Mathematics - 2023-08-01

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toodles

01:49

Hi. I am new to SE. I left an answer to a question a few months ago, and it appears someone marked it as a "low quality post." Two others reviewed it as "looks okay." I genuinely am curious what made the person mark it as low quality, as I am just looking to learn. Is there a way to contact this person about it?

1 hour later…

leslie townes

03:04

not that i am aware of. it is basically impossible to know why people do what they do on the site (and even if whoever that was was here, and they probably aren't, there is no guarantee that what they said would be honest or in any way helpful). thankfully the site is designed so that no one user's opinion has too much influence.

one potato two potato

03:18

If that's possible, Shaun won't show up in this chatroom. Or maybe he would for the upvote.

Soumik Mukherjee

Is asking someone's age on the internet considered inappropriate?

Ted Shifrin

03:37

Yes.

Soumik Mukherjee

Okay

Ajay

04:06

Is there a way to prove sin(x) is a contraction map on [-1,1] without MVT?

Ted Shifrin

04:27

That’s the right way. You can try the $\sin x - \sin y$ formula.

4 hours later…

Jakobian

08:30

@SoumikMukherjee just ask tbh

Soumik Mukherjee

08:48

@Jakobian lol, btw how old are you?

Ajay

09:05

@SoumikMukherjee i'm 34 years old.

Jakobian

09:28

@SoumikMukherjee 25

Alright now what

冥王 Hades

09:58

I hate the problem above.

So annoying

Soumik Mukherjee

10:29

@Ajay ooh

@Jakobian ooh, I may ask my topology doubts to you if you don't mind

Jakobian

I don't mind but I only really study general topology

I won't be much help with algebraic

toodles

10:49

@leslietownes Cool. That makes sense. Thanks for shedding some light.

Soumik Mukherjee

@Jakobian That's fine

Jakobian

11:24

@toodles the system is maintained by the community and community isn't always right. Just most of the time

Ajay

11:58

@SoumikMukherjee How old are u?

sunny

12:34

In computing $\lim_{n\to\infty}f_{n}(x)$, where $f_n(x)=\sqrt{nx}\arctan\frac{1}{nx}$, my approach is the following; write $\arctan\frac{1}{nx}=\frac{\pi}{2}-\arctan nx$, then $$\lim_{n\to\infty}f_{n}(x)=\sqrt{nx}\left(\frac{\pi}{2}-\arctan nx\right),$$ but now what? We have $[\infty\cdot 0]$.

Xander Henderson

Are you familiar with L'Hospital?

sunny

Yes.

Xander Henderson

Use that.

Ajay

@Jakobian Do you know of any good references for digital topology?

sunny

@XanderHenderson Is there a small-angle approximation for $\arctan$ like there is for $\sin$? I haven't found anything online on this, though it seems reasonable if I look on its graph.

Ajay

12:48

I would think arctan(x) = x

for small x

sunny

@Ajay yeah, me too, but what is the motivation behind that fact? For $\sin$ it's $\lim_{x\to 0}\frac{\sin x}{x}=1$, see here.

Jakobian

@Ajay No, I never heard of it

Soumik Mukherjee

@Ajay $22$, almost as old as this century :)

Xander Henderson

@sunny Sure. There is a small angle approximation. You could use that, too.

Just truncate the power series for arctan at the first term, I suppose.

Pretty sure you just get $\arctan(x)/x \to 1$.

Jakobian

Either way digital topology sounds like something touching more on manifolds or algebraic topology rather than general topology from what I read on wikipedia

Xander Henderson

12:56

I mean, $\tan(x) = \sin(x)/\cos(x) \approx x/1$ when $x$ is near zero.

σκουλήκι

what does the notation $C^{2+\alpha}$

Xander Henderson

arctan is just the reflection of tan across $y=x$, so...

@σκουλήκι Depends on context?

Where are you seeing that?

