand why you're directing this question at me.

@anon Yes, it's a noun, and I used it as a noun.

'A passionate of integrals' makes no sense.

interesting, it is a noun!

anyway, why the question at me? I might as well direct it at you for all its relevance to me.

It is?

I thought passionate was an adjective ...

so says wiktionary

It's definitely an adjective.

heh, oops

need the wiktionary link instead

@anon You don't intimidate me with my knowledge of English that is not very good, and I never pretended that.

@anon You can give me instead a nice infinite series related to the problem Paul Nahin posed in the excerpt above.

If you cannot, I can give you one (then you have the opportunity to understand his point, why is important to be interested in such questions).

@Chris'ssistheartist Certainly not, you knew of a cool word usage that I didn't, and I am thankful. What's your point? Why are you insinuating I'm trying to intimidate you?

Suppose that $\displaystyle x+\frac{1}{x}=\sqrt{\sqrt{\sqrt{3}+2}+2}$, then calculate $$\lim_{n\to\infty} \frac{\displaystyle \sum_{k=1}^n \left(x^k +\frac{1}{x^k}\right)}{\displaystyle \cos\left(\frac{\pi(n+1)}{48}\sin\left(\frac{n\pi}{48}\right)\right)}$$

heh, no idea

@Khallil you see now why it's important to know to answer that kind of question? You might meet anytime such a limit. Paul didn't put that at random. You should learn more stuff about integrals, series and limits before saying anything about his book. Or you can publish a better book.

@Chris'ssistheartist did Khallil ever question the mathematical usefulness of the example?

Let's not add fuel to the fire, @anon. ^_^

@anon He didn't say it directly.

It's a cool limit, @Chris'ssis.

Does it have any nice applications/has it arisen naturally from the study of anything interesting?

@Khallil: I agree, it's a rather condescending introduction. But from what I hear birds and feather flock together?

It's actually $$\lim_{n\to\infty} \frac{\displaystyle \sum_{k=1}^n \left(x^k +\frac{1}{x^k}\right)}{\displaystyle \cos\left(\frac{\pi(n+1)}{48}\right)\sin\left(\frac{n\pi}{48}\right)}$$

@Chris'ssistheartist the sine inside the cosine did look very concerning

now my first idea would be geo sum formula

@anon I wrote things in a hurry and let that inside.

Could you use complex numbers to simplify the limit?

It reminds me of De Moivre's theorem to do with complex exponentials, gathering terms and expressing them as sines/cosines.

(I guess that was the main purpose of the intro, @Huy. The author probably thinks that if you share the same sentiments toward such questions, you're probably better suited to his/her book than those who don't.)

@Khallil if someone asks themselves "of what earthly significance is this question?" then they might not have any incentive to "put the book down for another that addresses the same stuff but in a different way"

I am amenable to the interpretation that the author is condescending, but that is not my impression. Just because Chris's sis likes the intro doesn't mean the author is writing in the same spirit that Chris frequently does.

This guy simply made a very nice introduction to his book, and the point he wanted to emphasize is the curiosity for the mathematical stuff first of all. Why to criticize a book so fast without even reading it entirely? It's the author's style of writing the book, and I see absolutely nothing wrong with that. It's an amazing book.

@anon: Maybe not in the same spirit, maybe jokingly, but a bit over the top for my personal taste.

@Chris'ssistheartist: First impressions are often the most important ones, and after all you were the person who posted an excerpt of the introduction.

@anon I'd say, "good mystery novel", fine, "the latest Lincoln biography (there seems to be a new one every year - what could possibly be left unsaid?)", that's condescending. I'm not stepping into vegetaryan cookbooks.

@DanielFischer heh, okay, the word latest does seem to shine it in that light.

@anon My beef is particularly with the parenthetical remark.

@DanielFischer: Vegetarian, especially important for a German. ducks

@Huy That was quite intentional.

I did not expect it to be.

making fun of veg opinions and people seems to be a thing in pop culture.

@DanielFischer: Quite morbid if you write it that way, to be honest.

I'd give up meat, or some of it, for moral reasons if I wasn't so weak for it. I've not encountered a "vegetaryan" in real life, but supposedly they are a thing. (strangely, zero of the vegetarians/vegans/whatever I've met were so for moral reasons.)

