Mathematics - 2023-07-27 (page 2 of 2)

s.harp

23:03

@Semiclassical Re this picture: Chat GPT can string together coherent sentences. It can create poems and write songs about a theme (sometimes als do good ones about mathematical themes). Its really a different beast compared to a parrot. That said building a network that mimics the brain of a parrot would also be an incredible achievement!

Semiclassical

and if you ask it about any of those sentences or poems or songs 10 minutes later, it will have no bloody idea what you're talking bout

leslie townes

semi: huh?

Semiclassical

@leslietownes chatGPT can generate sentences etc. but if talk to it for 10 minutes and then ask what it was talking about 10 minutes ago, it has no idea

leslie townes

semi: what?

Semiclassical

...

s.harp

23:10

I asked it to write a poem about the slice theorem:

In topology's realm, hear the tale,
Slice Theorem unfolds, we set sail,
Lie groups act with grace,
On manifolds they embrace,
In slices, their orbits prevail.

At a point, we seek a smooth slice,
Transverse to the group's sweet device,
Tangent space aligns,
With stabilizer's confines,
Geometry entwined, oh so nice!

With G's action, M finds its way,
Local structure in a grand display,
The Slice Theorem's art,
Reveals a wondrous chart,
Where symmetries and manifolds play.

CroCo

Hi everyone, I need to convert improper rotation matrix $\det(R)=-1$ to quaternion. I haven't been able to find any resources regarding this conversion.

this is my question for more info

1

Q: How to obtain the quaternion from this Rotation matrix

Consider the frames in this picture to be stationary. The orientation of {1} with respect to {0} is obtained by inspection as $$ R^0_1 = \begin{bmatrix} 0&0&-1 \\ 1&0&0\\ 0&1&0 \end{bmatrix} $$ Let $p^1 \in\mathbb{R}^3$ represents a point in frame {1} (i.e. say $p^1=[1,0,0]^T$). I need to represe...

rotations

For mapping the position of the haptic device to a 3D robot simulator, I have constructed this matrix. In spite of that, the matrix performs well.

Xander Henderson

@s.harp Doggerel.

Pure and simple.

CroCo

The robotic book I'm reading tell me this

https://i.sstatic.net/MdiOT.png

This is exactly what I've been informed in the comment section in my question by others.

Jakobian

@Semiclassical more like 10 seconds

Xander Henderson

@CroCo GPT wrote your book?

CroCo

23:21

@XanderHenderson what is GPT?

Jakobian

"non-zero eigenvector"

really now

Xander Henderson

@CroCo A robot.

CroCo

The book is Robot Modeling and Control 2nd by Mark W. Spong and other.

Xander Henderson

Surprised you haven't heard of it.

Semiclassical

i mean. maybe it's referring to "an eigenvector which isn't annihilated by the matrix"

(i.e., an eigenvector whose eigenvalue isn't zero)

Jakobian

23:23

@Semiclassical non-zero eigenvector whose eigenvalue is zero

Semiclassical

...oh, hell

Jakobian

no wait it's 1

CroCo

@XanderHenderson A robot? I haven't heard of it before even though this is my field. What kind of robot is this?

Semiclassical

not a mechanical robot, to be clear

Xander Henderson

@CroCo I'd suggest a quick Google search.

Jakobian

23:23

either way they don't know that eigenvalues are non-zero by definiton

Xander Henderson

It has kind of been in the news a lot in the last half year or so.

Semiclassical

@Jakobian um

Jakobian

eigenvectors*

Semiclassical

right

CroCo

@XanderHenderson just did. Does it help me in my math though?

Jakobian

23:25

it's late and my eyes are closing

that's my excuse

CroCo

are you referring to this
https://en.wikipedia.org/wiki/GPT-3

Xander Henderson

@CroCo Sure. Or en.wikipedia.org/wiki/GPT-4 .

CroCo

@XanderHenderson As far as math is concerned, I am unsure if this kind of stuff will be useful.

s.harp

@XanderHenderson I mean the end result is a limerick that talks about the slice theorem in a correct way. I don't see how you can think that this isn't incredible. Nevermind if the poem itself is crap

Xander Henderson

@CroCo It isn't at all useful. I was making a joke.

CroCo

23:29

I see.

Semiclassical

i'll admit, i do like this chatGPT-generated summary of its own abilities here: reddit.com/r/ChatGPT/comments/13k9bje/comment/jkj30ho/…

Jakobian

@CroCo It can do some basic calculations but it breaks for advanced stuff

ChatGPT-4 is a little bit better but still sucks in that regard

those programs weren't made for math, so that's understandable

CroCo

Math teachers will be happy. No cheating I guess

which is good.

