Mathematics - 2017-05-23 (page 1 of 8)

I don't want to get into a big discussion, but grades are reality, and struggling with math is part of learning. So having answers available means that typical students won't learn s**t.

Daminark

Hai everyone

Ted Shifrin

Hi Demonark.

Eric Silva

yo

Ted Shifrin

You can find info on my profile page, I think, Bob.

Bob

you mention subject areas not the actual book but that should be good enough, thanks

Ted Shifrin

12:49 AM

Well, the diff geo notes are linked directly. But yeah, you can find 'em if you care.

Bob

what is the level of the diff geo notes?

Ted Shifrin

My books are challenging, and therefore not super popular. Although the free diff geo text is used all over because it's free. :P

Senior undergraduate.

I have selected answers at the back (even some proofs) in my last three books. None in the first.

Eric Silva

can never underestimate the power of free stuff

Ted Shifrin

Well, most free stuff is worth approximately what you pay for it :P

Daminark

Truly

Ted Shifrin

12:51 AM

There's a lot of free garbage out there.

Bob

I have found that there is a lot of good free stuff out there

Eric Silva

that's true

Ted Shifrin

Well, I'm pickier about what's good.

But, yes, there are some seriously good notes by good people out there.

A small percentage of what's out there free.

Semiclassical

hi chat

Bob

Ted, since you are a professor, let me ask you a Univ. Teaching Question

Ted Shifrin

12:54 AM

hi @Semiclassic

Eric Silva

hi @Semi

Bob

What do you think about a Computer Science Professor who gives out a programming assignment and then says to his students, that due to time constraints it will not be properly graded.

Ted Shifrin

Most homework is, IMHO, rarely "properly graded."

Eric, what's the most interesting geometry homework question this week?

Bob

I thank you for your honesty

Eric Silva

Uhh well there's one to prove a formula like $\Delta \nabla f = \nabla \Delta f + \text{Ric}(\nabla f)$, that's p cool

Ted Shifrin

12:57 AM

Most people here will give me credit for honesty, if nothing else :P

Is there a good application of that, Eric?

In general it's Bochner-type theorems when I care about things looking like that.

Eric Silva

You can use it to get a lower bound of the eigenvalues of $\Delta$ when you have Ricci bounded below by $ng$

No @Daminark

Santosh Linkha

hello everyone .. I have one quick algebra question. What is splitting field for $x^4+x^2+1$ over $\Bbb Q[x]$?

Ted Shifrin

Bob, I actually spent a ton of time grading my own homeworks in upper-level and graduate courses, but generally there are graduate students who grade for part of their support, and they are given not enough time to do a decent job. In lower level classes, typically nothing's graded at all, or else there's on-line homework.

Work it out, @Santosh.

Daminark

Triangles are flipped, now I see it

Ted Shifrin

What are the roots?

Demonark has vertical dyslexia.

Daminark

1:00 AM

Lol

Santosh Linkha

@TedShifrin I have already worked it out, I think it's $\Bbb Q[\omega, \sqrt 3]$. Just want to make sure

Ted Shifrin

We don't have these things memorized, @Santosh.

Eric Silva

that polynomial factors

Ted Shifrin

Better just to do quadratic formula.

And then find square roots explicitly.

Santosh Linkha

yeah but hint is given as to look for $(x^2-1)(x^4+x^2+1)$

Ted Shifrin

1:01 AM

@Santosh: What degree extension do you say you have?

Santosh Linkha

4

Ted Shifrin

I don't need that hint.

I say you're wrong, @Santosh.

Santosh Linkha

umm .. how?

Eric Silva

It should be degree 2

Santosh Linkha

hold on a sec

Ted Shifrin

1:02 AM

Yup, it's degree 2.

Santosh Linkha

6

Hold on, isn't $[\Bbb Q[\omega, \sqrt 3]: \Bbb Q] = 6$?

where $\omega$ is cube root of unity

eh .. no :(

Ted Shifrin

Whoa. Very much not 6.

Santosh Linkha

okay .. 2 sorry :( I wasn't careful

Ted Shifrin

Right, it's actually 2.

Eric Silva

It's the same as adjoining $\sqrt{-3}$

Ted Shifrin

1:06 AM

Right.

One needs to pay detention!

Santosh Linkha

*attention you mean :D

Ted Shifrin

No, I put you on detention when you don't pay attention.

Santosh Linkha

haha

Ted Shifrin

The regulars know to ignore my "humor."

Santosh Linkha

how does this relate to the hint $(x^2-1)(x^4+x^2+1)$

hold on a sec ... this has degree of TWO too!!