I don't think he makes fun of anyone, it's simply the author's style. He simply wanna create a pleasant environment while reading his book. You should read his book before saying more, I mean this guy really doesn't care of making fun of people. What's the gain of making fun of people in such a book, really? It's a book about integrals.

@anon I'm old enough to have certain experiences with earlier ideologists.

Pleasant???

I've heard some animal rights activists have ideas and actions that are extreme or incoherent. indeed the FBI identifies them (eco-terrorists in general) as the largest domestic terrorism threat, ahead of white supremacy and the radical islamic variety.

That's your opinion, @Chris'ssis.

@anon I know some vegetarians who are so for moral reasons. One will eat oysters because they have no nervous system so don't actually feel.

Of something like this.

@MikeMiller: But if the animals are dead they don't feel anyways, no?

@Khallil It's an opinion of good sense. You talk about his book without reading it.

they did when they were alive!

@MikeMiller: Aha!

@Khallil As far as it is commented by people that know far too less about integrals, yes, it is. Maybe you show me some brilliant proofs to his problems in his book.

@Chris'ssistheartist: Why does one have to be particularly smart at solving integrals to have an opinion on whether an introduction is condescending or not?

@Chris'ssistheartist + @Khallil

@Richard :-P

May I ask why you post a picture/gif of a hammer after intervening helping out, @Richard?

@Khallil - 'Tis Mjolnir. The Hammer of the G Mods.

Only those who are worthy may wield it.

What happened here?

@Khallil: Did you read in time what Chris'ssis wrote? I was AFK.

@Huy It doesn't matter. The messages were flagged and deleted.

@Huy: What have you been working on lately?

@MikeMiller: Preparing for an exam on diffgeo and Riemannian. I'll probably revise the Hopf fibration another time tomorrow, see if it makes more sense now. Also going through proofs more thoroughly because I wasn't very well prepared last time.

Ah, gotcha. When's the exam?

@MikeMiller: I've got one on Mon and Fri next week.

I heard that maths is actually quite easy. Discuss

/runs away quickly

Fly, you fool!

@Richard: Very. Mathematicians are the opposite of JFK.

@MikeMiller: I think the things bothering me most are bundles, orientations, and long computations. :(

"We choose to solve the Riemann hypothesis in this decade and get tenure and do the other things, not because they are hard, but because they are easy."

@Huy: Maybe I can say something to help for the first two, but I'm also very slow with computations.

@MikeMiller: I actually had a question on my mind the other day about bundles. I'll ask tomorrow if it comes to mind again, it's almost midnight over here.

Ok. Night.

Good luck.

Night.

@Chris'ssistheartist - If you persist in breaching the site's "Be Nice" policy, you'll be suspended for a much longer period

This chat room occasionally contains strong language (which may be unsuitable for children), unusual humour (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors). Nevertheless, we expect community members to treat each other with respect.

For the record, the unspoken part of "be nice" is "or go somewhere else"

A 3 hour suspension is overly harsh guys.

@Khallil what's goin on pal?

:(

@skullpatrol Let me refer you to the two messages just above yours. Given the number of suspensions and annotations on your account, I'd say 3 hours is fairly reasonable. Please do remember to be nice, even when others don't deserve it.

@skullpatrol I suspect it was an attempt at humour to lighten the mood.

Jesus, looks like I missed a maelstrom.

@Fargle - Nothing more than a storm in a teacup

As I added "in a math room" the mood is different @ArtOfCode

I suppose. I just quite often only look at the chat after some amount of proverbial feces has hit the proverbial fan

@skullpatrol - Rudeness is rudeness regardless of the room

Hi @Fargle. How's it going?

But the context here is different pal @Richard

Alright, I suppose. Trying to learn Galois theory, trudge further in Ted's notes, and write some music in the downtime. Yourself, @Mike?

@skullpatrol - No, it really isn't

The "be nice" policy applies across the entire estate.

@Richard You should try the EL&U room :-)

If you ever posted your ban hammer there they would laugh you out of there pal.

That's because they mood in there is a lot less serious than in here @Richard

The mood in here is serious in theory, but less often so in practice.

@skullpatrol Yet we're not talking about the hammer. The point is, you were rude and you were suspended for it. If you can't be nice, you'll continue to get suspended for longer periods.

@ArtOfCode It wasn't skull, he's been peaceful lately. It was (is) Chris's sis who is suspended.

