Mathematics - 2017-07-04 (page 2 of 9)

Typhon

05:00

@Daminark no you misunderstand. Teacher's aren't bothered. The institution itself has this weird idea in its head that they can only truly make sure teachers are doing their job well if every student is coming fresh into the material

Eric Silva

@Typhon basically sending me to administration at the slightest (or even nonexistent) provocation, constantly writing me up, giving me detention for things that i didn't do

Typhon

@EricSilva such as?

Eric Silva

and like, it's not like im making it up, i eventually befriended an adviser and they looked into it and had to reprimand a teacher

Typhon

sounds more like the teacher was generally an a**

Eric Silva

a band teacher gave me 3 weeks of weekend detentions for singing in class, i wasn't the one singing

and every time i would object he would add another week

Typhon

05:01

did they know it wasnt you or could it have been a mistake?

Semiclassical

Teachers are people, and people are occasionally assholes. hence, teachers are occasionally assholes.

Eric Silva

im fairly certain he knew it wasn't me

Typhon

@EricSilva what did you say to the administration

Semiclassical

I suspect I was a bit of a teacher's pet in high school? I'm not certain, though.

Typhon

To be honest, I keep getting in trouble but I also end up failing a lot of my classes as well. :-(

Eric Silva

05:03

i was a freshman and it was the beginning of the year and i was kind of a nervous wreck cause i went to a huge school far from home so i didnt really say anything

Typhon

and it definitely isn't undeserved

Eric Silva

but like eventually i found my backbone and didn't take this kind of nonsense without standing up for myself

i think i just had a few teachers who didn't like someone who did this (i was in this prestigious magnet program and teachers were generally used to very respectful students who didnt call them out when they were being unjust i think)

Daminark

Lol in high school teachers liked me quite a lot, since most students there didn't care about anything and were badly behaved so the fact that I did alright and didn't talk in class was enough for most of them. The teachers who were bad there were less unfair and more ineffective

Eric Silva

i also had a few who liked me for that who defended me a couple times so idk

@Daminark the thing that confuses me is that one of the dudes who was super out to get me was my senior year english teacher. But all i did was read books in his class. I never talked out of turn and i aced all his assignments. I honestly have no idea why he was so hostile to me

Daminark

I don't know how my teachers would respond to that

We mostly had either the people who paid full attention, or at least gave the guise of doing so, or people who were disruptive

Eric Silva

05:08

and the other thing that gets me is that the same teacher looooved my SO at the time, but she was super disruptive

Daminark

My English teacher in 11-12th grades would've likely been fine with it if you told him, like he'd plead that you do participate but in his class with the behavior problems, the statement "I don't think I gain much from our class discussion" would be a very fair case to make, and if you're not being disruptive he's basically happy

Twink

I've been ther all night

I've been there all day

Daminark

At one point he was reduced to saying "Look, I know you guys won't read this, so I'm assigning the sparknotes, please read it for the love of God"

Twink

and boy got me walking side to side :S

Typhon

heh

reading is boring

Semiclassical

05:11

I think I managed to sustain myself throughout high school on "school is what I'm good at."

Typhon

...

Daminark

But say, my physics teacher would be really upset if someone was just reading, even if it was physics, she wanted you to be paying attention to her lecture and nothing else

Typhon

even if it was the textbook?

Daminark

She didn't really operate out of the textbook directly

Had she been doing so, maybe she'd be alright if you glanced at it to understand something

Eric Silva

@Semi i honestly don't know what got me through high school, i do not think i thrive in that kind of environment, I was never really a traditional kind of classroom person

Semiclassical

05:14

What I don't understand is how I managed to get writing assignments done in high school.

Typhon

me neither

I hate writing

Daminark

Though even then she'd only likely go for it as far as, if the thing she's saying right now relied on a previous thing so you had to look it up for reference. Just reading the textbook and not focusing on the lecture didn't fly

Semiclassical

I suspect I just did it frequently enough that it wasn't so much a problem.

Eric Silva

i was always good at those

Typhon

^^^

Semiclassical

05:15

Which is perhaps why I've found it harder and harder to write at any length, except in an extemporaneous way.

