Mathematics - 2021-10-06 (page 1 of 2)

Ted Shifrin

00:00

Ah, that was his thesis.

In my paper, I worked with the second-order one in terms of osculating spaces and adapted frames.

The whole problem, in general, is that you have to use sheaves because (for embedded submanifolds) the osculating spaces may drop rank. But look at Pohl.

@leslie That looks like a horrid reference.

With a general (positive-definite) quadratic form, it's just the spectral theorem and the one-dimensional result.

leslie townes

that's done in the first bit of one of the solutions. i was selecting for paste-ability, not ease of reading.

i'm a low quality chatterbot interface to google.

Ted Shifrin

Well, we all know I set a high bar :)

I want to see a picture or three of the scary strawberry with paws.

leslie townes

we're still working out just how exactly this is going to work, but if it comes to fruition, ha ha ha, i will document it.

Ted Shifrin

Maybe a lingonberry garnish would be good.

Joe Shmo

00:28

@TedShifrin that one, but in $\mathbb{R}^n$. the 1D integral is by squaring, fubini, and then polar coordinates, I know.

@leslietownes, looks like it. Thanks!

leslie townes

look at the section beginning 'wick's theorem' where you diagonalize and it becomes a product of one-dimensional gaussians. i don't know what kind of physics sorcery is going on once you add in what they call a 'source term.'

Joe Shmo

oh no no I don't want any sorcery

leslie townes

physicists invented nuclear weapons, that's the kind of road you head down if you keep listening to them.

yeah.

Joe Shmo

I'm a peacenik

when it comes to physics

Ted Shifrin

There’s nothing $n$-dimensional about these questions. They do show up in harmonic analysis, though, integrating spherical harmonics.

Rithaniel

00:59

I hate that feeling when I'm trying to show something that my brain tells me should be easy, but I just can't find the right track to follow. It's like trying to climb something and just slipping down on the first step each time

I've got a couple of sequences in a Hilbert space, $\{x_n\}$, which converges weakly to $x$, and $\{y_n\}$, which converges to $y$. I'm trying to show $\langle x_n,y_n\rangle\to\langle x,y\rangle$. I know this has to rely on the fact that $\{y_n\}$ strongly converges, because this isn't true for two weakly convergent subsequences.

My search-fu has failed me again, though

(I found a thousand and one duplicate questions about showing that every bounded sequence in a Hilbert space has a weakly convergent subsequence, though)

user1115542

Real newbie query: Here's a simple proof from Spivak: $\frac{\left | x \right |}{\left | y \right |} = \left | x \right | \cdot \left | y \right |^{-1} = \left | x \right | \cdot \left | y ^{-1} \right | = \left | x y ^{-1} \right| = \left | \frac{ x }{ y } \right |$ I think I understand how this is derived, but what I do not understand is why $\left| y \right |^{-1}$ is different from $\left| y ^{-1} \right |$ Thanks.

Rithaniel

It's the difference between $\frac{1}{\vert y\vert}$ and $\vert\frac{1}{y}\vert$, which are honestly the same thing. It's why Spivak says they're equal

user1115542

So, he's just messing with me?? :-)

Ted Shifrin

No, didn’t he prove those equal in a previous part?

user1115542

@TedShifrin Correct...indeed he did. But was just wondering why he would go out of his way to show the two.

Ted Shifrin

01:13

That”s the point :)

Because that’s what’s going on at this point in the book. Proving everything that you always thought was obvious. Just using the properties.

user1115542

@TedShifrin OK... I think I get it. The logic, which I love mind you, takes a bit of getting used to. OK... I like that.

Ted Shifrin

For example, you’ve always said it was obvious that $0\cdot x=0$ for any $x$. Why is this?

leslie townes

the notation is just so familiar. people are trained in algebra class to perform those manipulations in their sleep, and not to notice the potential differences between them.

