Mathematics - 2023-08-04 (page 2 of 2)

Semiclassical

22:01

now suppose i take a slice with some plane, and project that 2D convex subset onto another plane. this is going to at least be a subset of what i'd get if i just projected the full 3D solid onto the other plane to begin with

Ted Shifrin

Projection and intersection are in some sense dual …

Semiclassical

but under what conditions would you expect that what you get is the entire 3D shadow?

i.e., that taking a particular slice first doesn't change the shadow on some particular plane?

Ted Shifrin

This must relate to the apparent contour we discussed before. It has to be a planar curve.

Semiclassical

(It's showing up because I constructed a certain convex set by projecting the entire solid, whereas an earlier paper does so by slicing first and then projecting. but somehow we get the same picture...)

hmm

Ted Shifrin

Its projection onto the plane is the boundary of the projection of the whole thing.

Semiclassical

22:05

right, that's a much simpler way to put it

@TedShifrin this is perhaps a good point to note that my solid of interest is known to have the following properly: its polar dual is affinely equivalent to itself

dunno if it's related, but since duality came up...

Ted Shifrin

Well, I’m not sure how to make my vague duality statement precise, but I think I used to know.

Semiclassical

projective duality?

Ted Shifrin

But your criterion holds iff the apparent contour curve lies in a parallel plane …. I think.

Semiclassical

there is some pretty neat math going on which i wish i understood better. there's actually three nested convex bodies here, which inclusions $\mathcal{C}\subset\mathcal{Q}\subset{\mathcal{N}}$

polar duality gives three additional bodies such that $\mathcal{N}^\circ \subset \mathcal{Q}^\circ\subset \mathcal{C}^\circ$

the fun bit is that each entry in the second list is affinely equivalent to the corresponding entry in the first

@TedShifrin this paper refers to section/projection-duality: sciencedirect.com/science/article/pii/S0022123611002965

the first line of the abstract: "In this paper, we show that the well-known duality operation in the context of convex bodies in RR^n is completely characterized by its property of interchanging sections with projections."

and the duality operation they have in mind is precisely polar duality

so i think we're talking about the same thing?

Ted Shifrin

Yeah, must be.

Shaun

22:22

Hi :) Does this work for infinite cardinals $a,b$? I believe so . . . I might ask a question about it but my grasp of infinite cardinals is weak.

leslie townes

at least some of the solutions seem to be bungling the "only if" direction, so if you're wondering about that, you might be right to wonder.

Shaun

@leslietownes Who, me?

leslie townes

yes, shaun, you.

Jakobian

@Shaun is this even true?

Shaun

I'm beginning to doubt, @Jakobian.

Jakobian

22:29

the author didn't mention surjective... eh

either way, $|F_\kappa| = \kappa$ for $\kappa$ an infinite cardinal, I believe

Shaun

They say "onto".

Jakobian

so assume AC, as any sane person would, this should be true in the infinite case

ah, they do say that

shintuku

choice is an illusion all is predetermined

Jakobian

If $a\leq b$ then a surjective homomorphism $F_b\to F_a$ obviously exists by restricting generators

Conversely if such homomorphism exists then $|F_b|\geq |F_a|$. If $a$ is infinite, we are done from previous result

Shaun

Awesome, thank you, @Jakobian :)

leslie townes

22:35

if a is finite maybe you abelianize and make some argument about rank to rule out a > b?

i dunno

Jakobian

well there's still the last bit, proving that a homomorphism $F_n\to F_\omega$ onto doesn't exist

leslie townes

oh and that

it's definitely nontrivial.

i'm not sure one would get that impression from any of the posted answers :)

shaun are trying to earn the 'necromancer' badge for doing something relating to a question from many years ago

because whether you are or not, this could be a golden opportunity

Jakobian

@leslietownes I think this boils down to something like "$F_\omega$ can't be written in terms of finite amount of generators" which should be a theorem in some book (about geometric group theory?)

just like how the answers tried to claim

Ted Shifrin

Number, not amount.