Typhon

i don't have that issue

Semiclassical

If you can get me talking and reasoning, I can get a nice stream of writing done.

Eric Silva

@Daminark i had a bio teacher who HATED when i didnt do bio in his class (i never did) but eventually we had a bet where he said if i get anything less than a 5 he'd get to give me a month of detention

Semiclassical

But sit me down and tell me to write something out, and ughhh

Typhon

i just write too much in the sense that i keep finding more and more and more and more sources until I am just writing about way too much to handle.

Daminark

05:16

I've generally been alright, but my writing has been described as convoluted

In large part because I always write in a train of thought style

Typhon

@Semiclassical i have that issue too I suppose. :-(

Semiclassical

@Typhon I tend to get stuck in my own head. So I'll read and absorb a lot, but

Typhon

oh

Semiclassical

getting me to actually pull it out is a problem.

Eric Silva

the timed writing is suuuuch bs

Typhon

05:17

me too, I suppose

Daminark

It's fine if I'm thinking of stuff and being like, alright I'll do this, now this, now this, blah blah

Semiclassical

@EricSilva uggghhhh yes

Eric Silva

it's also so like not what good writing is

Semiclassical

So, writing MSE answers? I can do that.

Typhon

nah

writing MSE answers is annoying for me

it is why I don't answer really ever

Semiclassical

05:18

Writing up something that needs to have an actual structure beyond just "this is the logic", though...

Eric Silva

like i remember having to write my first college paper and getting a month and i could actually say things and i was like taken aback

@Daminark, also what was your first uchicago paper

Semiclassical

I've always had a hard time saying whether I'm a good or a bad writer. I mean, I've always been reasonably happy with the writing I manage to do

Eric Silva

i feel like that's one of those things that might be memorable

Daminark

But anything which isn't train of thought just isn't my style. So I never really edited/revised much except locally, and often did in short bursts, so I often had 100+ word sentences at least once or twice per paper

Eric Silva

@Semi idt im a good writer but idk how to judge

like i can't go by grades cause that doesn't really mean anything?

idk man

Semiclassical

05:20

usually since once I managed to get myself to do it, the ideas had enough time to coalesce and form into something worth reading

What I am baaad at is documenting what I know.

Eric Silva

a theology grad student once told me my writing was good, but like another theology grad student once said it was mediocre so idk what to believe mannnn

Semiclassical

It's a bit like how I think about chess, I suppose. Set me up with a chess problem and I'll find it interesting.

But the strategy of an entire game? That's a different matter.

Analogously, ask me to talk about some element of a problem, and I'm happy.

Daminark

@Eric It was about how Genesis portrays the advancement of human civilization as being bad for our relationship with God

Semiclassical

Ask me to set it up? Hnnnnggg

Daminark

It was extremely short, that first paper was just 2-3 pages or smth

Eric Silva

05:22

@Daminark mine was about the relationship between Achilles and naturee or something like that

o that's really short

Semiclassical

Add to that the fact that I just don't feel like any of the writing I'd do these days has any intrinsic value

Daminark

Most of our papers in that class were 4-5 pages, our last one being 8-12

But our first was 2-3, second was 3-4, and thereafter 4-5

Eric Silva

o our first was like 10

our last was like 15

that class was a hard intro to college lol

Typhon

I cannot stop myself from having 7-8 sections with 5-6 subsections each most of the time.

but then each section is like... 5 sentences

Las Vegas Raiders

Concise writing is hard.

Typhon

05:25

@LasVegasRaiders huh?

the problem is that I couldn't write enough material

Daminark

My hum class was pretty reasonable, my sosc class first quarter had a lot of people not doing terribly well

Typhon

and I try to write about so many different things that my writing is useless

Daminark

Though it was more about our professor just grading really harshly (barely anyone got more than C+ on the first paper) than it was volume

Eric Silva

o wow that is harsh

Semiclassical

The message I've gotten from my advisor a few times, which I find absolutely frustrating, is

Eric Silva

05:27

I generally have done well on my paper classes, i think my graders have been very nice

Semiclassical

"This is just bureaucratic, don't worry about it."

with regards to writing.