Ted Shifrin

1:08 AM

Same splitting field, and easier polynomial.

Daminark

Preemptively put humor in quotes, I see

Ted Shifrin

But if you use the quadratic formula and then think about the unit circle, you can get there directly, @Santosh.

TheGreatDuck

@TedShifrin I'd put people on detention for blinking at me the wrong way.

:p

Ted Shifrin

doesn't roll any eyes at Demonark this time

Santosh Linkha

All right thank you :)

All

Eric Silva

1:09 AM

recognizing $x^{4} + x^{2} + 1 = (x^{2} - x + 1)(x^{2} + x + 1)$ also makes it easy

Santosh Linkha

I later found that feeding into wolframalpha

I thought it was irreducible

Ted Shifrin

My approach is minimally tricky :)

Eric Silva

True true

Ted Shifrin

Since I am sometimes fond of tricks, I wish to emphasize that point. :)

Santosh Linkha

I have another question, but I haven't really thought of it. I will come to it later to confirm.

see you later~

Ted Shifrin

1:11 AM

Bye

I wasn't so interested in that geometry problem, Eric. Next? :)

Eric Silva

There all pretty much like that unfortunately

Ted Shifrin

Meh.

Eric Silva

it's mostly just confirming stuff we skipped in class

Ted Shifrin

Right. That's called not taking much effort to write homework.

Eric Silva

and calculating coordinate expressions for all our things

Daminark

1:13 AM

Oh @Ted not sure if I told you this but Neves defined Euler characteristic out of vector field indices

Eric Silva

yeaaah

Ted Shifrin

That's a theorem, not a definition, Demonark.

I mean, it takes some work even to prove well-definedness ... and the rest of the world knows other definitions. That said, G&P define it in an odd-looking way, too (self-intersection number of the diagonal in $M\times M$) — but one can relate that to the usual one ... and to Neves's with some knowledge.

Eric, in fairness, some of my exercises I've sent you are of a similar nature, but a fair number are not.

(overuse of "fair" unintentional)

Daminark

Oh he told us that, he was all like "If you know what Betti numbers are this is a theorem, but you don't so it's a definition"

Ted Shifrin

He still needs to work pretty hard to prove well-definedness.

You need to interpret that as an intersection number in the tangent bundle.

Eric Silva

I mean doing these kinds of things are important I think, I usually do them regardless of being assigned them though, so it's a little disappointing

Daminark

1:16 AM

Also @EricSilva same he's running low on time to do stuff so he was just like "Yeah I'll just relegate results to the homework"

Ted Shifrin

well, no grad students bother to do 'em, Eric.

I'm OK with that, Demonark. That's not my complaint. I just want something more in homework, too.

Daminark

I haven't gotten a chance to look at the pset yet but I imagine that either he did put more stuff, or there were enough things we skipped to constitute a pset

Eric Silva

If i didn't do these sorts of computations id be so lost

Ted Shifrin

in diff top, I always have more problems to assign than students have time to work.

And Eric's seen my grad psets — they're way too long, too. I basically told grad students they'd get an A if they worked an average or 3 or 4 on each.

If I expected anything close to 90%, it'd be hopeless.

Daminark

3 or 4, out of how many?

Ted Shifrin

1:20 AM

usually 10 or a few more.

Daminark

Ah, I see. Wow

Ted Shifrin

that's for the grad course (my colleagues teaching advanced grad courses gave automatic A's, seriously).

So my deal was, if you come to class, I'll give you a B, but you have to earn an A by doing a reasonable amount.

I once had only undergrads take Riemannian geometry. They were damn good.

Eric Silva

they seemed reasonably sized to me but ive been taught by people who think 40 problems are reasonable

Ted Shifrin

None of them ended up with a Ph.D. in math, though. A lawyer, an economist, a preacher ... :D

not 40 problems at that level, Eric.

Eric Silva

that sounds like the set up to a joke

Daminark

1:21 AM

Neves' psets averaged 6 problems

Ted Shifrin

NO, it wasn't a joke.

Eric Silva

true true

Daminark

@EricSilva Souganidis intensifies

Ted Shifrin

My diff top was way more than 6, Demonark. Way.

And I graded 'em all.

Daminark

Lol in 207 we usually had 5 problems per pset graded

Ted Shifrin

1:22 AM

I'd send them to you, but you probably wouldn't want them.

Daminark

Well, 5 in analysis and 5 in linear algebra

TheGreatDuck

@TedShifrin Would there be any (realistic) way using just derivatives to actually calculate the motion upon an arbitrary parametric surface at a certain point in a certain direction?

