Mathematics - 2024-11-22

01:23

every knot complement on S^3 is homologous to trivial knot complement is because a knot is a codimension more than one object in S^3?

Ben Steffan

01:42

what do you mean by "homologous"

that all knot complements have isomorphic homology groups is a consequence of alexander duality and has nothing to do with codimension

one potato two potato

02:33

isomorphic homology groups >> in homology level, knot is same as unknot

I want to know why this happens in a more or less geometrical sense. To me, Alexander duality argument does not give me this kind of interpretation.

and also: Can a boundary map (connecting homomorphism?) on a long exact sequence of homology on manifold be interpreted as an actual topological boundary (of manifold representing the homology class) in some sense? I remember some interpretation like this before using triangulation but I don't remember where.

I lived well without really knowing this boundary map but at some point, I want to know.

Thorgott

02:49

@onepotatotwopotato Alexander duality is geometric, its about intersection numbers. But I believe you can also see it via a Mayer-Vietoris calculation.

Also I like interpreting H_1 by abelianizing a Wirtinger presentation

@onepotatotwopotato if the class is representable by a manifold with boundary, sure

one potato two potato

03:03

@Thorgott why is that true?

does that also mean if a representing manifold is closed then the boundary map is zero?

@Thorgott Hmm

psie

06:37

I am reading a section entitled "Differentiation on Euclidean space". Let $f$ be the Radon-Nikodym derivative of $\nu$ with respect to Lebesgue measure $m$, i.e. $d\nu=f\,dm$. Then $\nu(B(r,x))/m(B(r,x))$ is claimed to be the average value of $f$ on $B(r,x)$. I have a hard time understanding why this is so?

psie

06:51

Ok, something like this: en.wikipedia.org/wiki/Mean_of_a_function

I don't see its immediate uses, but OK.

nickbros123

09:09

My exams are finally over, phew

Can finally study now

X4J

10:07

@nickbros123 we've just proved at class the correspondence theorem

it's beautiful tho

nickbros123

10:18

It's a v useful theorem

Whenever u want to lookat normal subgroups of G that contain N, it's useful to look at all normal subgroups of G/N

Group theory is very interesting, I should study it more. Though I'm more of an analysis kind of guy

Thorgott

@onepotatotwopotato if $M$ is a compact $n$-manifold with boundary, it has a fundamental class $[M]\in H_n(M,\partial M)$ (coefficients being whatever as long as you're orientable w.r.t. them) and its image under the connecting homomorphism of the pair $(M,\partial M)$ is the fundamental class $[\partial M]\in H_{n-1}(\partial M)$ of the closed $(n-1)$-manifold $\partial M$ with the induced orientation

so if $f\colon(M,\partial M)\rightarrow(X,A)$ is some map of pairs, naturality of the pair sequence yields $\partial(f_{\ast}[M,\partial M])=f_{\ast}[\partial M]$

and if $M$ is closed, this is zero, but that's not surprising cause the element then factors through $H_n(X)$ and the composite $H_n(X)\rightarrow H_n(X,A)\rightarrow H_{n-1}(A)$ is zero

X4J

13:43

@nickbros123 yeah and I start getting the point with inner-automorphisms

those automorphisms "pass" to other structure under morphisms

Jakobian

"pass" to other structure?

Essentially, every automorphism is an inner automorphism in some larger group

they are inner because they happen inside the group

1

A: Why are "inner" automorphisms named this way?

Like the other answerers, I can only conjecture, but I would say this. Given a group $G$, there is always some larger group $G'$ containing $G$ as a normal subgroup and having the property that every automorphism of $G$ is induced by conjugation by a suitable element of $a \in G'$. In that case...

X4J

@Jakobian That's my first abstract algebra course, I said that in a loosely manner lol

oh, but what it means to happen inside the group

Jakobian

@X4J well it doesn't matter since it still doesn't make sense

X4J

I can be more precise

but that's chatting nere

here

Jakobian

@X4J every automorphism is a conjugation by some element, but possibly outside of your group. Those where conjugation is by element inside the group are called inner automorphisms

X4J

13:52

ohh

I see

thanks

So the inner-automorphism of G is in particular the conjugations for which the element that characterizes each one is in G?

Soumik Mukherjee

I can't write zeta☠️

Sine of the Time

14:31

@SoumikMukherjee why?

Jakobian

14:42

@X4J "for which the element that characterizes each one is in G?"?