σκουλήκι

what does the notation $C^{2+\alpha}(\mathbb{R^d}$ mean, it is a function space

Xander Henderson

@σκουλήκι Generally, $C^{n}(X)$ means the space of $n$-times continuously differentiable functions on $X$.

σκουλήκι

yeh but here $\alpha$ is in (0,1)

Xander Henderson

12:58

Which is why I asked for context.

Are you dealing with fractional derivatives?

σκουλήκι

the author does not give the notation

no definitely not

Xander Henderson

Hölder continuous functions?

σκουλήκι

much more likely

just weird its written with a +

Xander Henderson

Well, I still have no idea what the notation means, but perhaps it means functions with two derivatives, where the second derivative is $\alpha$-Hölder continuous?

σκουλήκι

yeh I think thats it

Xander Henderson

13:00

But, again, I would check the text. I would bet dollars to donuts that the notation is defined somewhere.

σκουλήκι

ok ill have a scower

get some donuts ready

Xander Henderson

Which book is it?

σκουλήκι

Analytical Methods for Markov Semigroups

Jakobian

13:18

@σκουλήκι that's the same notation that Gilbarg and Trudinger use in their book on Elliptic PDE's

one potato two potato

$\chi(\Delta) = -1/2$

2 hours later…

sunny

15:03

@robjohn why, in your answer here, are you stating "Since $\lim\limits_{x\to0}\arctan(x)=0$..."? Why is this significant?

Jakobian

15:21

you want to know what $\theta$ converges to

robjohn

@sunny Because $\theta=\arctan(x)$. Thus, we are looking at $\lim\limits_{\theta=0}$

user223626865

perhaps, adding that to the answer would help the next person with the same question

Xander Henderson

@user223626865 I think that this is one of those places where the student needs to read the argument carefully, and figure out why that half-sentence is important (or just skip by it as an unimportant detail).

While it is not my own personal style (I'm very wordy for a mathematician), there is an argument in favor of only writing out the details which are strictly necessary for solving the problem at hand. The reader is assumed to be capable of filling in small gaps.

It's a matter of style.

user223626865

In a textbook, yes.

冥王 Hades

@robjohn Hey John, could you take a look at the problem I posted? Should I post it again?

Xander Henderson

15:41

@user223626865 You seem to disagree with me. I maintain it is a matter of style, and perfectly acceptable on Math SE.

geocalc33

I have come up with a novel construction that I call the Twisted Poincare Web (TPW) It's when you take infinity copies of the Poincare groupoid and let the disjoint union of the objects form what is called the $\Delta$-base which may or may not be a discrete set. You link these Poincare groupoids together in nontrivial ways, by linking the morphisms

Jakobian

Today I thought that maybe we can do something interesting by doing analysis on something else than $\mathbb{R}$

but it turns out that $\mathbb{R}$ is the only ordered field that really works for such results like intermediate value theorem to hold

34

Q: Is the IVT equivalent to completeness?

Obviously we can use the completeness of the real numbers (least upper bound axiom, or one of the equivalent principles) to prove the IVT. Can we go in the opposite direction? This isn't a homework problem or something. I'm just wondering. If the answer is "yes", then I'm not really asking for ...

geocalc33

15:57

The problem is when you come up with incredibly good ideas like the TPW, you have a lot of work cut out for you to make everything rigorous. In fact you might find that halfway there, you can't make anything rigourous

user223626865

> If there is aught of good in the style, it is the result of ceaseless toil in rewriting.

Jakobian

thankfully those theorems of calculus still hold for polynomials over real-closed fields

so at least some of it is salvageable to polynomials

does anyone know of any sort of maps between real-closed fields that would act similarly to continuous functions on $\mathbb{R}$, other than polynomials?

Xander Henderson

@user223626865 That is non-responsive to my point.

Rewriting often includes deleting unnecessary detail.

Jakobian

@TedShifrin perhaps you know of anything like this? Since you know some algebraic geometry

user223626865

@XanderHenderson A non-response to your point is an agreement with it.

robjohn

16:09

@冥王Hades don't post again, but you can link to old comments

You are talking about this one?

user223626865

I merely predicted others having the same difficulty.