Eric Silva

grading takes soooo long

TheGreatDuck

or would that be a bit far fetched to attempt?

Ted Shifrin

@EricSilva: In some sense, it's the most important thing I did as a teacher. Not sure the students all appreciated it, though.

Daminark

1:23 AM

Depending on how many things I'll have going around in the void between class and the bootcamp that may actually be worth @Ted

Ted Shifrin

Duck, the first derivative is just the direction. What do you mean?

Derivatives never tell you global things unless you know things are real analytic and you have all derivatives.

Eric Silva

I think it's much easier to appreciate people grading and giving feedback once you've done it

TheGreatDuck

err.. wat?

Ted Shifrin

Yup, Eric.

TheGreatDuck

parametric surface

not a parametric curve

Ted Shifrin

1:25 AM

The direction tells you to compute a directional derivative, so that gives the tangent vector to the surface in that direction.

As usual, your question is totally vague.

TheGreatDuck

I never said the derivative in that direction

Ted Shifrin

You said "a certain direction."

Daminark

Oh @EricSilva there's apparently reason to believe that next year Soug might not be as bad as he was last year in functional

Ted Shifrin

That is a tangent vector to the surface.

TheGreatDuck

NO

Eric Silva

1:26 AM

I always give really extensive feedback on homeworks, last quarter I remarked how a problem was related to QM on the pset of a physics major and he came to me and had a discussion about it, and it was honestly the highlight of my quarter

TheGreatDuck

I said a motion on the surface in a given direction

Daminark

Apparently part of the reason he assigned so many problems is that he felt the first quarter guy didn't do enough.

TheGreatDuck

i.e. a geometric motion

Ted Shifrin

That makes no sense.

After you take multivariable calculus, we'll talk again about it.

TheGreatDuck

...

alrighty

thanks for the insult

Ted Shifrin

1:27 AM

@EricSilva: good for you for doing a beyond-the-usual job.

Daminark

Nice @EricSilva

TheGreatDuck

@TedShifrin you do realize I'm three classes past that already, right?

Eric Silva

it made me think I'd probably like teaching a lot

TheGreatDuck

I'm referring to surface geometry. Like spherical geometry

Ted Shifrin

No, I have no idea. You have a propensity to ask ill-posed and vague questions.

I wrote a differential geometry text, Duck, so I know a little about surface geometry. "Like spherical geometry" is not helpful.

TheGreatDuck

1:30 AM

surface geometry. Spherical geometry is an example of surface geometry.

Eric Silva

Oh @Ted looking at it now one of the geometry problems is like a gateway to something interesting

It asks to show that you get eigenfunctions of $\Delta$ on the sphere by restricting certain linear functions

So it's like a gateway to spherical harmonics

Semiclassical

Spherical harmonics, ugh. Hard to like those after doing E&M.

Daminark

Did they give you a headache @Semi?

Semiclassical

They're just kinda painful

Daminark

Or is it that E&M gave you some other way of doing life that worked and are preferable in that context?

Ah

Eric Silva

1:40 AM

I remember getting really excited about them in high school chem

Daminark

Neat

Eric Silva

I remember getting super wigged out when my teacher told us we could predict the shapes of the orbitals using math

Daminark

Oh so is that how it goes? I see

I remember getting explanations at like, so many different levels for orbitals at various times and I'm like, wait hold on wat?

Eric Silva

The teacher ended up giving me a book on PDEs and quantum chemistry as a graduation gift that explains it properly but i haven't read it

TheGreatDuck

@TedShifrin this might help a bit. I was trying to shy away from an isometry as I recognize that not all mapping like this are necessarily isometries. Hopefully a diagram clarifies things.

if anyone else happens to have any insights that would be greatly appreciated.

Dair

1:55 AM

"...is it possible to find B if we know the direction V and distance d." it sounds like you've already found it...

you sound like you need a hillclimbing algorithm

TheGreatDuck

@Dair not necessarily. This is distance along a curve. Tricky stuff.

Dair

but like your question is not exactly that... however, I agree with Ted: your question is kind of ill posed.

TheGreatDuck

@Dair exactly how is it ill posed?

Dair

You have A and the distance d and what I would assume is the directional unit vector, you can find B's coordinates trivially by adding the two.

TheGreatDuck

no

d is the arclength between a and b

not the euclidean distance in space

Dair

2:00 AM

and V is in euclidean space?

