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10:00 PM
Hi @Adeek
 
@Antonios-AlexandrosRobotis you should read Ravi vakil
ohhh @MatheinBoulomenos
how are you my algebraic bro :)
 
aka alge-bro
 
haha
 
I'm doing fine, thanks. And you?
 
@BalarkaSen ok algebra was a mistake
 
10:01 PM
lmao
@Daminark How are you doing alge-bro?
 
@MatheinBoulomenos I am good finished first 20 page of my thesis. So I am happy. Can't wait to go until Grothiendieck-Riemann-Roch
 
@Eric Don't hate on my alge-bro nation
 
I was also solving Yoneda lemma in Ravi-Vakil I am close to solving it
 
seethes with hate
 
I would like start working on higher K-theory
@Antonios-AlexandrosRobotis Actually I think in order to define any abstract notion of geometry you need sheaves. I think geometry is just how classes of function behaves in your space.
I think how the classes in very abstract sense behave gives you geometry.
 
10:10 PM
I won't hate on that idea, but I don't think it's mere "function space, cool shit, sheaves - bam, geometry". Alain Connes thinks this way, but his work involves nontrivial connection between the actual geometry of the space as we perceive it and the algebra/functional analysis of the function space of that space
It's not as trivial as what you make it sound like
 
@BalarkaSen dabs on the haters
 
@Daminark vlogs on them haters
 
There's lots of geometry that totally doesn't fit in this description is the thing
 
Translating between the two sides of the story is a nontrivial field of study I mean :)
It's called noncommutative geometry
(Eric knows this of course)
 
is there a trivial field?
 
10:16 PM
what is a trivial field
 
Is this a pun or a serious inquiry
 
i didnt notice the pun lol
i was thinking about alain connes so automatically thought Leaky was speaking about F_1
well rip me then
 
I also immediately thought it was a joke about fun
 
lol
7 mins ago, by Balarka Sen
Translating between the two sides of the story is a nontrivial field of study I mean :)
 
My advice about studying F_1: don't
 
10:20 PM
what is $F_1$?
 
nobody knows
 
Idk what the deal with f_1 is
 
it's a mystery
 
what is it ?
 
It's a thing that some people think should exist
 
10:20 PM
what things does it relate to ?
 
who knows if it does
 
@MatheinBoulomenos Hey do you want to take a look at my proof?

I have one question though. I'm fine with the fact that if the limits of the real and imaginary part exists then the limit of their sum exists which is just the linearity property of the limit. But I'm not comfortable with the other direction, since for ex. $\lim 0 = \lim (x-x)$ exists, but $\lim x + \lim -x$ not?! So is there a problem with the proof?
Let $f: \mathbb R \to \mathbb C$ be a function. Then

$$\begin{aligned} f'(x) &:= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \\&= \lim_{h \to 0} \frac{\Re(f(x+h))+i\Im(f(x+h))-(\Re(f(x))+i\Im(f(x)))}{h} \\&= \lim_{h \to 0} \frac{\Re(f(x+h))-\Re(f(x))+(\Im(f(x+h))-\Im(f(x)))i}{h} \\&= \lim_{h \to 0} \frac{\Re(f(x+h))-\Re(f(x))}{h} + \Big(\lim_{h \to 0} \frac{\Im(f(x+h))-\Im(f(x))}{h} \Big)i \\&= \big( \Re(f(x)) \big)' + i \big( \Im(f(x)) \big)'\end{aligned}$$
 
haha @MatheinBoulomenos @BalarkaSen
 
@philmcole there's the exact problem with your proof that you were pointing out: you don't know if the limits for real and imaginary parts exist. I think it would be a good idea to convince yourself that a limit (of a single-valued function) in $\Bbb R^n$ exist iff the limit of the components exist. This is not related in particular to $\Bbb C$ or derivatives
 
10:25 PM
@MatheinBoulomenos Oh I totally forgot about that theorem. Thanks for the hint!
 