Ted Shifrin

Rithaniel: You need an “add a clever zero” proof.

user1115542

Yes...funnily enough, just did that last night. As you say, not at all "obvious"

Ted Shifrin

01:17

But we all assume it to be gospel truth. Yet, it’s not a standard property that we get to assume.

user1115542

What I like about it, to be honest, is the rigor that needs to be applied. As you say, it's not rote by any stretch of the imagination.

Ted Shifrin

Spivak (or, indeed, proof math) isn’t the right choice if you want just to assume that everything “obvious” is true.

leslie townes

if someone said here are functions f and g, show that f(g(x)) = g(f(x)), you'd think hmm, that's not a general property that all pairs of functions have, so this might have to do something with these specific f and g, and what do we know about them, etc. but if instead of f and g you write | | and ^(-1) suddenly it just looks like "duh."

user1115542

Agree... that's why I am trying to go through his book.

Ted Shifrin

Nice comment, @Leslie.

leslie townes

01:19

we need to unlearn the 'duh' so we can relearn the higher 'duh' and step closer to enlightenment.

Ted Shifrin

@Leslie Was my advice to Rith right?

Rithaniel

@TedShifrin I was thinking that might be the case. Something like $$\vert\langle x_n,y_n\rangle-\langle x,y_n\rangle+\langle x,y_n\rangle-\langle x_n,y\rangle+\langle x_n,y\rangle-\langle x,y\rangle\vert$$ was what I was originally considering, but I dislike the middle

user1115542

Thank you Ted and Leslie. That's a really nice summary of what I was actually trying to ask. :-)

Ted Shifrin

Can you do just one zero?

user1115542

Is that for me?

Ted Shifrin

01:22

@user1115542 Through Spivak (and my books too) you are expected to use what’s been proved in previous parts. Always look for that.

No, the one zero was for Rith.

user1115542

Yes...that I have figured out...hence my reluctance just to "plow" on and hope for the best.

@TedShifrin I apologize if I do not realize who you are.

Rithaniel

I don't see a "one zero" idea immediately. Are we trying to build it into the form $\langle x-x_n,y_n-y\rangle$ or summat?

leslie townes

@Rithaniel try the version indicated here: math.stackexchange.com/questions/2952507/… note that the hypotheses there are not your hypotheses, but you have stronger hypotheses. you can control <u - u_n, v_n> using the strong convergence of u_n to u and the fact that the weakly convergent sequence is norm bounded.

Rithaniel

Again, leslie has found a relevant question almost immediately

2

Ted Shifrin

I wanted $x_n,y_n$, $x_n,y$, $x,y$.

leslie townes

01:26

as ted suggested, just one zero is enough. that answer shows why the result can fail if in the absence of the strong convergence assumption.

Ted Shifrin

Oh darn. I was too too lazy to type inner products. I like to think instead of searching. Ahem.

Rithaniel

@TedShifrin Oh yeah, weak convergence

I like to think, too, but there is more than one way to bake an apple pie from scratch

One is called "research," and the other is called "reasoning." I do want to understand where and how the strong convergence assumption comes into play, though

Ted Shifrin

You certainly need it for my first term!

Thinking is ultimately research.

2

I’m serious here.

Rithaniel

Very true

Ted Shifrin

The skills for becoming a researcher.

2

I’m not a fancy functional analyst. I use my basic Spivak skills.

leslie townes

01:30

you cheated by writing half of his exercises.

Ted Shifrin

Not half

Rithaniel

Also, clicked over to that link that leslie had. I did find that one already, but I dismissed it because it was trying to prove this to be the case with both sequences being weakly convergent

Ted Shifrin

Ha ha

FB may go to hell. Zuckerberg deserves it.

Even though I’m an addicted subscriber.

Rithaniel

But, I suppose the argument would work if one of the two were strongly convergent, then?

Also, I heard about the whole FB thing

Ted Shifrin

So work out what I wrote, dammit.

If I’m wrong, show me.

Rithaniel

01:33

I will, Ted, don't worry

But I was gonna share my ambivalence to FB going down

Ted Shifrin

Grrr.

leslie townes

rith: yes, if one is strongly convergent you can control the term he couldn't control using the schwarz inequality.

that argument breaks down if only weak convergence is assumed because x_n going to x only weakly doesn't force ||x_n - x|| to go to zero.