TheGreatDuck

v is a real vector, yes

it's a tangent vector

(there probably is a small bit a of vagueness in that I am simply meaning V to be any clear way of distinguishing a direction)

Daminark

How am I sleepy at 9PM? Something does not compute right now

TheGreatDuck

wat

Dair

@Daminark You are a sloth how are you not sleepy all the time?

TheGreatDuck

"$f(tV + A)$" <=== that doesn't make sense

Dair

2:02 AM

Wait I messed up. yeah.

Daminark

Good point @Dair

TheGreatDuck

basically, I'm trying to determine the general formula for rigid motions on a parametric surface

Eric Silva

@Daminark how many of you sloth fanatics at uc have ever held a sloth

TheGreatDuck

the problem with the term "rigid motion" is that it claims it is distance preserving with respect to arc length.

Daminark

shrugs

TheGreatDuck

2:03 AM

obviously, I only care about precisely one point

Daminark

The sloth thing was just a prank at first

Eric Silva

I found it incredibly confusing

Daminark

I didn't anticipate the direction it went

My original intention/idea was that we'd all change our profile pictures to sloth photos and then never speak of them again

Eric Silva

i dont understand

Daminark

Then on the morning when everyone came in I was like, wait I need to delete our original post lest everyone realize what we're doing here

Couldn't reach it due to all the sloth photos, very pleasant surprise

Eric Silva

2:06 AM

i never spoke to anyone from the page, so i just thought that people who went here weren't funny

Daminark

Is it that you were on the page and didn't interact?

Eric Silva

i commented a couple times but i never really tried to speak to anyone

Daminark

So what led to that particular impression?

Eric Silva

cause the sloth thing was super inside joke-y

Daminark

Ah

Eric Silva

2:11 AM

i generally didnt think most of the humor of that page was funny :P

Daminark

Yeah I was really surprised when so many people in round 2 adopted it and it became a meme. Of course I welcomed that but like, originally it was supposed to confuse people and probably die off in a week

Lel, yeah I was posting a lot along with a few other people. We have a pretty particular brand of humor

Eric Silva

yeah: anti-funny

Dair

@TheGreatDuck I feel like you just need to take <x, y> component of V and just travel d in that direction... i'm too tired to justify though.

Daminark

Lol, "funny" is entirely subjective, anyone could easily say the complement of that is what's anti-funny

TheGreatDuck

@Dair that I can verify with a strong NO.

Eric Silva

2:14 AM

No, I am the only arbiter of funny

there can be no other

TheGreatDuck

@Dair go get some sleep.

Daminark

Darn shakes fist

Dair

@TheGreatDuck I'm tired... but i don't want to go to bed.

Daminark

@Dair that feeling is real

Dair

Oh, you could have some weird normal vector thing that doesn't work... ugh...

Ted Shifrin

2:17 AM

Duck, I have a brief response to your post, but I don't want to get into a whole long thing. First of all, a sphere has the most symmetry you can have. Most surfaces have no motions at all. A surface of revolution has rotational symmetry. ... Next, close to a point A, there is always a shortest path to nearby points. The shortest path is called a geodesic on the surface. You can certainly follow a geodesic in direction $\vec v$ a distance $d$.

TheGreatDuck

@Dair it just fails in general to be frank.

Ted Shifrin

However, if the distance $d$ you specified is greater than the shortest possible distance to B, then there will be zillions of possible paths, all even with the same initial direction $\vec v$.

Hi @Dair. Even for an old guy like me, this is early for bed :P

Dair

@Ted Virginia tho... haha.

TheGreatDuck

@TedShifrin B is an unknown point. Rather the direction and distance are known and I wish to traverse the geodesic in that direction.

Ted Shifrin

Oh, well, a little more excuse, then, @Dair.

TheGreatDuck

2:19 AM

I know motions as in distance preserving mappings don't exist, but this mapping need not be "distance preserving"

Ted Shifrin

Well, fine, then, Duck, there's a unique geodesic (but they're very hard to find in general) in any given tangent direction.

You don't have a mapping.

TheGreatDuck

well fair enough. bad choice of words. :p

is there any known algorithm for doing so that only requires the derivative operation and elementary functions?

Ted Shifrin

If you specify $\vec v$ and distance $d$, then there's always a unique mapping that takes you from $A$ to the unique point along the geodesic in direction $\vec v$ a distance $d$ along it.

TheGreatDuck

(assuming that the surface's parameterization only uses elementary functions)

Ted Shifrin

Once you get past a sphere, geodesics are incredibly complicated and there are no closed-form formulas in general. For surfaces of revolution, there are.

TheGreatDuck

2:20 AM

and tube plots

Ted Shifrin

No, in general definitely not elementary functions.