Semiclassical

is that the same as not being finitely-generated?

leslie townes

22:40

jakobian i think so, i just wonder how many of the answers were even aware of this

or if this is one of those deals where, "for it to be otherwise would be contrary to the nature of a straight line"

Jakobian

Rank of a group

In the mathematical subject of group theory, the rank of a group G, denoted rank(G), can refer to the smallest cardinality of a generating set for G, that is rank ⁡ ( G ) = min { | X | : X ⊆ G , ⟨ X ⟩ = G } . \operatorname {rank} (G)=\min\{|X|:X\subseteq G,\langle X\rangle =G\}. If G is a finitely generated group, then the rank of G is a nonnegative integer. The notion of rank of a group is a group-theoretic...

free group on $\kappa$ amount of generators has rank $\kappa$

according to this article

leslie townes

i find that easier to see with abelianization than i do at the level of free groups

although i'm wondering if any treatment i've ever read of rank of abelian groups treated the infinite case with any subtlety

general abelian groups are nightmarish

Jakobian

@TedShifrin wdym. This is the same word

leslie townes

don't poke the beast

Ted Shifrin

Learn some English.

leslie townes

22:44

ted, you might like this. the other day my wife asked munchkin a question and munchkin responded: "there was something in your talk that i don't know"

(the question involved the word 'toiletries,' which munchkin did not know)

Ted Shifrin

She didn’t want to say ”speech,” for fear of suggesting politics.

Xander Henderson

@Jakobian No, they are not the same word.

leslie townes

ted doesn't like it when we don't talk good

Jakobian

@XanderHenderson they mean the same thing

Xander Henderson

@Jakobian Not quite, not.

Jakobian

22:46

stop trolling me

leslie townes

Jan 15 at 19:15, by Ted Shifrin

Number, not amount.

Xander Henderson

"Amount" is used with collective or mass nouns, "number" is used with count nouns.

leslie townes

Sep 22, 2022 at 1:20, by Ted Shifrin

Number, not amount.

Ted Shifrin

Interesting how stubborn some people can be out of ignorance. Speaking of politics.

Xander Henderson

"Number of generators" vs "amount of salt."

leslie townes

22:48

jakobian ted is 100% right about this, but right in the way that only an english teacher can be right

Ted Shifrin

Well, you can’t teach math to a point-set topology nerd.

leslie townes

here he is doing it with me

Jan 29, 2022 at 19:56, by Ted Shifrin

BTW, certain number of hours, or certain amount of legal education. Grr.

2

Semiclassical

@XanderHenderson vs number of salt crystals

Xander Henderson

@Semiclassical Indeed.

Jakobian

ah. I see the difference

Ted Shifrin

22:49

Nice citation, leslie.

Semiclassical

to not spam with images, i'll make a separate room

leslie townes

if the world of english speakers were to fight a war over this, jakobian, our side would win, but we would put "it was number, not amount" on ted's gravestone, and we would know in our hearts that ted was right about it

Ted Shifrin

If you even have hearts …

Semiclassical

my go-to for this

2

Jakobian

@leslietownes so it's a language purist type of thing?

Ted Shifrin

22:53

No.

This is not ain’t versus isn’t.

Every language I know has the same distinction.

leslie townes

jakobian: it's a distinction that does exist in the language, even if many speakers are unaware of it (most people) or do not acknowledge it (me). it's a "real" difference and not just a schoolmarm thing

Xander Henderson

@Jakobian No, in this case, native speakers are going to hear what you say, and it is going to sound wrong to most of them.

Ted Shifrin

Well, there are plenty of American native speakers who can’t tell what’s correct.

Xander Henderson

Descriptively, the vast majority of native English speakers simply wouldn't say "amount of generators", and would find it subtly wrong.

Ted Shifrin

Probably not so true in Britain. But I don’t know.

Jakobian

22:56

@leslietownes why do you not acknowledge it then?