Rithaniel

Oh, I see. I suppose that would have been the step that I would have figured out with the thinking step. (Sorry Ted. Next time you're helping me I'll do it purely myself)

leslie townes

i really should use mathjax.

Ted Shifrin

LOL … Or you can follow the advice not to ask here and find it on main. I prefer making people think.

@leslietownes Duh.

Rithaniel

01:38

To further my "climbing up a slippery surface" metaphor, this is like a person pointing out to me that there's a staircase

Ted Shifrin

Read Camus’s The Myth of Sisyphus!

Or, hell, google.

Rithaniel

Sisyphus was the guy rolling the boulder up the mountain, right?

hyper-neutrino

yes

Xander Henderson

@Rithaniel And when he was up, he was up. And when he was down, he was down. And when he was half-way up, he was neither up nor down.

Oh, wait... that's the Duke of York, not Sisyphus.

Rithaniel

I know that one! Also, Tantalus is a guy that might have a parallel to the experience of research

Ted Shifrin

01:44

My metaphor comes to life!

@XanderHenderson No, he kept sliding down.

Joe Shmo

02:03

I've been off FB for years. and I never had IG either

but they have all my data anyway because I use whatsapp

leslie townes

i've never used FB, but my cat and daughter have IG accounts. FB thinks i am interested in cat toys. the algorithm basically advertises to me as if i am a cat.

Joe Shmo

02:20

its creepy that "the" algorithm had become an institution in our society

Semiclassical

02:48

i've got a facebook account but like

i haven't used it in RL years

so i'm not so aware of that algorithm

Leaky Nun

@JoeShmo 1984

Semiclassical

by contrast, i'm very familiar with the shenanigans of the Youtube algorithm

Joe Shmo

@LeakyNun, Yes. An Orwellian institution, indeed.

user178758

03:15

an artificially intelligent Orwellian institution has emerged from the apocalyptic pandemic

Ted Shifrin

Greetings, Leaky.

Leaky Nun

hi

Rithaniel

The year is 19^84. Long ago, we started using exponential notation to keep track of the year because it got too long otherwise

The stars went dark many eons ago, but still people are posting dumb stuff on FB

3

leslie townes

03:31

and my cat is still cute on instagram

Ted Shifrin

In a strawberry costume?

leslie townes

possibly. last year the daughter did go as a black cat. we forgot to get a pic of the two of them.

user178758

04:04

\o @AudenYoung

Auden Young

04:38

@user178758 o/

user178758

how are you?

robjohn

04:56

@leslietownes still wanting to be a strawberry ghost?

leslie townes

yes. with paws.

dc3rd

Ted popping up because he sensed I was calling to him through ESP.....

user178758

05:25

and claws?🧐

leslie townes

gotta have claws

Koro

professor Ted, while writing an answer, I wasn't aware that you were answering that question of dc3rd. Please let me know if I should delete it.

dc3rd

too late since I've read all of it. :), I'm still working it out on my own though...

Koro

@dc3rd ahh, I see. I was excited to answer it as soon as I saw the question as I've been studying linear algebra these days :)

dc3rd

no worries....it is all helpful.

still using Axler's book for linear algebra Koro?

Koro

05:33

yes, alongwith Hoffman &Kunze's

dc3rd

I've been working through Friedberg, Inse, et al'

user178758

"Lang's Algebra changed the way graduate algebra is taught..."

Koro

Last year, I'd read D.Poole's and Gilbert Strang's linear algebra.

dc3rd

I'm just now stumbling into the fact that linear algebra is a lot more vast than an undergraduate course.....

Koro

@dc3rd Do they use matrices or linear maps or both?

dc3rd

05:44

Insel does both. as should all undergrad linear texts. since they are one in the same.