冥王 Hades

@robjohn yep, that one

robjohn

Is $\angle ABC$ a right angle?

Ted Shifrin

16:24

@Jakobian I worked over $\Bbb C$. I have no idea.

g'morning, @robjohn. How's the pup?

冥王 Hades

@robjohn yes.

Xander Henderson

@user223626865 I didn't disagree that others might have the same difficulty. I suggested that it was intentional, and a matter of style. Many authors choose to elide details as part of their pedagogy.

robjohn

@冥王Hades That should be marked in the diagram (the usual right angle symbol would be great).

冥王 Hades

@robjohn yeah I had the same complaint as well but the person who sent it just told me it’s a right angle

Jakobian

@XanderHenderson books which do that are great to read from

user223626865

16:26

@XanderHenderson Indeed, we are in the realm of pedagogy here.

Jakobian

well, not all of them, that depends on how forgiving an author is

robjohn

@TedShifrin Good morning. The pup is fine. My older dog and I were attacked by yellow jackets Sunday evening. We spent both days this weekend in the emergency vet

Ted Shifrin

Oy. Did the vet take care of you, too?

robjohn

On Saturday, there was a 2" gash on her shoulder that we have no idea how it happened.

Ted Shifrin

Sometimes too many details make it more difficult to read and understand.

@robjohn Yes, you told us about that. That's why I asked how she was.

robjohn

16:28

Ah, that pup. Yes, she is much better today.

Still on meds, but better

冥王 Hades

I slipped on a banana peel a few hours ago and had my cola spill partially onto me

is this karma? I still laughed though

Ted Shifrin

If only it had been a chocolate-covered banana peel.

Jakobian

I'd say a good book is one that finds balance between not giving answers to the reader too much, but not leaving everything for the reader to figure out

冥王 Hades

@TedShifrin I would’ve eaten it

robjohn

@Jakobian That balance could well be different for each reader.

Xander Henderson

16:31

@robjohn This.

Jakobian

That's true. It depends on experience somewhat

user223626865

We all agree.

Xander Henderson

Personally, I like verbose books. But I have good friends who actually like Folland's style.

Ted Shifrin

Verbosity does not equate to clarity.

冥王 Hades

Too much verbosity can be a pain though

Ted Shifrin

16:32

Often, it's the opposite.

Xander Henderson

@TedShifrin No, not always.

And yes, one can be overly verbose.

But the same can be said for terseness.

Ted Shifrin

Agreed.

Off to the dentist. Bubye.

user223626865

cya

冥王 Hades

It makes me go “Is this a Math textbook or a history textbook?”

Xander Henderson

Personally, I prefer authors who err on the side of saying too much. But, again, it is a matter of personal preference.

@TedShifrin Good luck.

Break a leg.

robjohn

16:33

@TedShifrin put the dentist to sleep with math, for a change.

冥王 Hades

@XanderHenderson Wow, I understand you disagree with him but that’s too much

Even I’m not that apathetic

Jakobian

I don't really like Folland because I find his arguments somewhat not careful enough for me

冥王 Hades

@robjohn 💀💀

Jakobian

which is probably a conscious thought given the subject

Xander Henderson

"Break a leg" is a typical English idiom used in the context of theatre or other performing arts to wish a performer "good luck". An ironic or non-literal saying of uncertain origin (a dead metaphor), "break a leg" is commonly said to actors and musicians before they go on stage to perform or before an audition. Though the term likely originates in German, the English expression is first attributed in the 1930s or possibly 1920s, originally documented without specifically theatrical associations. Among professional dancers, the traditional saying is not "break a leg", but the French word "merde...

Jakobian

16:36

Right now I'm reading Gillman and Jerison, and I think that's a good example of a good book. It's pleasure for me to read

it's a book I'm planning to finish to the end

冥王 Hades

@XanderHenderson I can’t believe this is real

Well I hope we all break our legs

user223626865

Believe it.

robjohn

@冥王Hades It is so.