Eric Silva

what does that complaint mean

Typhon

idk

I honestly despise writing

Semiclassical

This is mostly in the context of stuff for various qualifying requirements.

Eric Silva

ah ok

Twink

05:29

I know I should've fought it, at least I'm being honest

Semiclassical

so the message is "it's not worth caring about, but you still need to do it."

Daminark

I did better on hum papers than sosc ones, I think in part because my procrastination on sosc papers was of an iffy sort and because my focus was just bad

I procrastinated hard in hum too, like I think I probably did a bit under a page the night before and then 3 pages in the 3-4 hours before it was due

Semiclassical

Fun story: I wrote an undergrad research paper for a behavioral psych/econ course.

I did it on academic studies of procrastination.

I got it in a day late :P

Typhon

@Semiclassical how big was it?

Eric Silva

i think my hum teachers were all nice, so i never really had a harsh grade in those classes, my history papers have all been really good too, but my ethnomusicology papers were just t r a s h

@Semi lmaooo

good topic

Semiclassical

05:32

I feel like...10 pages? But I could not tell you at this point.

Typhon

ugh

that's not even worth mentioning. XD

Daminark

I took longer to do sosc papers but that was amidst distractions, and it was often because I developed my argument as I went, so at one point I'd say something and be like "Shit I can't counter that and it was supposed to be devil's advocate... But if I am to negate my thesis I'd need to rewrite this paper... Fugg!"

Typhon

my teachers give me assignments to write 30 page papers

Eric Silva

@Daminark rip.

tfw you alienate yourself from your Gattungswesen

Daminark

Then an hour later I'd figure out how to counter it, but there would be an hour left before it's time to submit so I'd have write 2 pages in that time

Which often meant that the quality of writing was just bad, I'd have way too many comma splices and such

Eric Silva

05:35

u played urself m8

Daminark

Or I mean, I'd write it such that each sentence corresponded to a given train of thought, so any given comma separated 2 things which chained well, but it would still be annoying to read on the whole

Semiclassical

My usual strategy with my papers was to somehow turn them into writing about things I was already interested in.

Typhon

17

Q: Is being a right triangle both necessary and sufficient for the Pythagorean Theorem to hold?

I'm hoping that this isn't a dumb question, but I encountered a Stack Overflow question in which the OP was testing for whether a triangle was right by whether or not it "met" the criteria of the Pythagorean Theorem (i.e. whether or not the square of the hypotenuse is equal to the square of the t...

this is by far the dumbest duplicate I have ever seen.

they actually duplicated another question

Eric Silva

i remember i wrote my final paper on my modernism class in like 2 hrs @Daminark. I was so tired from a long quarter that i basically just didnt write a conclusion and then i submitted it

Daminark

Okay conclusions mess me up so bad

Eric Silva

05:37

i ended up getting an A- on the paper and the comment "this paper would be an A but i feel like there should be more"

and i was like rip me

i played myself so hard

Typhon

i fail on the introduction, the body, and the conclusion

Semiclassical

@Typhon Eh, I think that's a bit of a leap to judgment absent the relevant SE question

Typhon

uuuh

Semiclassical

(which they should have included)

Daminark

I've started to get the hang of them but it was like, what's there to say at this point? "Yeah, so that's how it goes, I guess. Kthxbai"

It always just felt cheesy

Eric Silva

05:38

it's different depending on what kind of paper it is though

Typhon

a question asking if a triangle is a right triangle by confirming the pythagorean theorem directly implies and asks the question: "does the converse hold"

therefore, this question is an exact duplicate

Eric Silva

like you don't end papers in history classes the same way you end literary or formal analysis papers

Daminark

Lol I remember in my TAPS class I had this one assignment which I really underestimated how long it'd take

Semiclassical

Without the other question to compare with, you only have one source of evidence for that.

