Mathematics - 2024-04-25

Jakobian

00:52

@leslietownes more like cateboring

YourLordJoyBoy

01:29

@Obliv Figures that when I ask for help I'm kicked out where I got the question.

Anyways, I've got another one. Its involving finding difference, but as I've learned its a trick. I got this far with the problem, the invisible 1 changed some of the signs. $-2d^3 + 6d - 9 + 7d^2 - 5d - 13$
Originally it was $(-2d^3 + 6d - 9) - (-7d^2 + 5d + 13)$. I got this far alone and want to confirm my thought process, so I'd like a second opinion.

leslie townes

the core principle with polynomial arithmetic is recognizing what the monomials are. in one variable d, that's powers of d, in a multivariable setting it is subtler.

the addition is defined so that you can "collect" like monomials and "add their coefficients." but none of this is well-posed without a lot of background understanding. "6a^4 b^4 +(8a^4 b^4 - 9b)" is not in anyway disqualified as an answer to an ill-formed question that just asks you to "add" those polynomials

Xander Henderson

Agreed. That is an ill-posed question.

leslie townes

same reason that "31 - 26/2" is a perfectly good response to "what's 31 - 26/2." specifying the form of an answer to the question is 99% of the question.

Xander Henderson

Were I to give a similar question to my students, the instruction would likely be something like "Simplify the following polynomial expression so that there is only one of any term of the form $Ca^m b^n$, where $C$ is a real number, and $m$ and $n$ are natural numbers."

I would need to word smith that a bit more, as it is a bit hard to grok, but I think that the general intention of the question is to get you to "combine like terms".

In other news, it is currently 77°F inside. Is it time to turn on the AC?

YourLordJoyBoy

@XanderHenderson THANK YOU! Thats what I thought but wasn't sure. As soon as I finish these, I'm goin to sleep.

Xander Henderson

01:42

@YourLordJoyBoy Again, I don't know that this is what the question writer is asking, but that is my best guess.

Obliv

@YourLordJoyBoy Yeah so you basically distribute the minus sign. Also the first parentheses is just another implicit $1(...)$ so we just have $-2d^3+6d-9-1(-7d^2+5d+13)=-2d^3+6d-9+(-1)(-7)d^2+(-1)(5)d+(-1)(13) = -2d^3+6d-9+7d^2-5d-13)$

YourLordJoyBoy

@XanderHenderson @Obliv No worries guys, I got it. :D I combined em and it ended up being $−2d^3+7d^2+d−22$

Xander Henderson

Actually, I would probably give the instruction "Rewrite the polynomial in standard form," where "standard form" is something that would have been discussed previously in class---roughly, write the terms in order of decreasing degree, and if two terms have the same degree, write them in lexicographical order; no two terms should contain precisely the same combination of variables, i.e. collect "like terms").

YourLordJoyBoy

@XanderHenderson I believe that was in standard form, the final answer I mean. And now I'm doing a similar one, and trying to combine $9/7z with 15z$. Other than that the problem is cake.

$9/7z with 15z$

leslie townes

yeah, generally speaking, you make a problem like that unambiguous by specifying the desired form of the answer

we can all enjoy the ambiguity of "9/7z"

YourLordJoyBoy

01:49

@leslietownes a fraction combining with a number, each with the variable z

$9/7$

Xander Henderson

That clearly means $$ \frac{9}{z}\cdot 7.$$ Doi.

Hrm... or maybe $$\left\{ \frac{9}{7}, \frac{9}{z}\right\}.$$ Yeah, okay. I agree. It's ambiguous.

YourLordJoyBoy

@XanderHenderson How do you do the fraction bit? And I'm trying to combine the fraction that's multiplied by z with 15z.

Xander Henderson

@YourLordJoyBoy I think that, perhaps, the joke has flown past you a bit too quickly. Something that you should keep in mind is that Xander lies.

YourLordJoyBoy

@XanderHenderson To be fair, any joke'll pass me quick when I'm this tired. XD I'm tempted to dub you Usopp now with the lying habit though.

Xander Henderson

@YourLordJoyBoy Sadly, your joke flew over my head.

Is Usopp some gen alpha thing?

YourLordJoyBoy

01:56

@XanderHenderson Consider it vengeance XD

I'll probably explain it in time

Unless someone else gets it and decides to.

Xander Henderson

Google says its some shonen BS.

YourLordJoyBoy

@XanderHenderson Yes, its a character who is always telling tall tales. XD

Xander Henderson

(The question is, how many otaku / weebs do I piss off by calling One Piece 'some shonen BS'?)

YourLordJoyBoy

@XanderHenderson Not me since I could care less who hates it XD

$$\frac{9}{7}{z} + {15}{z}$$ Wonder if I did this right

$$\frac{9}{7}{z} + {15}{z}$$

ABSOLUTELY SCREWED THAT UP! :o

FTR, only adding a star to those two messages.