Semiclassical

thing is, i don't hate chatGPT itself. i just hate how people frame it

Jakobian

@CroCo my opinion is that people cheating won't benefit from the course anyway

s.harp

23:33

Wonder what will happen to mathematicians once somebody writes a bot that can properly navigate LEAN

Jakobian

you have to want to learn to learn something

math teachers shouldn't worry about cheaters

Dannyu NDos

My most awkward moment with ChatGPT was it claiming the terminal object in the category of integral domains is the rationals.

The object really doesn't exist.

CroCo

As a kid, I made a terrible mistake by relying on calculators. There was no one to correct me, which is unfortunate. There is never a mistake I make again.

Dannyu NDos

4

Q: Is there the terminal object in the category of integral domains?

I wondered what the initial object and the terminal object are in the category of integral domains. Simple argument: Since integral domains do not put an additional restriction on the definition of a ring homomorphism, integral domains should inherit the hom-set of rings (with unity). That means ...

abstract-algebra ring-theory category-theory integral-domain terminal-objects

Xander Henderson

@Semiclassical This. It is an interesting toy, but it isn't magic.

Semiclassical

23:38

exactly

if it convinces people it's intelligent, that just demonstrates the human capacity to infer intention from experience

CroCo

Is there a way to handle this matrix?
$$
R = \begin{bmatrix} 0&0&-1 \\ 1&0&0\\ 0&1&0 \end{bmatrix}
$$

leslie townes

one thing chatGPT seems particularly good at is getting people on here to talk about chatGPT

Semiclassical

yes

leslie townes

croco: "handle" it how?

CroCo

@leslietownes I need to convert it to quaternion but $\det(R)=-1$. Online software doesn't yield the correct version, even Matlab.

I'm not able to find a proper way to convert it.

this my question in here
https://math.stackexchange.com/questions/4743537/how-to-obtain-the-quaternion-from-this-improper-rotation-matrix

leslie townes

23:42

i don't know exactly what "convert it to quaternion" means but if you compose it with a reflection that has det -1 (e.g. the diagonal matrix with -1, 1, 1 as its diagonal), could you convert that to quaternion? and could you convert the diagonal matrix representing the reflection to quaternion?

Xander Henderson

There, now it has a handle.

Dannyu NDos

That said... Anyone been annoyed by Fraleigh and Wikipedia mismatching on ring terms?

What Wikipedia refers by "ring" is "ring with unity" in Fraleigh.

Xander Henderson

@DannyuNDos If you didn't like it, why'd you put a ring on it?

leslie townes

what's that, whether rings have 1 in them or not? that goes beyond fraleigh.

CroCo

@XanderHenderson I've laughed so hard. Nice ear.

Dannyu NDos

23:43

Good one

leslie townes

i think ring = 'ring with unity' is the majority rule in 2023, so if we have to be annoyed by someone, it should be fraleigh and not wikipedia. it's also the objectively right definition.

Semiclassical

plus rng = ring without identity is a good enough name for the other case

Xander Henderson

@Semiclassical Ugh... I hate that joke-which-is-now-taken-seriously. X(

Words need vowels!

Dannyu NDos

Yeah... Guess "rng" is such a good pun on eliminating the requirement of identity.

"rig" and "rg" also

CroCo

@leslietownes see this

Semiclassical

23:46

rig already shows up for rigged hilbert spaces tho

hmm

does y count as a vowel

i mean literally no

if you don't count y there's plenty of words without vowels, but that does seem contrary to the spirit of it

CroCo

@leslietownes for my rotation matrix, I get $\boldsymbol{r}^0_1=0.5-0.5\hat{\imath}+0.5\hat{\jmath}+0.5\hat{k}$ which doesn't yield the correct result as the case with rotation matrix.

$\boldsymbol{p}^0 = \boldsymbol{r}^0_1 \boldsymbol{p}^1 (\boldsymbol{r}^0_1)^*$

in rotation matrix, $\boldsymbol{p}^0=R^0_1 \boldsymbol{p}^1$.
with proper definitions for $\boldsymbol{p}$

Dannyu NDos

What about Munkres and Wikipedia mismatching on separation axioms? Munkres' "regular" is Wikipedia's "regular Hausdorff" and so on.

And there's also Steen & Seebach. Dang.

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