Daminark

It was actually to write a scene this time, not an essay, but I basically started 2-3 hours before class

Typhon

05:39

the op flat out admits their question is a duplicate by stating that

they describe an exact duplicate

hence "lol, this guy just made a duplicate and admitted it"

Eric Silva

@Daminark sounds hard. I actually haven't taken a "paper class" since this past fall... I need to get back to it

I miss proper history classes

Daminark

I got an A- on that and the comment was "Yeah this is fantastic but like, this was supposed to be a 6-8 minute scene and this was like, way shorter"

Semiclassical

Let me put it like this. Suppose the person hadn't included it. Would you think it's a duplicate?

Typhon

@Semiclassical that doesnt make any sense

i wouldn't know the other question existed...

Semiclassical

Have you seen the other question?

Typhon

05:42

you're not making sense

Semiclassical

If not, how do you know it exists? How do you know their description of it is accurate?

Typhon

the person has no link to the duplicate

Semiclassical

That's my point.

Typhon

they described a duplicate

Daminark

I mean, reading it, it might not necessarily be a duplicate. I actually get the vibe that this was more, hey in the process of doing some question, the OP used this method, now I'm asking if that method was valid

Semiclassical

05:42

^^^

Typhon

and it is isn't possibly a failure to interpret

Eric Silva

@Daminark i write tersely and im always scared i have no content because i always feel like i have said nothing

and my papers end up on the lower end of length

Typhon

they say a question was asked to confirm a triangle was a right triangle by using the pythagorean theorem

Daminark

This is the point, it's not that someone is claiming to have a duplicate. If someone were to explicitly say that, that'd be pretty pathetic

Typhon

so their question is already asked by the other question since someone must confirm the test works to answer it

Daminark

05:44

"Hey, can someone show me how to integrate this function by using a trigonometric substitution?"

"Hey, how do we know that change of variables is true?"

Typhon

@Daminark yet they ask if one can use the pythagoren triangle to confirm a triangle is a right triangle in response to seeing a question use that "test"

@Daminark they are the same

also the latter is just

"is change of variables true"

Eric Silva

questions can be logically the same without being duplicates

Typhon

it doesn't ask for justification

it's just "yes or no"

Daminark

I said "how do we know"

That's asking for a proof

Typhon

but the question i linked asks

"is it true"

Semiclassical

05:45

what I really find it quite silly how you resort to calling this question 'dumb'

Typhon

yes or no

Eric Silva

i think it's actually quite a good question, cause it leads to an interesting formula

Semiclassical

It's not a dumb question. It may be a lazy question but it's not a dumb one.

Daminark

In answering the first question, people often black box the fact that change of variables is true and just use it. The second is distinct in what it's requesting you to do

Typhon

@Semiclassical I meant that it is the dumbest duplicate I've ever seen as someone actually managed to create one off of another question and not just due to lack of research.

Semiclassical

05:46

And, once again, you are entirely taking them at their word for that.

Typhon

@Semiclassical If they claim it to be true, then they should be held to their word.

Daminark

@Semi I mean, I don't think anyone would lie about a question being duplicate when it really isn't, that would make no sense, but this person doesn't sound like he's claiming this at all

Typhon

If I wrote "this answer was copied from quora" would you not punish me even if it were not true?

regardless

Semiclassical

No. I'd ask for you to show me the question, and then ask what about that question's answers was not satisfactory. I'd then tell you how to modify your question to reflect that.

Typhon

asking for some help finding the supposed post isn't wrong

Eric Silva

05:48

The question the OP described wasn't even necessarily asking if the converse of the pythagorean theorem was true

Typhon

at least it can then be sent through the queue and verified one way or the other

Semiclassical

To merely say "haha dumb duplicate" is at the very least unproductive.

Daminark

He's saying, a test was used in another question, which doesn't mean that the answer went and proved what was being said.

Typhon

@Semiclassical you didn't read my last post. I said that if I just claimed "my answer is copy/pasted from another site" would you not punish me?

Semiclassical

And I answered. No, I wouldn't necessarily

Eric Silva

05:49

@Daminark yeah, and honestly from reading the question i actually would assume from the language that it wasn't proved.

Semiclassical

Or, at the very least, I would not simply take you at your word.