Xander Henderson

@YourLordJoyBoy I don't hate it. I simply have no feelings toward it at all.

I bounce off of most shonen BS pretty hard.

YourLordJoyBoy

02:05

In other words, @XanderHenderson just no care.

For me however, it heavily impacted meh life during hard times.

And you're still Usopp to me XD

But seriously its one of those things, you're going through crap times irl and something crazy pops up that proves to be a fantastic escape from the reality bs in your life.

Thorgott

02:20

@XanderHenderson I don't care about One Piece, but I'll take issue on general grounds of reductiveness

Xander Henderson

03:08

@Thorgott Good. That was the reaction I was hoping for. :D

Semiclassical

me: modifies someone else's Tikz code to make a mass spectrometer

result:

Xander Henderson

@Semiclassical I might try to learn Tikz someday. But I learned pstricks such a long time ago, and Tikz looks so very annoyingly not at all the same.

Semiclassical

Xander Henderson

That looks like a butt.

Semiclassical

i'd agree, except for the shaft going upward. but that makes it worse

Xander Henderson

03:11

Oh, sure. It could be a dick'n'balls.

I see that, now that you mention it.

Maybe it's both things!

Right. I need to go to bed. It is so late.

It is after EIGHT PM!

Silly Goose

05:15

when people say the killing form is "ad" invariant, do they mean little or big adjoint?

concretely, with matrices little ad is the commutator over lie algebra $[X, -]$ and big adjoint is lie group conjugation over lie algebra $g(-)g^{-1}$

John Zimmerman

09:50

modular form on disk

keyword: j-invariant

Jakobian

11:04

@SineoftheTime Do you like Caparezza?

Soumik Mukherjee

13:24

What is this figure called?

Xander Henderson

@SoumikMukherjee A "book cover".

Soumik Mukherjee

Heh

YourLordJoyBoy

@XanderHenderson There goes Usopp again XD

Soumik Mukherjee

@XanderHenderson Sorry about my behaviour the other day

Due to the current political scenario of my country, I get tilted very easily

Lucky Chouhan

@SoumikMukherjee don't discuss politics as we know it is total mess in our country. We have best laws but worst implementation.

@SoumikMukherjee @XanderHenderson doesn't care about it.

Have some Maaza!!

Soumik Mukherjee

13:33

@LuckyChouhan General people not discussing about politics enough is what that led to this situation

Lucky Chouhan

Three Potato

Soumik Mukherjee

Also we don't have the best laws, eg uapa

Lucky Chouhan

:(

Jakobian

@SoumikMukherjee its called a link

its a collection of knots

Soumik Mukherjee

@LuckyChouhan Anyway I am sorry to you as well. I shouldn't have talked to you in that rude way.

Jakobian

13:36

@SoumikMukherjee the name here has an error. Its Karol Sieklucki

YourLordJoyBoy

I'm really frustrated with the political bs we're dealing with on my end. But yeah...

Soumik Mukherjee

@Jakobian oh thanks

Jakobian

Link (knot theory)

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a trivial reference link, usually called the unlink, but the word is also sometimes used in context where there is no notion of a trivial link. For example, a co-dimension 2 link in 3-dimensional space is a subspace of 3-dimensional Euclidean space (or often the 3-sphere) whose connected components...

Lucky Chouhan

Government has nothing to do with climate change, biodiversity, education, etc. They just focus on Trading and GDP lol, I don't know where we're going. Do you know what recently happened in Dubai? and few months ago in Joshimath. Many more such incidence yet to come. Anyway, who cares?

Soumik Mukherjee

@Jakobian Ooh, never noticed that

Lucky Chouhan

13:37

@SoumikMukherjee Never mind,

Xander Henderson

@Jakobian It's more than just a link. A knot is homeomorphic to a circle, and that figure contains more than just circles. I looks more like the complement of a link in a handlebody.

Lucky Chouhan

@XanderHenderson 🤗🤗

Jakobian

@XanderHenderson nah its a link in $\{x\in\mathbb{R}^3 : 1 < \|x\| < 2\}$

Xander Henderson

I don't buy that. Based on the overlaps, the bits between the two circles are meant to be embedded into a three dimensional space (there are four circles, linked together), and the surrounding annulus looks like a handlebody, i.e. a solid torus.

I think it is mean to be a link complement.