Thanks @DanielFischer

@skullpatrol You're still weird ;)

:-)

Ah, I see. I've been looking at links to the wrong user. I'm sorry, @skullpatrol - mistaken identity.

Np pal

Teach me to get involved where I don't usually stick my nose.

@ArtOfCode A little castigation is not wasted on skull.

:O

I just want to say, for the record--I don't enjoy her ego very much, and she most certainly has been rude, so I'm not defending her on those things or anything, but Chris's sis is also the target of rudeness from people who are annoyed by her ego. I'm not proposing anyone get punished or anything, I just want that to be noted. Two wrongs, et cetera.

if someone had been suspended for 3 hours they wouldn't be complaining about it 15 minutes later from the same account #maths

Still, I'd rather not pick on the wrong people too much.

#notonthissite

Good plan @ArtOfCode

Left to get some popcorn and already the fight is over.......

@KevinDriscoll Crisps are superior to popcorn.

That^

Is in uncalled for

Don't worry, I'll make sure to post it if I ever have to ban you :)

:D

@DanielFischer Ya I prefer chips. That's what I get for tryin to be hip.

Anyway, I'm about to have breakfast for dinner. I'll be afk for a bit.

@Fargle: Sorry, my hands were full for a minute there. That all sounds fun.

I'm doing ok. running some errands.

Have you started your class yet @Mike?

That was yesterday. I've got office hours but I only teach section once a week.

is that like a sixth of a chapter?

There's 8 mods in here.

-_-

Well aren't you all lucky then? :)

There's a reason for that...

@anon that or a typo

8 mods for what

For your viewing pleasure

Hi all, I was wondering if anyone could help me with a TeX question. I want to use $\dot\sim$ and $\sim$ in similar contexts, but $a \dot\sim b$ and $a \sim b$ seem to have different spacing. Trying to space out $\dot\sim$ with \ or \, doesn't seem to work: $$a \sim b \\ a\ \dot\sim\ b \\ \\ a \,\dot\sim\, b \\ a \dot\sim b$$ Does anyone know how to get the right spacing?

Hm, the second one looks a lot like the first now using MathJax...

It looks slightly different in TeXPaste: texpaste.com/n/kek9fw4p

@AntonioVargas Make a new operator, \DeclareMathOperator{\dsm}{\dot{\sim}} or something, that gives you some automatic spacing. If you want the exact same behaviour as for $\sim$, probably TeX - LaTeX.

Let's see: $\DeclareMathOperator{\dsm}{\mspace{2mu}\dot{\sim}\mspace{2mu}}$ $$a\sim b\\ a\dsm b$$

Different spacing.

Thanks @DanielFischer, that put me on the right track. If you use \newcommand{\name}{\mathrel{\dot\sim}} you get the right spacing

\sim is a relation operator, so you just need to tell TeX that \dot\sim is one too

@AntonioVargas Ah, \mathrel. I always forget the proper way.

Oh no @PedroTamaroff is here. Time for my lifetime suspension :-)

@KarlKronenfeld As Mike said, no. Note that there are lots of maps $\partial:A\to A$ for $A$ a ring that satisfy the Leibniz rule and are not $2$-nilpotent!

@skullpatrol You can always ask for it! =)

Morning @DanielFischer.

@MikeMiller Night.

@PedroTamaroff =)

I'm serious. I would have to oblige.

It's always dichotomies here. Nobody ever says "Mid-afternoon."

@MikeMiller Mike check your mobile.

You falsified that statement in making it.

@ArtOfCode Who are you addressing?

over 9000 hours later...

Good after-late-afternoon-before-early-evening, everyone

@PedroTamaroff Mike's claim that "nobody ever says" :)

@ArtOfCode: Take it as a metaphor.

this chat is really rowdy

I thought the math people talked shop

@0celo7 Is that worrying you?

Math chat is hours and hours of unrelated chatter, followed by short bursts of actual math

I thought it was hours and hours of math

Try the homotopy theory room

I'd rather not

For hours and hours of math and short bursts of chat

it would seem the physics chat is more intense than this once

huh

Yup

physics chat isnt as active as this chat

It is

well, perhaps it is now, I haven't been in the physsics chat in some months

If you want to see master trolls in action come to the EL&U room :-)

EL&U?

English language & usage

I even convinced robjohn to visit once :-)