Koro

In Axler's I have noted that, linear maps are highlighted more but matrices are discussed less in Axler's book.

dc3rd

mainly linear maps though because the idea is that everything can be converted to a matrix so make's it less clunky to write

leslie townes

yeah, axler covers the bare minimum of matrix stuff.

dc3rd

Insel has tossed it in here and there and I'm almost done the book. It is more so along the lines of "you should be aware that the linear maps can be turned into matrices", so for ease of typesetting I'll mainly use linear maps unless necessary

user178758

Would you agree with the quote above @leslietownes about Lang's Algebra changing the way graduate algebra is taught?

leslie townes

05:53

i dunno, i did not live in a pre-lang world. many of lang's books are somewhat divisive. they have devotees and people who hate them. if you look at early-mid 20th century algebra books they are sometimes a little crusty and obsessed with special cases, to my mind. and they predate the widespread adoption of basic frameworks and terminology.

is lang responsible for bringing that in, or did he just happen to write a book that became popular while that was happening generally in algebra, i dunno.

dc3rd

is graduate algebra built off of linear algebra?..........I have a feeling this is a rhetorical question...

leslie townes

sort of, if you interpret 'built off' very loosely. in a formal sense, not really. a lot of notions that are very interesting and useful in algebra become trivial in linear algebra, so linear algebra does not have analogues of, or stepping stones to, a lot of algebraic topics.

dc3rd

hmmm........one or two of my profs before had called linear algebra a "terminal" course as in once you learn it there is nothing more to learn about it and only can be applied.....

leslie townes

that is probably more reflective of the course offerings at wherever that is than some objective reality. although maybe it depends on what 'linear algebra' means. there's a sense in which even calculus classes are "terminal." you can learn more about the theory, the generalizations, etc. but they are probably not going to make you any better at calculus.

linear algebra over finite fields can have a very algebraic flavor that you don't see too much of if you only work over R and C. and linear algebra + topologies + no finite dimensionality enters into PDE, functional analysis, etc.

BannedUser

today I learned 3 laws of newton in a fist fight

user178758

06:12

Did you use the third one your advantage? 👊🏼🤼

BannedUser

it is disadvantage since if I hit someone I get hit too

user178758

that's when speed comes in...

p=mv

talkin' about fights, there's a good heavyweight fight on this Saturday

It could go either way

BannedUser

06:44

are you talking about ufc or wwe?

user178758

07:17

🥊 boxing

Tyson Fury vs. Deontay Wilder III

Tyson Fury vs. Deontay Wilder III, billed as Once and For All, is an upcoming professional boxing trilogy fight between WBC and The Ring heavyweight champion, Tyson Fury, and former WBC heavyweight champion, Deontay Wilder. The bout is set to take place at T-Mobile Arena in Paradise, Nevada on October 9, 2021. == Background == Fury and Wilder first fought in December 2018, with the bout ending in a controversial draw. The pair had a rematch in February 2020, with Fury emerging victorious via seventh-round technical knockout to capture the WBC and vacant Ring magazine heavyweight titles. Wilder...

user178758

07:33

\o @Slate Welcome 🙏

love_sodam

08:23

Let $X$ be a finite set and $\mathcal{F}\subset\mathcal{P}(X)$ i.e., subcollection of subsets of $X$, such that any $F_1,F_2\in\mathcal{F}$, $F_1\cap F_2$ is nonempty i.e., pairwise intersects. I want to show that by deleting a point $x\in X$ (so deleting any $F\in \mathcal{F}$ with $x\in F$), we can color $X-x$ by two colors such that for any $F'\in\mathcal{F}'$, $F'$ contains two different colors. Here, $\mathcal{F}' = \{F\in\mathcal{F}\mid x\notin F\}$.

Any thoughts? I think the choice of $x$ is important so I need to delete a point $x$ with some special property like a point that contained $F$'s the most.

SMC not Sven Magnus Carlsen

09:00

Hi @BalarkaSen, no never seen a Siefert surface of the trefoil knot?

This is what you meant right? mathcurve.com/surfaces.gb/seifert/seifert.shtml

Balarka Sen