Ted Shifrin

But we are not the uneducated, ignorant masses in this chatroom.

leslie townes

jakobian: no reason, i just don't pay too much attention to how i use language outside of formal settings

Xander Henderson

@TedShifrin Yeah, I'm not going to follow you that far. "Amount of generators" is wrong, but only because nearly everyone who speaks English natively would perceive it as wrong. There is no reason that the language couldn't shift in such a way as to make it "right"; it just hasn't.

leslie townes

trying to imagine a world in which the vast majority of english speakers have any reason whatsoever to talk about generators

Ted Shifrin

What amount of tires do you need to buy?

leslie townes

22:58

i'm not fighting the underlying point :), it's just funny how it's arising in a context that would only arise in math discussion

Xander Henderson

And telling people that their language is "wrong" or "uneducated" is often used as a cudgel to discriminate against people. The "amount" vs "number" thing is pretty universal, but "ask" vs "aks", or "needs washed" vs "needs to be washed" are cultural / regional.

Ted Shifrin

I agree with that.

Jakobian

Thank you for pointing out my error. Leslie doesn't need to care because he is a native speaker, but I should

Semiclassical

it's wrong in practice vs wrong in theory, more or less

Xander Henderson

@Jakobian Yeah, learn to f'in' English, bruh!

Semiclassical

23:01

amount of generators is wrong in practice, b/c it would perplex native speakers

leslie townes

my wife sometimes thinks i choose incorrect words to annoy her, because she is very much "team ted" when it comes to using language correctly when you know it is possible to do so, but most of the time i'm just not paying attention

Xander Henderson

Sorry, I mean "Lern to English".

Ted Shifrin

As I said, this is a universal issue across most (if not all) languages.

leslie townes

and maybe only 15% of the time deliberately choosing the wrong word to annoy her

but yeah if i were in an environment where people might be judging me for how i used language, i would talk more like ted and less like myself

Xander Henderson

@leslietownes I used to do the same thing. Maybe that's why she left? :/

leslie townes

23:01

certainly if i were editing a written document i would correct the amount/number thing if it was wrong

Xander Henderson

I still do the same thing to my daughter.

Ted Shifrin

Munchkin will learn from you.

leslie townes

oh boy did my daughter pick that up. one of the first things she learned to do after learning how to sing the ABCs, was how to get laughs from me by singing the ABCs wrong (or as ted would say "incorrectly")

Ted Shifrin

That versus which is something I correct in my writing, not always in my speech.

leslie townes

munchkin would start out A B C D E and then start doing letters at random and it would just drive my wife up the wall

Ted Shifrin

23:03

She’s a smart twit!

Xander Henderson

@TedShifrin Yeah, I have a really hard time with that one. I second guess myself every single time I have to write one or the other. Same with affect and effect.

Ted Shifrin

I’m a language nut …

Xander Henderson

@TedShifrin hash tag me also

Ted Shifrin

Almost always affect is a verb only, but effect can be both.

Xander Henderson

Which is why I know that this is a problem that I have.

Ted Shifrin

23:05

The effects and effect a change, but you affected the outcome.

Xander Henderson

@TedShifrin Oh, intellectually, I know. But "affect" as a noun is a thing, too. But it describes behaviour (more or less).

Semiclassical

@TedShifrin images done: chat.stackexchange.com/rooms/147686/…

Ted Shifrin

Yes, that’s rare, though.

Xander Henderson

In any event, I get affect/effect right about 99 times out of 100, and then I look at it and go "Wait, that isn't right..."

The change is back and forth 17 times before looking it up on Google and determining that I got it right the first time.

Ted Shifrin

@Semiclassic Did you check out my apparent contour claim?

Semiclassical

23:07

trying to figure out how atm

but wanted to get the images down there

Ted Shifrin

Cool.

Semiclassical

i do know what the boundary curves for the slice image are

so that's a start

Ted Shifrin

Where is the surface normal orthogonal to the normal of the projection plane?