You need to be more precise. This is important for mathematics

Jakobian

14:56

@SineoftheTime Soumik probably means that they can't write zeta on paper

I used to struggle with aleph myself, but I got used to it

Sine of the Time

I've never tried to write aleph

Soumik Mukherjee

15:22

@Jakobian Yes, I struggle with Xi also

X4J

15:40

@Jakobian Sorry. Anyways, I've read the post you've mentioned above regarding the topic. Thanks

Soham Saha

Found an easter egg: Try typing “teapot” in Desmos 3D

And here’s a modified version: desmos.com/3d/mflzdc5lhh

Sine of the Time

@SoumikMukherjee xi is the most satisfying letter to write

ILikeMathematics

@BenSteffan

No group theory in Algebra I at Bonn?

Or am I not seeing something

Also for some reason there is no single webpage for an Algebra II course

nickbros123

15:55

@SohamSaha wow this is cool

can I just look at the proof for "Axiom of Choice if and only if Zorn's Lemma"?

Xander Henderson

@SohamSaha I assume this will cause it to render the Utah teapot?

think_meaning_buildß

but @XanderHenderson your avatar already has glasses :^)

yesterday, by Xander Henderson

Ugh... I need reading glasses... :(

ILikeMathematics

16:11

Oh, it's in Einführung in die Algebra

Xander Henderson

16:24

@think_meaning_buildß My day-to-day glasses are for myopia. But they don't work that well for reading. I am going ot need to get reading glasses soonish, I think.

Sine of the Time

Jan 17 at 20:30, by leslie townes

time to update that avatar image.

psie

17:00

I'm working an exercise called Lusin's theorem (see above). What troubles me is that I'm unsure what kind of measure $\mu$ is supposed to be? Is it a Lebesgue-Stieltjes measure? In Theorem 2.26 (not shown), it says if a function $g$ is in $L^1(\mu)$ and $\mu$ is a Lebesgue-Stieltjes measure, we can find a continuous function that approximates $g$ in the $L^1$ metric. I feel like I need $\mu$ to be a Lebesgue-Stieltjes measure.

ModularMindset

17:35

Happy Friday!

psie

I think $\mu$ has to simply be Lebesgue measure on the Lebesgue $\sigma$-algebra, since $f$ is Lebesgue measurable.

think_meaning_buildß

@SineoftheTime thanks for that link

copper.hat

17:50

@psie it is certainly true for the Lebesgue measure, i looked at some books and they all have different takes on Lusin's (Luzin's) theorem. Rudin has the vague $\mu$ is a measure on a locally compact Hausdorff space which has the properties stated in Theorem 2.14 (seems to be inner & outer regular, finite on a compact set, complete).

think_meaning_buildß

the future of AI looks bright

copper.hat

Durrett is even more vague, and it is a exercise. Kantorovich & Akilov requires the function to be ae. finite on a compact Hausdorff space (with no mention of regularity, strangely, but it is possible that they have some additional assumptions embedded earlier on).

@think_meaning_buildß is there a particular time that you think is interesting? that is almost 60m long.

think_meaning_buildß

40 min was interesting for me

copper.hat

i'm not a real fan of ai, much overhyped by folks who do not understand what it is.

think_meaning_buildß

this guy appears to know what he's talking about

copper.hat

17:58

statistics on steroids.

Jakobian

what are some good accessible books

math books

copper.hat

the problem is that you get wealthy successful people talking about things that they may or may not understand at a fundamental level. clearly these are folks that got something right in life, but that does not mean that their perspectives off piste are meaningful.

Soham Saha

@XanderHenderson yep

Jakobian

I've learned basics when it comes to Wallman-Frink compactifications and realcompactifications

now onto another chapter and I will hopefully get to construction of continuous pseudometrics, which is what interests me

copper.hat

people send me pdfs from time to time, i just accumulate them.

Jakobian

18:03

the book is really bad but I think I can benefit from it

think_meaning_buildß

yeah, he's quite the successful business man

Soham Saha

@copper.hat Ha, here are some pdfs I ‘accumulated’: drive.google.com/drive/folders/…

copper.hat

@SohamSaha i'm slightly uneasy with the provenance of some of my pdfs, so i don't make them public.

Soham Saha

No prob

Sine of the Time

copper is an undercover CIA agent

Xander Henderson

18:10

@copper.hat Admit it: they're from Trump's bathroom, aren't they?

copper.hat

indeed. if i was in the intelligence community, i would in j2, not cia

@think_meaning_buildß Musk has started spouting on all sort of stuff, but has a fairly loose grasp of history & economics. i'm not saying that one needs to be an expert to have an opinion, but he certainly brings the sycophants & gullilble out.