Xander Henderson

@冥王Hades Well, I did just create that Wikipedia page to mess with you. In the last two minutes. Obviously.

Jakobian

the chapter that assumes knowledge of real-closed fields is a bit worse but it's understandable that they assumed it

It's not that big of an issue since I can complete my knowledge by studying some field theory, and then a bit from a book about real algebraic geometry

冥王 Hades

16:38

@XanderHenderson Now that’s much more believable

user223626865

Don't believe the hype

Jakobian

I find it interesting that to prove uniqueness of real-closure of an ordered field one has to prove such things as Sturm's theorem for real-closed fields

something I did know for polynomials over R (but forgot)

user223626865

Never forget that polynomials are our friends.

Xander Henderson

@user223626865 Blueberries are our friends.

robjohn

Friends don't let friends drink and derive.

Jakobian

16:45

It's just surprising to me because real-closure is something I feel like is fundamental, but I don't feel the same way for Sturm's theorem or related

they're just weird results about polynomials

robjohn

I think Sturm's Theorem is cool.

I have a Mathematica function that computes the chain and ancillary functions that apply the Theorem to the chain at various points.

Jakobian

ancillary functions?

robjohn

@Jakobian yes, support functions.

Jakobian

ah. I never heard of this word "ancillary" before

toodles

@Jakobian Thanks for the note. I wasn't looking to contest it, but I was just interested in hearing why their reasoning, in case I could potentially learn something. All good, though.

冥王 Hades

16:57

@robjohn I managed to prove the $AD=DE$ part by construction a cyclic quadrilateral. I suspect the latter has something to do with constructing side lengths that themselves are equivalent to $AE$

user4539917

@robjohn have you seen this argument mapping app

robjohn

17:15

@user4539917 I haven't.

123

Hello Everyone...

Xander Henderson

Hi, Doctor Nick.

shintuku

18:02

room temperature superconductors

i tell you, this year is unusual in scientific advances

冥王 Hades

18:31

I sense early symptoms of a cold. Irritated/sore throat

Ted Shifrin

19:05

or Covid ...

冥王 Hades

19:21

Would suck if I got Covid

I’d have to attend online classes until I am cured

1 hour later…

sunny

20:37

Hmm, I have the following sequence of functions $$f_n(x)=\left(\frac{x}{1+x}\right)^n\frac{1}{1+x},\quad n\geq 1,$$ and I'm trying to determine their uniform convergence on $[0,\infty)$. I have calculated the pointwise limit to be $0$ for all $x$ (hopefully this is correct), but how do I tackle computing $\sup_{x\in [0,\infty)}\left|\left(\frac{x}{1+x}\right)^n\frac{1}{1+x}\right|$. Any hints are appreciated.

PM 2Ring

20:47

@shintuku It would be amazing. But the early reports are that the paper is rather shoddy, and doesn't give much confidence in the quality of their work. Ok, maybe they rushed to put the paper out, but still...

leslie townes

sunny: in general remember that you don't necessarily need to be able to evaluate that sup as a function of n to show the uniform convergence (or disprove it). you just need to bound it in such a way that you can show it goes to 0, or analyze it in some fashion to find that it cannot go to 0.

sunny

@leslietownes true

leslie townes

sunny: here however you can use calculus to see that f_n is increasing on [0,n] and decreasing on [n, infty) and so sup_x f_n(x) = f_n(n) which is at most 1/(1+n) and that is enough to show it goes to 0.

if you stare at the formula for a while you can see that f_n(n) is not just "at most 1/(n+1)" bbut about 1/e 1/(n+1) when n is large.

but i would not be able to expect to find formulas for these sups in general, even in otherwise tractable problems

sunny

@leslietownes did you calculate the derivative to conclude it is increasing and decreasing on the intervals you wrote respectively?