冥王 Hades

23:23

Japanese is even more painful to learn than English.

It is so heavily dependent on context that some words and phrases can mean something entirely different to their literal meaning just because context.

Semiclassical

also tonality matters a lot

Ted Shifrin

Yeah, and intonation in Asian languages completely changes meanings.

Semiclassic beat me.

Xander Henderson

I'm pretty sure that Japanese is not a tonal language.

Jakobian

what Xander said

Ted Shifrin

I was thinking about what I’d been told about Chinese.

Jakobian

23:26

yeah, Chinese, Thai

Chinese meaning, something like Mandarin

@冥王Hades yep, that's part of why google translate works so poorly with Japanese

冥王 Hades

Uchu means universe in Japanese, Uchuu Kukan means outer space, but uchu kukan literally translates to “space of the universe”, but then most Japanese just use uchu to describe outer space…

I don’t know why I even bother

Jakobian

even more difficult is learning the Japanese alphabet though

冥王 Hades

Hiragana and katakana are easier but kanji is a nightmare.

robjohn

@XanderHenderson My wife and I cringe every time the commercial comes on that talks about "less diaper changes".

Xander Henderson

@robjohn Ouch.

冥王 Hades

23:35

Also when someone says “Hoshi-tachi” they can either mean planets or stars because….Japanese uses the same word for both

Xander Henderson

Here, the grocery store gets it right: "15 items or fewer".

robjohn

That is an enlightened store

I know our grocery store gets it wrong.

冥王 Hades

And don’t even get me started on the million different uses of the word ちょっと

robjohn

I have 14.7356 items

@冥王Hades like this?

冥王 Hades

Grrr

robjohn

23:40

:-)

Jakobian

@冥王Hades In Polish we also omit person pronouns, but we still keep them as suffixes on certain words

in other Slavic languages this can also happen, but iirc less often

Semiclassical

@TedShifrin found the contour which forms the relevant boundary from the (2,0,-1) direction: it's the curve traced by $(x,y,z)=(4t^3-3t,t,t)$

Xander Henderson

@Jakobian It happens a lot in Russian.

Semiclassical

which indeed lies in the $y=z$ slice plane

Jakobian

also there's no such thing as a gender neutral "they" unless you want to call someone an "it"

Xander Henderson

23:46

I would imagine that it is not uncommon in a lot of flexive langauges.

Jakobian

@XanderHenderson I heard that it happens less often in Russian

Xander Henderson

@Jakobian I don't speak Polish, so I couldn't comment on the comparison.

Jakobian

I don't speak Russian, so it's not my opinion either

Xander Henderson

But Russian will often omit pronouns for the subject of a sentence.

Since the verb conjugation clear that up.

Jakobian

well, the lack of "they" is unique to Polish, I think

as a way to refer to one person, that is

Semiclassical

23:49

blah, ignore the above. i'm rechecking it

Jakobian

I heard that this is partly because of hatred for communism in Poland

Xander Henderson

@Jakobian In Russian, он / она / оно are third-person singular (masculine "he", feminine "she", neuter "it"; neuter is not generally used for people, though my recollection is that the word for "baby" is grammatically neuter). они is third-person plural ("they").

I believe that you are suggesting that Polish doesn't have a third-person singular pronoun.

Jakobian

yes

Xander Henderson

A quick Googling suggests that детя (child) is grammatically neuter, which gels with my recollection.

So children are neuter.

(Not used to seeing that word used in the singular...)

Jakobian

I think I mixed things up

we have the word "wy" which is second person

in the past it could be used as a polite way to refer to someone

a singular person

Semiclassical

23:56

@TedShifrin got it now, i think, in that same room. bottom line is that the boundary curve for the $(0,-2,1)$ direction lies in the $y=z$ plane

which i think is precisely what you said it should do

Jakobian

nowadays in Poland rather than saying "you" to one singular person, you should say Pani/Pan which is more like "mister/miss"

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$Mathematics$ Mathematics