Okay so the idea with $F_1$ is that we're not really looking for a specific field, we're looking to replace "fields" as something that we do stuff over (like linear algebra or studying varieties) by something new such that there exists an object $F_1$ that has certain properties that we want. The idea that this should be possible comes from several analogies. E.g. $\Bbb Z$ is kind of the polynomial ring over something, so we would want that $\Bbb Z = \Bbb F_1[x]$ if the RHS is suitably defined.
 
hiiiiiiiiiiiiiii
 
In combinatorics, there are various analogies (cf. en.wikipedia.org/wiki/Q-analog) that involve some ordinary sets and linear algebra over finite fields, so want that "vector spaces over $\Bbb F_1$" are just pointed sets, and affine spaces over $\Bbb F_1$ are sets
 
If it exists meaningfully, the point is it would rigorously establish number fields as spectrum of certain $\Bbb F_1$-varieties
 
yes, that's the point
it's a huge techincal difficulty that $\Bbb Z$ is not an algebra over a field
 
10:30 PM
That would settle the "dictionary" that goes between algebraic geometry and algebraic number theory
Right
 
In particular, for varieties over (certain) fields, we have solved the analog of the Riemann hypothesis (see Weil conjectures), so some hope that a proper theory of $\Bbb F_1$ might even settle the Riemann hypothesis if you could adapt Deligne's proof in a suitable way
 
(I meant the other way around above, I think. Spectrum of number fields would be F_1-varieties)
 
So far, various complicated theories were proposed for $\Bbb F_1$ but afaik no one behaves precisely like we want
 
spooky stuff
 
do you know why it's so hard to pin down
 
10:43 PM
Would someone give me a little algebra hint? I need to isolate $y$:
$(y-1)e^{y} = -e^{-x}-\frac{1}{3}e^{-3x}+c$
 
transcendental equation
no, you can't isolate $y$
 
D: My professor is mean
He does this from time to time haha
 
lol
 
How would I go about showing it can't be isolated, just out of curiosity?
 
With pain
 
10:47 PM
whips out a bottle of tylanol I'm ready for the pain
 
It's not trivial to proof. Not even trivial to write down what it means to not being able to isolate $y$, let alone
 
@BalarkaSen so fast
 
He's got a set of hot keys to all the different arxiv papers he likes :P
 
@CookieToast this actually isnt a bad idea
 
I wrote "proof" instead of "prove"
someone clobber me to death please
kthnxbai
 
10:51 PM
@EricSilva What would be even better would be software that offers you possible arxiv papers based on the text in this chat
Probably wouldn't be the hardest machine learning project to pull off
 
please
i only read vixra papers
 
i only read vixra papers that are made in MS paint
 
Oh please. I only read them if they're written on 500 year old tablets found in caves
archaeology meets arxiv
 
arxivology
 
historical math texts are actually my kink tho
 
10:54 PM
I only look up at the stars like ancient cave peoples and count stars.
 
What is actually the difference between the logic of philosophy and logic of maths?
 
^^^ @LeakyNun :P
My cousin taught a few semesters of intro logic at a CSU and her lecture notes looked pretty darn similar to intro logic in most math departments. But I wonder if the similarities stop there.
 
I'd say philosophers care a whole lot more about the semantics of logic, but I'm no expert so take this with a grain of salt
 
@EricSilva 2000 years ago
(Pythagoras theorem in Euclid's Elements)
@CookieToast I don't know what logic in philosophy looks like
 
10:59 PM
the real proof of pythagoras is dope
 
do they have semantic-completeness? first order logic?
 
Please, I read my ancient Greek without translation on the side
 
lol
 
I'm pretty sure, but I don't think they use the same nomenclature
 
11:00 PM
@BalarkaSen lol nope
@CookieToast how do they call it?
 
@LeakyNun It's right up your alley
 
ya i have a greek copy of euclid and it's one of my fav possessions
 
my reply to the comment before the edit would have been "kinky"
 
I'd have to ask my cousin to be sure. I'll see if she'll send me her lecture notes too
 
11:02 PM
holy fk, fta proved in metamath
 
@BalarkaSen lmao
 
@BalarkaSen #freudian
 
#justFreudianthings
 
@MatheinBoulomenos do you see how fta is proved in metamath?
 