Jakobian

Sure its more probable that this is inside a cyllinder without the center of the cyllinder

but its a link, just embedded somewhere

that's how it looks like

$\{x\in \mathbb{R}^3 : 1 < x_1^2+x_2^2 < 2\}$ lets say

one potato two potato

13:44

One of the most interesting problems I've seen in AT is that $\Bbb CP^n$ is not a covering space of any manifold except itself. It's not very intuitive at first but if you know any continuous map on $\Bbb CP^n$ to $\Bbb CP^n$ has a fixed point (Lefschetz I guess) then it's believable

Xander Henderson

Given that knot complements are related to the classification of 3-manifolds, I would expect that the image is a link complement. But, at this point, we are each saying "No it isn't!" "Yes it is!" with no references, so this seems unproductive.

one potato two potato

now it's time for Thorgott to say that it's a consequence of some more general fact

2

YourLordJoyBoy

Also, I dare @XanderHenderson to look up what a Joy Boy is

one potato two potato

there're two types of general methods to create closed 3-manifold, one is dehn filling link complement and the other is heegaard splitting. Any closed 3-manifold can be realized in both two ways

Jakobian

@XanderHenderson I mean it might be a solid torus but I'm not sure why you brought up link complements

one potato two potato

13:49

the latter is quite straightforward if you know that any 3-manifold has a triangulation

Xander Henderson

4 mins ago, by Xander Henderson

Given that knot complements are related to the classification of 3-manifolds, I would expect that the image is a link complement. But, at this point, we are each saying "No it isn't!" "Yes it is!" with no references, so this seems unproductive.

Jakobian

I mean there's no real way from a picture to tell between a set and its complement. Both are drawn, necessarily, if one is

one potato two potato

I want to read the conformal dimension book you suggested @XanderHenderson long time ago but first the book is not very easy and second I'm studying different stuff so hard to make a spare time

YourLordJoyBoy

Was working with Gwen, she's really helpful with the factoring :D

Jakobian

@SoumikMukherjee @XanderHenderson I found exactly what the picture represents

Its the construction of the so called Antoine necklace

Antoine's necklace

In mathematics Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeomorphic to each other. It was discovered by Louis Antoine (1921). == Construction == Antoine's necklace is constructed iteratively like so: Begin with a solid torus A0 (iteration 0). Next, construct a "necklace" of smaller, linked tori that lie inside A0. This necklace is A1 (iteration 1). Each torus composing A1 can be replaced with another smaller necklace...

so we were wrong that those were links

(the same illustration is under an exercise in the book which asks to verify details of this construction)

one potato two potato

14:06

seems the construction is not very hard

Jakobian

@YourLordJoyBoy its a relatively new user and no one will know who you are talking about when you say "Gwen". Its not a regular user in this chatroom for sure

YourLordJoyBoy

Yes I'm aware

Gwen: math.stackexchange.com/users/1188598/gwen

When I say Gwen, thats who I speak of @Jakobian

Jakobian

@LuckyChouhan discussing politics is allowed, its just about peace of the chatroom itself. Since this can lead to controversial opinions and arguing with each other, which is what the actual problem is.

YourLordJoyBoy

@Jakobian Which is why I hardly speak of it. I'm surprised I didn't drive everyone into a frenzy when I talked about how humans and making stuff XD

I'm quite thankful for that actually.

Jakobian

how humans and making stuff

YourLordJoyBoy

14:12

How humans didn't make anything without taking something that already exists to make it.

We simply turned something existing into something else, but made nothing new except to us.

Jakobian

"but made nothing new except to us." this is where you're wrong

when an artist makes a stone statue, they only carve the stone that already existed

and its making something new. And "except to us" makes no sense

its new for us just as it is new for a pigeon or any animal

what is "except to us" supposed to mean here? Eh

YourLordJoyBoy

It's new to us. But the materials were there to begin with.

We didn't just make something out of nothing, that would be mind blowing.

Jakobian

so what

YourLordJoyBoy

Simply stating.

Nothing more or less.

Jakobian

In a sense we did make something out of nothing if we give some kind of structure to an already existing thing

YourLordJoyBoy

14:20

I suppose that's true/

I do wish I could post my study guide on here.

Jakobian

Not truly out of nothing since it requires resources. But the point is that changing something still counts as creating something

YourLordJoyBoy

@Jakobian Yes. Agreed. But what if we could make something into existance without any previous resources?

Jakobian

why am I supposed to care about that

YourLordJoyBoy

@Jakobian I just figured if it could be done, it would be AMAZING!

psie

Basic question. Are singleton sets closed in $\overline{\mathbb{R}}^n$?

The reason I'm asking is because it's claimed $A=\{(x,x)\in\overline{\mathbb{R}}^2:x\in \overline{\mathbb{R}}\}$ is measurable since it's closed. If we replace $\overline{\mathbb{R}}$ with $\mathbb R$ in $A$, call it $A'$, I know how to show it's closed. However, I'm not sure about $A$. If the singleton sets are closed in $\overline{\mathbb{R}}^2$, then $A$ would simply be a finite union $(-\infty,-\infty)\cup A'\cup(\infty,\infty)$, which is indeed closed.