Typhon

@EricSilva but it is directly implied that it be verified as true. The op didn't ask for a proof. He just asked "is it true, yes or no".

Daminark

To return to the example I gave, if I ask for help on computing this kind of an integral, I'm not asking for a proof of this fact necessarily.

Semiclassical

To reiterate:

2 mins ago, by Semiclassical

No. I'd ask for you to show me the question, and then ask what about that question's answers was not satisfactory. I'd then tell you how to modify your question to reflect that.

Typhon

@Semiclassical "No. I'd ask for you to show me the question, and then ask what about that question's answers was not satisfactory. I'd then tell you how to modify your question to reflect that." isn't a fairly logical response to copied answer.

that sounds like a response to copied question

Semiclassical

05:51

That was in regards to what you said about a self-reported copied question from quora

Eric Silva

@Typhon where did it imply that it was verified as true in the original stack overflow post.

Typhon

ok

@EricSilva they ask "is it true, yes or no". Therefore any answer to the other post must at least claim the test was either true or false when answering.

i hope that made sense

Eric Silva

no it might just use it as an intermediary step without justifying

Typhon

@EricSilva that is verifying

i never said they proved it

Eric Silva

if it justified it well, this person wouldn't have asked this question

Typhon

05:52

this question isn't asking that either imo

they just want someone to say "yes or no"

@Semiclassical self-reported copied answer from quora

you misread it

anyways whatever

i thought maybe you guys could help find it

Daminark

@Typhon
1) This still isn't what the original question was necessarily asking, the original question wanted to use the test. It's possible that someone decided to take it at face value that this worked and just apply it, or assumed that the OP knew.
2) This interpretation of the wording as just asking for yes or no, and not verification, is literal to the point of being complete and utter nonsense
3) Why would you punish even if they falsely claimed to copy?

Eric Silva

if i ask someone "is this true" in math i usually want them to justify, and i think it's fair to assume that this question is more than a yes or no question given that this is a pretty normal thing to do in math

Typhon

@Daminark with regards to 3, if someone was claiming they posted plagiarized material, I would immediately delete the material and that's that.

@EricSilva actually, asking "can X be done" is considered wrong for the reasons I stated on virtually every computer programming based site here.

because "you're just asking for a yes or no"

Daminark

1) Why? You punish people for plagiarizing, not for saying they plagiraized
2) No one in their right mind would do that anyway, and the wording of this question is in no way explicit

Leaky Nun

@EricSilva depends on the question. some really do expect to have a yes/no answer.

Typhon

05:57

@Daminark it is an analogy and i only wish the post to be closed.

Daminark

Still doesn't make sense

Typhon

what is wrong with me asking for people to help find the post?

Eric Silva

@Leaky Nun, sure, but i'd say it's not the norm, and idt the post that Typhon linked has language that suggests this is the case, i think it actually is suggestive of the opposite

like this person seems like they want more than a yes or no answer, they don't say it explicitly but again it's like, not an unreasonable assumption.

Daminark

"Hey, this question referenced another so I'm interested in seeing the source"
vs
"Oh wow what a lame duplicate question even though you have to be interpreting it in a very specific and likely wrong way to arrive at that conclusion"

These are sharply different things

Typhon

it just seems like the person already has the answer to their question

as the statement being true or false is confirmed by the other post's answers

Daminark

06:01

As I said before, interpret it from a reasonable standpoint

Eric Silva

if they had a satisfactory answer, why would they ask

Daminark

The fact that the OP was curious about this suggested that he didn't want to take it on faith that this was true just because you wouldn't have answered it/asked it like that if it were false

But then you won't want to take it on faith if someone's like "Yeah dw"

Typhon

anyways

we can agree to disagree

let's do number theory

Daminark

So even if the word-by-word interpretation doesn't tecHNICAlly 0_0 ask for a proof, if the person wasn't doing that it'd make no sense, while there is an option to interpret this such that things do make sense.

Typhon

actually

it was an SO question

flips table

this discussion is a moot point

MSE questions cannot be duplicates of SO questions

intra-network duplicate tracking doesn't exist yet

Daminark

06:06

Okay now look at the revised question

Typhon

i just came from there...