Sine of the Time

lower than you Erdos number

copper.hat

if i was in the intelligence community, it would be j2, not the cia.

that statement, of course, means i am in neither.

my Obama number is 2.

Xander Henderson

How do you determine an "Obama number" or "Trump number"?

Sine of the Time

My copper number is 1

copper.hat

18:14

i know someone who knows Obama, that makes 2.

Xander Henderson

Ah.

copper.hat

@SineoftheTime do i know you?

Sine of the Time

No, but I know you :)

think_meaning_buildß

@XanderHenderson if you stormed the Capitol your number is zero

copper.hat

its transitive :-)

Sine of the Time

18:15

Six degrees of separation

Six degrees of separation is the idea that all people are six or fewer social connections away from each other. As a result, a chain of "friend of a friend" statements can be made to connect any two people in a maximum of six steps. It is also known as the six handshakes rule. Mathematically it means that a person shaking hands with 30 people, and then those 30 shaking hands with 30 other people, would after repeating this 6 times allow every person in a population as large as the United States to have shaken hands (7 times for the whole world). The concept was originally set out in a 1929 short...

Xander Henderson

@SineoftheTime Yes, I am aware. But both Eros and Bacon numbers place restrictions on what it means to be connected to a person.

think_meaning_buildß

Do we admit negative numbers

copper.hat

it was a joke some time ago, this whole Trump/Obama number thing

my dad used to do business with Gadaffi

think_meaning_buildß

surely storming the Capitol was irrational

copper.hat

i agree, but i'm staying off discussion that directly. only innuendo.

the Italian suppository.

think_meaning_buildß

18:18

and dare I say complex

copper.hat

maybe quaternion...

many folks, MTG in particular, were able to do some peculiar logical rotations of position and memory about such events that could only be explained by quaternions.

ModularMindset

my steve jobs # is 2

copper.hat

was never a fan of jobs.

i was/am a fan of Woz

ModularMindset

woz # is 3

copper.hat

apple used to be much more open and contributing, like in the homebrew days

jobs is one of those people i was rambling about above.

ModularMindset

18:23

My cousin is friends with Jobs neice

copper.hat

linkedin used to have a degree of separation thing

think_meaning_buildß

now it's a meet market

copper.hat

i worked in silicon valley starting in the late 80's, you meet all sorts of people.

i believe it was Tim Cook nixed the permissions for my company (long time ago) to put the apple logo on our site as a customer (saying Apple was ok, but the logo no).

but it still amazes me that it is hard to find a crisp statement of Lu[sz]in's theorem. one of the few cases that Wikipedia gets it right

my dad used to do the finances for an Apple manufacturing plant in Ireland

the gm was mad at jobs, when jobs came to give a talk, he insisted that the whole operation shutdown because it made a little noise that bothered him. cost 2 weeks to bring the line up again.

after Gaffadi. before he rose to power

they bought Irish beef.

they had to be careful not to mix with some other groups, notably the rabbis from Israel.

think_meaning_buildß

I see.

copper.hat

the world was a crazy place back then. just not published on ig/twitter every second

psie

18:30

@copper.hat yeah, but I like the way it is stated in Folland's :) makes it more straightforward if the function is Lebesgue measurable. Then we know that we are dealing with Lebesgue measure on the domain, no need to worry about any other measures (for which the theorem might hold)

copper.hat

@psie i would deal with Lebesgue, in practice all you will need, i imagine.

too much generality sometimes makes things useless because it becomes hard to remember.

psie

indeed

copper.hat

i personally find it a bit aggravating (albeit understandable) that every text has mistakes.

psie

it's annoying, but it's human

copper.hat

the bible says that one should not kill, but has many admonishments to do the opposite...

not that it would be my guiding light.

Xander Henderson

18:34

A better translation would be "don't murder".

copper.hat

my Aramaic is a bit sketchy

Xander Henderson

Hebrew, n'est-ce pas?

Or did Jeebus have more to say on the matter?

copper.hat

i could generalise and say that I am a bit sketchy

The first time I saw Hebrew, I thought it was something from Startrek

ModularMindset

Jesus said to the disciples: "Drink this blood eat this bread of life. One does not survive on mana alone."

copper.hat

my understanding was that most of the Old Testament is Hebrew, with some parts Aramaic. but could easily be wrong.

psie

18:38

writing 300 pages or more in math, boy, that must be "many dots to connect"...as some CEO used to say once :-)

Xander Henderson

@copper.hat The Torah is Hebrew, and I assumed you were referencing Exodus (which is in the Torah).

copper.hat

I think Musk seems to be the Second Coming

@XanderHenderson i am skating on microscopically thin ice here

personally, i go for Ogham stones

think_meaning_buildß

That bullet grazing his ear made him the chosen One.

copper.hat

curious to see how that bromance will evolve

Xander Henderson

@copper.hat Trusk?