Jakobian

14:26

that will never happen so its pointless and meaningless to ponder it

@psie yes

psie

ok, thanks

Jakobian

$\overline{\mathbb{R}}^n$ is a metrizable space

YourLordJoyBoy

@Jakobian Or say, if we could make stars. Or a planet.

Jakobian

@psie you mean one-point compactification here by the way, right?

psie

yeah, probably

Jakobian

14:28

oh you mean product of extended real lines

@psie no you don't mean that

psie

yeah, I'm unsure

I can post a screenshot

Jakobian

$\overline{\mathbb{R}}$ is a metrizable space

it fact its homeomorphic to $[0, 1]$

and the product you wrote is homeomorphic to $[0, 1]^n$

still metrizable

all good properties you expect from metric spaces hold in this case

psie

ok

Jakobian

@psie $A$ is closed because $\overline{\mathbb{R}}$ is Hausdorff

having closed singletons is not enough

your argument is wrong here

For a space $X$, the diagonal $\Delta_X = \{(x, x) : x\in X\}$ is closed in $X\times X$ iff $X$ is Hausdorff

psie

ok, I should look up a proof of that

Soumik Mukherjee

14:34

@Jakobian Thanks

Jakobian

@psie Its pretty simple, if $x, y$ are distinct points find open disjoint $U, V$ with $x\in U$ and $y\in V$, then $U\times V\subseteq (\Delta_X)^c$, the complement of the diagonal

and this shows that the diagonal is closed

and converse is not needed for you anyway

(but the converse is also really simple, just take $(x, y)\in (\Delta_X)^c$ and then there is a basic open set $U\times V$ with $(x, y)\in U\times V\subseteq (\Delta_X)^c$)

basic means element of the basis

psie

alright, neat, thanks

Jakobian

you know for product topology on $X\times Y$ those are products of open sets

psie

indeed

YourLordJoyBoy

15:03

Another day, another attempt at editing posts.

Thorgott

16:39

@onepotatotwopotato I don't know any!

all arguments I've ever seen involved in showing that a space does not cover other spaces non-trivially were usually rather ad hoc

Koro

17:23

how to solve 2?

John Zimmerman

19:20

0

Q: analytically continue functions on independent planes in $\Bbb R^3$

I know techniques to analytically continue some functions off the positive real line. I am less sure given I have 2 independent euclidean planes sitting in $\Bbb R^3$ with some function $f$ defined on the positive real lines of the respective planes. For ex. I can continue $f$ on each plane indep...

I still don't get this

John Zimmerman

19:38

I know the concept exists already but I don't know where to find it online

John Zimmerman

19:50

Let $n \ge 4$. Take $X=(0,1)^n.$ Fix all points $p,q$ s.t. sup $\text{dist}_n(p,q)=\sqrt{n}.$ Does there exist a smooth regular codim. one foliation of $X$ whose leaves are topologically $J=(0,\sqrt{n})\times S^{n-2} $ accumulating to $p,q$? For all $n\ge 4$ is $J$ convex?

Lucky Chouhan

20:12

@robjohn @XanderHenderson why Michael Hardy was suspended? He has answered more than 7000 questions on MSE.

Can moderators suspend themselves?

John Zimmerman

you can request to be suspended

Xander Henderson

@LuckyChouhan We don't publicly discuss the reasons for suspensions.

@LuckyChouhan In theory, sure, I guess? Maybe? I have no idea.

YourLordJoyBoy

20:43

Anyone ever try to recall information from earlier in the year when studying only to struggle? That's what's happening to me.

John Zimmerman

21:38

@YourLordJoyBoy I can't remember

Jakobian

21:58

@LuckyChouhan people with a lot of reputation are usually suspended for answering low quality questions

Jakobian

22:19

@YourLordJoyBoy not surprising

knowledge is not eternal

YourLordJoyBoy

@Jakobian The more recent stuff is especially easy to remember because we just did a quiz on it yesterday.

And I know, but not having said knowledge can and will cost me if I don't do something.

Jakobian

only way to remember something good is to reinforce that knowledge

YourLordJoyBoy

That can be a challenge when you're in college.

YourLordJoyBoy

22:36

I'm pretty much having to go back and rewatch previous videos of my class from this year, all of which feature me.

@Jakobian On another note, picture this. The right answer to a quiz or exam question is not available on said exam/quiz.

Jakobian

@YourLordJoyBoy what are you talking about

YourLordJoyBoy

@Jakobian Something that happened on one of my quizzes.

The right answer wasn't a choice!

Oh it made me so mad.

YourLordJoyBoy

23:10

Um guys, does this meet site standards and ruling? math.stackexchange.com/questions/4905709/…

Transcript for

$Mathematics$ Mathematics