Daminark

You have an answer which indicates VERY CLEARLY that this was in no way a duplicate

Semiclassical

just for reference, I think the SE question they refer to is this one: stackoverflow.com/questions/44890678/…

I say that both at the level of the SE question itself, and also the fact that one of the commentators there is the OP here.

...I'd love if they'd actually bother to say as much, though.

Typhon

"its been deleted"

post that link and see if they say it is the question

maybe they lost the link

Semiclassical

"The Stack Overflow question (which has since been put on hold and will likely either be deleted soon "

Typhon

06:09

read the latest replies

0

A: Proof for elements of $\textbf{Z}[\sqrt{3}]$ regarding the existence of the norm.

There is nothing wrong with Akiva Weinberger's post. However, since this post was made in the context of a somewhat more rigorous and formal setting, I'm posting a formal proof that I believe would most likely be considered acceptable in that setting. Let $z$ and $y$ be extended integers such...

oops

XD

(i was in the process of asking for verification for my answer in chat)

math.stackexchange.com/questions/2345327/…

Daminark

So looking at Semi's link, it wasn't even clear what that other post was asking, especially since it was a Java context

So now he goes to a math community and says "Yo, some question involved some test that I'm curious about"

That's absolutely alright

Semiclassical

Ahah, he confirmed it.

Typhon

true

but originally it was suspect

so I don't see why I got flak for asking for some help

(finding the context)

Daminark

There was potential for it being a duplicate but the most viable possibility was that it was not. And you were jumping toward being judgmental about it, so you got flak for being condescending when it was in no way called for

Semiclassical

careless question, yes. "dumbest duplicate ever"?

Typhon

06:15

well it certainly appeared that way

and that comment was made in a joking manner

not meant to be rude or judgemental

hence "dumb" as obviously it wasn't malicious

Eric Silva

lolwut

Semiclassical

...you and I have very different definitions of 'judgmental', it would seem.

Daminark

You had to be trying to interpret it like so. And it wasn't obvious that it wasn't malicious at all, in fact it seems unlikely even now, since you were literally searching for a way to get him closed and being condescending about it.

Semiclassical

Regardless, it's 1:20 in the morning and I have no idea why on earth I'm still awake. night.

Eric Silva

night @Semi

Daminark

06:17

Good night!

Typhon

@Daminark no i was not claiming malicious

malicious != dumb and/or not-well-thought-out behavior...

Daminark

You were not claiming he was malicious, I'm saying you were kinda being malicious insofar as you were being condescending

Anyway, let's put this to rest and stop making these snap judgments, capiche?

Typhon

@Daminark alrighty

but before then

-11

Q: Pythagorean theorem to syntax

This is just a sample code for the proble. int a,b,c; system.out.println("Enter sideA"); system.out.println("Enter sideB); if ( c == a * a + b *b) system.out.println("This is a right triangle"); else system.out.println("This is not a right triangle"); [formula](https://i.sstatic.net/1NqAP.p...

java

i dont have edit privileges and this looks like vandalism

XD

could someone revert the edit?

Daminark

I don't have editing privileges, at least on a question which is on hold

But this is bizarre

Brown Ninja

08:19

Hello all,
Let $P$ be an $n\times n$ doubly stochastic matrix and $S$ be an $n \times k$ orthogonal matrix. Is $S P S^T$ also a doubly stochastic matrix? Can someone comment on this please?

Daminark

Hmm, I don't know much about this stuff but I do wonder if there's a way to think about this as a property of linear transformations

Balarka Sen

Hi chat

Las Vegas Raiders

Hi chat

Daminark

Probably not, given that I'd guess that this isn't a property that holds under similarity

Hey @Balarka and @LasVegas

Brown Ninja

Right

Las Vegas Raiders

08:25

Hi pal @Daminark

Secret

A stochastic matrix is one where columns or rows sum to 1. That sounds like for a right stochastic matrix, the column vector (1,1,1,1,1) is an eigenvector with eigenvalue 1 whle for the left stochastic matrix, the row vector (1,1,1,1,1) is an eigenvector with eigenvalue 1. So a doubly stochastic matric will have both of theses vectors as eigenvectors with eigenvalue 1

Daminark

Wait hold up a sec what's your definition of an orthogonal matrix, just to be sure?