Mump?

What is their couple name?

copper.hat

18:45

Mumps seems more appropriate

think_meaning_buildß

The UFC couple.

copper.hat

just to be clear, i am no fan of their political opponents who would not recognise an oncoming train if they were standing looking at it

as i have pointed out multiple times over the past few years to relevant folks

ModularMindset

Have you tried the Trump Musk ? It smells patriotic

copper.hat

seems like a marketing opportunity

along with the Trump Bible

Xander Henderson

My prediction is that bromance doesn't last more than 6 months. Two big egos in the same room? No way.

copper.hat

18:48

i think you are being generous with 6mo

i gotta say, Gaetz lifted my spirits yesterday

Xander Henderson

@copper.hat I said it wouldn't even last that long.

It is still 2 months from inauguration. They won't break up until after then.

Ben Steffan

@ILikeMathematics no group theory in algebra 1

algebra 1 is commutative algebra

copper.hat

i used to commute

Ben Steffan

basic group theory is taught as part of the Einführung lecture

otherwise there is nobody really doing group theory here so the chances for learning advanced group theory here are small

although there's somebody here doing geometric group theory, which is at the intersection of group theory and topology

not sure why nobody seems to ever make web page for algebra 2, but it really doesn't matter

what algebra 2 is about is pretty much entirely up to who teaches it

often it's something along the lines of algebraic number theory

but it's not a very popular course and not that many people take it

the standard route is to take algebraic geometry 1 after algebra 1

ModularMindset

19:17

I have a question

Let $X=[0,1]^n$. For all $n>2$ does $X$ admit a unique (up to permutation) codimension one surface of revolution, $L$, with a complete metric (away from the cone points) and an embedding $e :L\hookrightarrow X$, which maximizes volume while retaining constant positive sectional curvature?

Addendum: sup dist_n (p,q) = sqrt{n}

where p,q are cone points in $L$.

Xander Henderson

20:15

Wait... The Onion bought InfoWars!? Ha!

psie

20:28

It's been a long week. Let $B(r,x)$ be an open ball of radius $r$ and center $x$ in $\mathbb R^n$. It is claimed that $$\chi_{B(r,x)}\to\chi_{B(r_0,x_0)},$$pointwise as $r\to r_0$ and $x\to x_0$ in $\mathbb R^n\setminus S(r_0,x_0)$, where $S(r_0,x_0)$ is the sphere $\{y:|y-x_0|=r_0\}$. How does one show this?

Here's my attempt. Case 1: $|x - x_0| < r_0$, i.e. $\chi_{B(r_0,x_0)}(x) =1$. How do I proceed to show that for large enough $n$, $\chi_{B(r_n, x_n)}(x) = 1$?

Case 2 is; $|x - x_0| > r_0$, i.e. $\chi_{B(x_0, r_0)}(x)= 0$. But I'm having the same issue here. I don't see how to conclude for large enough $n$, $\chi_{B(r_n, x_n)}(x) = 0$.

psie

20:48

In my previous message ^, I meant to put $x_0$ after $r_0$ in the arguments for $B(\cdot,\cdot)$. That's silly.

copper.hat

@psie it is not true. $B_n=(0+{1 \over n}, 1+{1 \over n})$, $B=(0,1)$, but $1_B(1) =0$ whereas $1_{B_n}(1) = 1$ for all $n$.

it is true for points in the exterior or interior of $B(x_0,r_0)$.

if $\|y-x_0\| < r_0$ and $x \to x_0, r \to r_0$ then the inequality $\|y-x\| < r$ will hold close by.

psie

@copper.hat yeah, that is what I was sniffing

but why will it hold close by?

Claudio

I did the midterm exam today and I must say it did go pretty good, at least I think so, those exercise sheets really paid off :p (and I wanna thank the chat in general for the help I got as well :P)

Jakobian

If a professor doesn't answer any of your emails, does that mean they think you are a crank

psie

@psie that was stupid of me, see how I misused $x$ here. Jeez...

copper.hat

20:58

@Claudio hope it works out for you!

