Mathematics

Erik Humphrey

3:00 AM

yes it should be 12

MatheinBoulomenos

Well, finite abelian groups are kinda boring

Leaky Nun

a fundamental flaw, in my opinion, is that you can get something non-abelian from abelian groups... i.e. automorphism

MatheinBoulomenos

I like non-abelian groups as well

Leaky Nun

but that can't compare to the greatest flaw of maths

or rather, limit

i.e. the incompleteness theorems

MatheinBoulomenos

but yeah modules over PIDs tend to behave quite nicely

Leaky Nun

3:00 AM

we'll never know if our system is consistent

Erik Humphrey

$\frac{1}{n} + 12 \leq n^2 + 12n^3$

$\frac{1}{5000} + 12 \leq 5000^2 + 12(5000)^3$

orbit-stabilizer

@ErikHumphrey What are you trying to do?

anon

4 mins ago, by anon

in any case if $n>1$ then $A\le B$ implies $An^3\le Bn^3$ so indeed $1/n+12\le (1/n+12)n^3$ if that's what you mean

yes

Daminark

I've definitely had a lot of fun with algebra so far, though I've found my favorite parts to be the conceptual bits, as well as counting-type problems

Erik Humphrey

$12.0002 \leq 1500025000000$

evaluates to True

anon

3:01 AM

showing $1/n+12$ is $O(n^3)$

simplest to observe $1/n+12\le 1+12=13\le 13n^3$ is $O(n^3)$

Erik Humphrey

$\frac{1}{n} + 12$ is $O(n^3)$

QED

Daminark

Moreso than playing with cycles, and especially more than playing with $D_n$

Leaky Nun

I've heard that cohomology algebra is nice

orbit-stabilizer

I hear calculus is amazing

MatheinBoulomenos

I never hear about cohomology algebra, perhaps you mean homological algebra

orbit-stabilizer

3:03 AM

/s

Erik Humphrey

Okay, 7 more to do like this.

Leaky Nun

@TedShifrin what was the greatest news in maths throughout your entire career?

Erik Humphrey

I have calculus next semester, but I doubt it includes big O notation

Daminark

@orbit-stabilizer ewwwwww

Erik Humphrey

it's super elementary

orbit-stabilizer

3:03 AM

@Daminark differential geometry isn't nice?

Leaky Nun

@Daminark manifolds are cool

I swear

MatheinBoulomenos

Differential geometry is confusing

orbit-stabilizer

@ErikHumphrey wait, you haven't taken differential calculus?

Daminark

Differential topology is aight but my experience with geometry was just not fun

MatheinBoulomenos

yeah, that was my experience as well

Daminark

3:05 AM

I can't process the geometric bits and the calculus computations are just no fun

Erik Humphrey

differential calculus? what's that?

anon

I'd rather every manifold was assumed a submanifold of euclidean space for proving purposes.

Erik Humphrey

In Canada, you only take one math course in high schools with a breadth of units

they don't have specific titles like "probability and statistics I"

anon

then just keep the fact the abstract manifolds are realizable as submanifolds of euclidean in the background

Erik Humphrey

I took "Calculus and Vectors"

orbit-stabilizer

3:05 AM

Ah, you're in high school

Leaky Nun

uniformaskdjf theorem is dank

Erik Humphrey

No

First year university

Leaky Nun

uniformization?

Erik Humphrey

The calculus course I will take is "Calculus I"

I have yet to take any other calculus in university

perhaps that will include differential calculus?

orbit-stabilizer

Calc I = differential calc

Daminark

3:06 AM

@anon my difftop course did assume everything was a submanifold of R^n

Erik Humphrey

is that where 2x -> 2 and 4x3 = 12x2?

derivatives?