leslie townes

yes, using the quotient rule [regarded as the quotient x^n/[(1+x)^(n+1)] and bearing in mind that as we care only about the sign of the derivative for this analysis, we care only about the numerator, and the common factors of x^(n-1) and (1+x)^n (which are nonnegative due to the domain of x) that appear in various terms there do not affect the analysis.

i.e. while the algebra looks goofy, on (0,infty) analyzing the sign of f_n'(x) is the same thing as analyzing the sign of n-x

冥王 Hades

21:27

user image

Success! @robjohn

i can die in peace now

shintuku

@PM2Ring infighting between the authors apparently

berkeley material sciences confirmed it works in a simulation. dunno what that means tho

leslie townes

this would be a hell of a way to find a new floating point bug in intel processors

Xander Henderson

@冥王Hades Okay. Bye.

Ted Shifrin

I hope the cold or Covid isn’t terminal.

shintuku

at leslie: lmao

Shaun

21:42

Hi :) In linear algebra, there's algebraic multiplicity and there's geometric multiplicity. I remember from the undergraduate days that there's at least one other type of multiplicity. Please would anyone refresh my memory? I can't seem to find the other(s) anywhere . . .

Xander Henderson

@Shaun Well, there's "double secret multiplicity", but, uh... I've already said too much.

Shaun

@XanderHenderson Lol :)

Ted Shifrin

@Shaun Certainly not in this context.

leslie townes

do you mean a 'multiplicity' as specifically associated with an eigenvalue of a linear map on a finite dimensional vector space? or just some other use of the word? the word is used elsewhere, e.g. for polynomials or other functions, even when one has no linear map in mind.

probably a zillion senses of it in algebraic geometry.

Shaun

I mean the multiplicity of an eigenvalue. I think it might be "minimal multiplicity", but the Google searches are inconclusive.

Ted Shifrin

21:49

Nothing I know as a linear algebra “expert.”

If you’re referring to the algebraic multiplicity in the minimal polynomial, perhaps some author coined a term for it, but it’s not universal.

Shaun

@TedShifrin Yes, I think that's the definition! Thank you! I'm arranging for my undergraduate linear algebra notes to be sent my way, so I might update you on what it is if you want.

Ted Shifrin

As I said, not a universal term.

It’s the size of the largest Jordan block for the eigenvalue.

Xander Henderson

In Soviet Russia, Jordan blocks you!

Shaun

@TedShifrin I believe you; I just thought you might be curious, s'all.

Ted Shifrin

What I just explained is the only reason someone might care about it

i guess it’s surprising there isn’t a term for the size of the largest Jordan block. Maybe there is and I have just never seen it.

Xander Henderson

22:02

@TedShifrin There is a term! "The size of the largest Jordan block". :P

*runs away and hides*

leslie townes

^ largest

Xander Henderson

Yeah, sure. I wasn't paying attention to the context, to be honest.

Ted Shifrin

You’re too busy competing with shin for being maximally annoying.

Shaun

Anyway, thanks for the help. See you later! :)

leslie townes

ted's so old that when he uses chalk to lecture, he sometimes recognizes the plankton minerals in it from when the plankton were alive. this leads to some poignant moments at the beginning of his multivariable calc videos.

6

robjohn

22:17

@冥王Hades I'm glad. I had to go out for some appointments, so I didn't get to work on it much. I think it can be solved via trig fairly simply, but as I said, I haven't worked on it.

Semiclassical

trying to think through the following geometry problem, specifically to convince myself that it actually makes sense. (i can substantiate it algebraically but not geometrically)

Suppose I have two pairs of vectors $(a_1,a_2), (b_1,b_2)$

bleh, shouldn't have started before i was confident of it

Xander Henderson

@Semiclassical Okay, you've got two vectors. They look real nice. I think I want them. Hand 'em over before things get... violent!

Semiclassical

@XanderHenderson call them scissors for the full effect of the violence

Ted Shifrin

Forget coordinates.