Claudio

@copper.hat Thanks :) I'm forcing myself to stay positive today ahahah

psie

@copper.hat I don't understand what you mean by close by. Do you think you could elaborate?

copper.hat

suppose $x_n \to x_0, r_n \to r_0$. if $\|y-x_0\| < r_0$ then there is some $N$ such that if $n\ge N$ then $\|y-x_n\| < r_n$. So, if $y$ is in the open ball, then $y$ will be in the 'moving' ball for $n$ large enough. Same if you replace $<$ by $>$. Its just continuity.

leslie townes

@Jakobian honestly, i would assume something got stuck in a spam filter before that

copper.hat

also, not everybody responds immediately

psie

21:08

@copper.hat ok 👍thanks

leslie townes

@copper.hat yes they do

copper.hat

i'll have to think about that

trying to decide if i want to go legal on UCSC

my son is waiting for a class to finish which they told him would be next quarter, now they are vacillating

i may have an aneurism before then

pie

Are there any books you recommend that are used for preparing for an integration contest?

leslie townes

copper good luck with that, i am sure that the fine print of any contract says that the UC owes you nothing, and they probably have sovereign immunity even if it didn't

pie

I hope I won't be ignored this time:)

leslie townes

21:13

pie just so that you feel that people aren't ignoring you (which may have been your impression), i had never heard of such things until being on this chat

so i do not have books to recommend

copper.hat

@leslietownes i suspect so. then it will have to be social media...

leslie townes

my best friend has been trying to finish an undergrad degree for, it feels like 20 years? and a different UC just canceled the program she was hoping to graduate from

so yes, let's name and shame on social media

copper.hat

wow, unbelievable

makes oxford look like a bargain

pie

@leslietownes How do people prepare for college-integral contest? like MIT integration bee for example? If it doesn't exist then my worst fear is true, one enters with all of tricks he accumulated before, there isn't books for training like how there is book for IMO.

maybe a better a question is "do this type of books exist?"

think_meaning_buildß

Have you searched YouTube

Jakobian

21:19

@copper.hat how many months do I need to wait then

pie

@think_meaning_buildß no, I have not, can you recommend YT channels ?

copper.hat

@Jakobian :-).

think_meaning_buildß

Just a general search will get you there :-)

leslie townes

@pie i honestly don't think these things existed when i was in college but for something like that i'd start with the higher numbered homework exercises in a standard issue calculus book

and maybe haunt web forums for the rest

Sine of the Time

@pie Are you interested in this book?

pie

21:28

@leslietownes Maybe it would be a good idea to make such a book, If I became a mathematician I will try.

Sine of the Time

leslie townes

lists of past problems for X are a good way of preparing for X

pie

@SineoftheTime Interesting...

Sine of the Time

I have a pdf copy, if you don't find it I can email it to you

pie

@leslietownes I don't think this would work for IMO

@SineoftheTime I appreciate that, If I couldn't find it I would email you.

think_meaning_buildß

21:30

Contest math is a race against time.

leslie townes

yeah, take anything i say with the qualifier that i don't really understand contest math at all and basically didn't know that world existed until i had a degree in it

think_meaning_buildß

No human can come close to the speed of chatGPT

RIP contests

pie

WOAH

chat gpt can solve Olympiad level problems!!!???

It can't even solve integrals(not the super elementary one).

think_meaning_buildß

And the Putnam.

pie

WOAH

HOW? WHEN? I don't believe it.

think_meaning_buildß

21:42

Oct 17 at 8:56, by think_meaning_builds

https://www.youtube.com/live/2xYosqVjROc?si=dPoaxgIry_-Tc8Xk

pie

AT this rate 2030 AI can solve Riemann hypothesis

Maybe mathematician would be something from the past?

Sine of the Time

No

think_meaning_buildß

No

pie

@SineoftheTime I hope so, but why are you sure?

Sine of the Time

the name AI is an oxymoron

think_meaning_buildß

21:48

Intelligence is not well understood.

Xander Henderson

22:07

@copper.hat UCSC.... So he's waiting on his advanced hacky sack class?

leslie townes

hahaha

Binky

Hi

think_meaning_buildß

Hi

Sine of the Time

hi

Xander Henderson

No.

think_meaning_buildß

22:13

Full stop.

Binky

but is there a way to understand if a seemingly indeterminate function limit is possible to solve without using Hopital or Taylor expansions?

Sine of the Time

@Binky that's too broad

Xander Henderson

22:28

@Binky "solve" is not the right word here.

You are talking about evaluating limits.

copper.hat

22:44

@XanderHenderson I would be soooo happy if that was the case.

He took this quarter off because the class was not offered. Supposedly it was to be offered in the upcoming quarter, but apparently he is getting mixed messages now.

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$Mathematics$ Mathematics