Brown Ninja

Doubly stochastic -> all rows sum to 1 and all columns sum to 1

orthogonal matrix -> all columns are orthogonal

Daminark

Normally I think about it as having its inverse as its transpose, but you're defining it such that it's not necessarily square

Balarka Sen

@AkivaWeinberger That's an interesting question.

Brown Ninja

08:27

to each other

Secret

This is because for any matrices that multiplies to the vector of all 1s, it is effectively the same as summing up each rows and columns, and the summand is given by the entry of the resulting vector

Daminark

Well, don't you want them to be orthonormal?

Brown Ninja

No they should not be orthonormal. Let me explain the bigger problem I am dealing with.

Secret

So I guess, the question boils down to whether $SPS^T$ still have the column and row vectors of all 1 s as an eigenvector with eigenvalue 1

Brown Ninja

I have an $n$ element vector on which I am applying summation over every $k$ elements

to obtain an $\ell$ element vector

this can be written as $\vec{x} S$ where $S$ is of the form
$$
\begin{bmatrix}
1 & 0 & 0 & \cdots & 0\\
1 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \cdots & \vdots\\
0 & 1 & 0 & \cdots & 0\\
\end{bmatrix}
$$

where the first column has $k$ ones at the first $k$ locations

Balarka Sen

08:33

@AkivaW Well, for one, you can still define linking numbers for maps $f : S^m \to S^n$ by taking two generic regular values $x$ and $y$, and looking at linking number of the submanifolds $f^{-1}(x)$ and $f^{-1}(y)$ in $S^m$. Check this question out for the definition and the meaning of this.

In any case, if you do this, the linking number of the quaternionic and the octonionic Hopf maps turn out to be $1$ too; in fact these are the only maps with linking number 1 - this is the famous Hopf invariant one problem I think.

So there's that genericness theorem; if you find a map $f : S^7 \to S^4$ with linking number 1, it's the Hopf map. I think that answers our question by similar logic as before?

Brown Ninja

I have another vector $\vec{y}$ which majorizes $\vec{x}$

So let $\vec{x}' = \vec{x}S$, and $\vec{y}' = \vec{y}S$ Does the majorization relation still hold between $\vec{x}'$ and $\vec{y}'$?

Liad

im trying to find the volume of the set $T$ where $T$ is the set of points that satisfy $x \ ^ 2 + y \ ^ 2 + (z - 1) \ ^ 2 \lt 1$ but not in $z = \sqrt{x \ ^ 2 + y \ ^ 2 }$, someone can help ?

Brown Ninja

I hope I made some sense.

$S$ is not $n \times k$, it is $n \times \ell$

Leaky Nun

@Liad try to draw a 3D diagram

or picture one in your head

Balarka Sen

@AkivaW Actually that question does not give the definition in full generality so let me say it; if $M$ and $N$ are two closed submanifolds of $\Bbb R^{n+1}$ such that $\dim(M) + \dim(N) = n$ the linking number $\ell(M, N)$ is defined to be the degree of $f : M \times N \to S^n$ given by $f(x, y) = (x - y)/\|x - y\|$.

Leaky Nun

08:42

@Liad the last one is a cone

Liad

@LeakyNun i did , but i want to use spherical coordinates and the fact that the ball is not around the orgin make it difficult to me.

we want $x \ ^ 2 + y \ ^ 2 + (z-1) \ ^ 2 \lt 1 $ and $z \ ^ 2 \gt x \ ^ 2 + y \ ^ 2 $ , right?

Balarka Sen

@Daminark Did you understand why, if $f, g : M \to S^n$ are homotopic maps with regular value $p$, then $f^{-1}(p)$ and $g^{-1}(p)$ are cobordant?

i wanted to talk about pontryagin thom a bit :3

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