Semiclassical

(because each pair is like a pair of scissors blah blah blah)

wasn't planning to use coordinates if i could help it (though i do know one way)

Suppose I have two pairs of 3-vectors $(a_1,b_1), (a_2,b_2)$, with dot products $x=a_1\cdot b_1$ and $y=a_2\cdot b_2$. There is obviously quite a bit of freedom in this, since i can reorient either pair while keeping its angle fixed

Can I always choose these pairs so that $a_1\cdot a_2=b_1\cdot b_2$?

leslie townes

22:26

choose them subject to what constraints? what stops you from choosing them all to be 0 all of the time?

Semiclassical

oh. unit vectors

blah

(I guess there's probably a way to restate this as a problem of a spherical quadrilateral...meh)

leslie townes

would anything stop you from choosing a_1 = a_2 = b_1 = b_2 all of the time (so the dot product is 1 and you still get to choose which unit vector they all are)? are there other constraints?

Semiclassical

@leslietownes i mean, i haven't said what $x,y$ are

so they dont' have to be 1

i fix the angles between $a_1,b_1$ and $a_2,b_2$ (not necessarily the same angle)

Ted Shifrin

None of this is intrinsic? It’s all about coordinates in some basis?

Semiclassical

what?

dot products are intrinsic

leslie townes

22:29

okay, so you're definitely fixing x and y (which other than "being the dot products of unit vectors", which does impose some constraint, might be arbitrary)

and then looking at the 'other' pair of dot products

Semiclassical

right

leslie townes

hrm

Ted Shifrin

You’re writing dot products of coordinates, it seems.

Semiclassical

no, i'm not. $a_1,a_2,b_1,b_2$ are 3-vectors

Ted Shifrin

Oh, now these are vectors? Ugh.

leslie townes

22:30

your notation has offended ted

do you know what gutenberg used to charge per subscript? do you?

Ted Shifrin

No, he switched in the middle.

Semiclassical

meh. two pairs of vectors = 4 vectors altogether

so i don't agree that there was any ambiguity

Ted Shifrin

I’m paying attention to news … a yuge indictment just broke.

Xander Henderson

@Semiclassical For what it is worth, I was confused, too.

Semiclassical

yeah, fair enough

I think the spherical quadrilateral way to say this is: suppose I tell you two side lengths of a spherical quadrilateral. is it possible that the two remaining side lengths are equal?

and if it is, what possible values are there for these two remaining side lengths (in terms of the given side lengths)

Xander Henderson

22:33

@TedShifrin I'll be shocked if it actually hurts Trump's standing in the elections, however. It's not like anyone who was going to vote for him is going to change their minds because a clearly bought-and-sold court handed down some phony charges. :/

Semiclassical

the only way it matters is if it somehow legally prevents him from being on some ballots...in which case things get messy for other reasons

Ted Shifrin

Of course not. Even if he actually is found guilty. The hell with the Constitution.

The hell with the right to vote.

leslie townes

it will definitely help him in the midterms. i don't know about the general election.

who these "swing voters" are, with something like that on the ballot, i can only imagine.

Semiclassical

my brain sorta throws a fault when i think about 2024

Xander Henderson

@leslietownes Midterms? Do you mean primary?

I think I need to change my affiliation to "Republican" for this election, so that I can actually vote in the primaries. Since it doesn't look like anyone is going to attempt to primary Biden. :(

Ted Shifrin

22:39

@leslietownes Midterms? You mean primaries?

Wrong, ass Kennedy is primarying Biden.

Semiclassical

@XanderHenderson except for people who have zero credibility

case in point

Ted Shifrin

And we have third party sabotage being paid for by the neo-Nazi who pays for Thomas in his pocket.

leslie townes

yes, the primary.

Semiclassical

the whole notion of "swing voters" strikes me as dubious. it assumes that each party has roughly equal blocks of voters they can reliably turn out, and thus a small percentage of uncommitted voters can make a difference

leslie townes

i'm gonna vote for the wagner group guy. he is firm but he gets results.

Semiclassical

22:42

i guess i specifically mean: not the existence of swing voters, but their relevance

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