TheGreatDuck

I know that from how I built the 3D models last fall. :p

Dair

@TheGreatDuck Do you need a precise representation? There might be approximate computational methods?

Ted Shifrin

No, geodesics on a general tube are not much easier. Only certain ones are.

There are always numerical methods, but almost impossible to solve the system of nonlinear second-order differential equations.

TheGreatDuck

@Dair it needs to be precise and efficient.

Ted Shifrin

2:22 AM

Nope.

Dair

but does it need to be accurate and efficient? :P

TheGreatDuck

@TedShifrin I don't think people would want to wait a while just to have something take one step in space. :p

Eric Silva

I imagine we have some pretty fast ways to find approximate geodesic paths, it sounds like something that should be pretty common in computer graphics operations

TheGreatDuck

@Dair I tend to use the two words interchangably. I just mean it needs to give decent results.

Daminark

@Dair The algorithm spits out exactly 7 no matter the input, runs in $O(1)$

TheGreatDuck

2:24 AM

@EricSilva Actually not at all. Where do you get that from?

Eric Silva

pls @Daminark

Ted Shifrin

They can be solved numerically quite fine, @EricSilva.

Anyhow, I'm gone.

TheGreatDuck

cya

Eric Silva

Duck, hhoppe.com/geodesics.pdf

see ya Ted

Dair

cya Ted

Daminark

2:25 AM

@Duck Accurate means that you're not spitting out 12 when the answer is 190, precise is when you have a lot of resolution regarding decimal places

See you @Ted!

Dair

@Daminark Precisely.

Daminark

@Dair but what about accurately?

TheGreatDuck

@Daminark a balance of both is always good.

@TedShifrin I'm referring to tube plots which are those weird 3d models I showed you a while back. Pretty sure those things work pretty well. Sharp corners are always the tricky thing.

Eric Silva

Of course you have fast accurate methods, we have loads of fast accurate numerical methods for ODEs, we've had good methods for a long time

Daminark

In general you don't want to be lacking in either, just that perhaps if cutting the error range by $35$ percent brings you from linear time to quasipolynomial (to be ridiculous), then merp. You don't want to ever be inaccurate though

Anyway, I'll leave this discussion to experts more because I don't know algorithms yet

Eric Silva

2:28 AM

@Daminark, it depends a lot on your applications at the end of the day

Dair

unless the leading coefficient is ridiculous for the algorithm...

Looking at you matrix multiplication

Daminark

I guess that depends on the problem size

Wait is that process slow?

I mean I guess this doesn't surprise me

Semiclassical

Eh, matrix multiplication isn't bad if you can afford to diagonalize first.

Dair

Matrix multiplication algorithm

Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in many fields including scientific computing and pattern recognition and in seemingly unrelated problems such counting the paths through a graph. Many different algorithms have been designed for multiplying matrices on different types of hardware, including parallel and distributed systems, where the computational work is spread over multiple processors...

Eric Silva

the obvious one is like $O(n^{3})$

Daminark

2:30 AM

@Semi Wait what would you do to diagonalize quickly?

Semiclassical

Didn't say diagonalization was quick.

Dair

@EricSilva You can do better. The best asymptotic one has a ridiculous leading coefficient though which makes it impractical for most applications...

also, cache hits are pretty important too.

Eric Silva

Oh im very aware you can do better

Semiclassical

I said that, if you're in a position to do it easily, then matrix multiplication isn't too bad. (Though I really have in mind taking powers of a given matrix.)

TheGreatDuck

@Dair the problem with those sorts of approximations in my situation is that this is basically the math that makes a guy walk on a parametric surface in space and so literally that one mapping is one step. It needs to be realistically fast so that there is no lag.

Dair

2:33 AM

@TheGreatDuck What approximations are you referring to?

Daminark

@Semi Ah, I see

TheGreatDuck

@Dair the approximations to find that B point in my surface geometry problem. I'm trying to make a guy walk.

perhaps the better way to do it would be to do the calculations ahead of time...

Dair

@TheGreatDuck You could probably construct a super wonky surface that makes it difficult for a computer. You could also have nice surfaces that you could heavily optimize the algorithm for.

TheGreatDuck

yeah

Dair

if you don't need any user interaction you might consider precomputing.

from what I understand you can try it naiively, see if it works. If it does, you're basically done. If it doesn't you have to take 1 of 2 routes: 1. Precompute. 2. Understand what surfaces you are analyzing and optimizing the algorithm for that particular set of surfaces.

Transcript for

$Mathematics$ Mathematics