orbit-stabilizer

No, it's moreso limits

Erik Humphrey

I know some limits though this part of asymptotic analysis does not seem to invovle it

orbit-stabilizer

Here

Leaky Nun

$\mathcal I (\mathbf V (I)) = \sqrt I$

Erik Humphrey

3:08 AM

I will solve the rest of the questions and ask if they are right

MatheinBoulomenos

ah, the Nullstellensatz

orbit-stabilizer

Wait

Erik Humphrey

yeah no worries i got the splitscreen thing going on

Leaky Nun

@MatheinBoulomenos ja, der Nullstellensatz

orbit-stabilizer

If |f(n)|/|g(n)| -> 0 as n-> inf then f(n) = O(g(n)). Pretty sure that's right

Leaky Nun

3:08 AM

der Null stellt

orbit-stabilizer

So, you can use your limit knowledge to solve these questions

MatheinBoulomenos

Nullstellensatz is an established term in English :P

Leaky Nun

@MatheinBoulomenos sure

MatheinBoulomenos

"Null" is feminine in German

don't ask me why

Erik Humphrey

I struggle to understand why there is another function g(n) involved

Leaky Nun

3:09 AM

der die Null stellt

orbit-stabilizer

That's your n^3

Erik Humphrey

ahh

orbit-stabilizer

That you were looking at previously

Leaky Nun

der Nullstellensatz, der die Null stellt

Erik Humphrey

then...then what's f(n)

orbit-stabilizer

3:09 AM

that's your 1\n + 12

Erik Humphrey

i see

i will save this for later and try to make sense of it

Leaky Nun

50 mins ago, by Leaky Nun

@TedShifrin is there a polynomial map $\Bbb C^n \to \Bbb C^n$ which is easy to check injective but not surjective?

anon

$\sum n+1/12$ is $O(0)$ tho

5

Leaky Nun

@MatheinBoulomenos

@anon lol

MatheinBoulomenos

@LeakyNun I heard there's a simple proof of the Nullstellensatz using the completeness of algebraically closed fields of a certain characteristic

Erik Humphrey

3:11 AM

0

Q: Is it ever possible to have Big O less than O(1)?

Possible Duplicate: are there any O(1/n) algorithms? Is it ever possible for your code to be Big O less than O(1)?

cool embed

orbit-stabilizer

@anon haha

he's making a joke

Leaky Nun

@MatheinBoulomenos :o

orbit-stabilizer

@anon, don't you mean 1/n?

MatheinBoulomenos

but I don't know the actual proof

Leaky Nun

@MatheinBoulomenos but you can't state it in first-order crap

MatheinBoulomenos

3:12 AM

we proved it -without model theory- in our algebra lectures

orbit-stabilizer

i forget the -1/12 thing

analytic continuation of some function?

Ah it is n, mb

Daminark

Zeta function

anon

analytic continuation of $\zeta(s)=\sum_{n=1}^\infty 1/n^s$ from its abscissa of convergence ${\rm Re}(s)>1$ to $\Bbb C\setminus\{1\}$ yielding $\zeta(-1)=-1/12$

Erik Humphrey

wait

is this a typo

anon

no

Leaky Nun

3:14 AM

> abscissa

Erik Humphrey

shouldn't it say n^2 + 42n + 7 = O(n^2)

Leaky Nun

when you're trying to use professional words

anon

@ErikHumphrey did you read the second bullet point?

Erik Humphrey

Yes

MatheinBoulomenos

@LeakyNun there are several forms of the Nullstellensatz

anon

3:15 AM

unless that's the last bullet point on the slide, I guess it should say $n^2+42n+7$ in that case

Erik Humphrey

yes

the last line should be what you're proving, shouldn't it?

anon

IIRC I saw strong null proved using weak null elegantly, and weak null proved very inelegantly

@ErikHumphrey eh, it's a pp slide

Leaky Nun

why does it matter how a theorem is proved?

MatheinBoulomenos

there's the "weak Nullstellensatz" which states that if $f_1, \dots f_n$ don't generate unit ideal, then they have common zero in $K^n$, (K algebraically closed)

that's first-order

Erik Humphrey

What is the number below my username on the left?

anon

3:16 AM

@LeakyNun mathematical morality

Leaky Nun

@anon proof irrelevance

anon

@ErikHumphrey 465

Erik Humphrey

Thanks. How was this number determined?