@LeakyNun I don't know anything about that kind of stuff
 
11:04 PM
What is metamath?
 
> Lemma for fta 20540. There exists some r such that F has magnitude greater than F ( 0 ) outside the closed ball B(0,r).
 
I prefer my proofs to be human-legible
 
hmm, looks like the standard topological proof
 
Does it proof theorems with computer?
 
@MatheinBoulomenos That's why you don't do analysis then
 
11:05 PM
@AlessandroCodenotti exactly!
 
@AlessandroCodenotti noooooooonseeeeeeeennnnnsssssseeee
 
@Mathei is secretly a topologist I'm telling you
I have not seen any algebraist speak such blasphemy
 
why? algebra proofs are perfectly human-legible
 
> You can be extremely confident that the proofs follow from their axioms. All reasoning is done directly in the proof itself rather than by algorithms embedded in the verification program. Computer verification programs never get tired and rigorously check every step.
 
they are human-legible to mutants
not humans
 
11:07 PM
@quallenjäger so I think they are written by hand and checked by computers
afterall, proving things is not polynomial
 
lmao, this book is listed under the category "non-fiction books for children" (= "Kindersachbücher" in German) ebay.de/itm/…
 
Jungendbücher
 
probably didn't get his 10 year old children all that exited with the book
 
@MatheinBoulomenos Well, that's weird, abstract algebra is clearly fiction
 
@AlessandroCodenotti Lol
 
11:16 PM
I need some heavier and more atmospheric metal ... I have ran out of to-listens
 
Heavy metals are generally defined as metals with relatively high densities, atomic weights, or atomic numbers. The criteria used, and whether metalloids are included, vary depending on the author and context. In metallurgy, for example, a heavy metal may be defined on the basis of density, whereas in physics the distinguishing criterion might be atomic number, while a chemist would likely be more concerned with chemical behaviour. More specific definitions have been published, but none of these have been widely accepted. The definitions surveyed in this article encompass up to 96 out of the 118...
 
oldjoke
 
@BalarkaSen Sure, can you be more precise on what kind of metal are you looking for?
 
I have been super into doom metal lately. I like progressive things, blacky stuff, post-rock influences, etc
Anything which feels trippy
 
If I have $f(x) = \frac{1}{x} + c_{1}$, $c_{1} \in \mathbb{R}$, I can rewrite it to $\frac{c_{2}}{x}, c_{2} \in \mathbb{R}$ without any issues right?
 
11:19 PM
how can you do that?
The limit is $c_1$ in first case and 0 in second
they are different functions.
 
@CookieToast $\dfrac{1}{x} + c_1 = \dfrac{1}{x} + \dfrac{c_1 x}{x} = \dfrac{1 + c_1x}{x}$
there is a problem there, no?
 
i need an interesting function
thats bounded on [-1, 1]
 
Ahhh because the numerator still varies with respect to $x$ @xander?
 
@MeowMix $f(x) = 47$
 
that's a line
 
11:22 PM
lines are not interesting?
 
im sure my riemann integral demonstration will look very interesting with constant functions
 
$f(x) = \sin\left(\frac{1}{x}\right)$ for $x \ne 0$, and $f(0) = 0$
that is interesting
 
bounded in domain?
 
and bounded
 
or in image?
 
11:23 PM
image
 
@Balarka @Mathein algebra proofs are, by definition, more legible than topology proofs
Because they actually contain words
 
or $x^2 \sin\left( \frac{1}{x} \right)$ if you want something absolutely continuous
 
How about $f:\mathbb{R} \to \mathbb{R}$, where $f(x) = $ the average of all the digits of $x$ @MeowMix
 
uhh
also integrable
 
Not even sure what that means for $x$ $\in$ $\mathbb{I}$
 
11:25 PM
man, you're just putting on more and more conditions
that sucks :(
 
You'd take the limit of averages but that might not converge
 
rather heavy, definitely not as heavy, instrumental post metal and another one from the same group because they're great. They're all more progressive/post rock/post metal than doom, but maybe you'll find something you like @Balarka. If you tell me some bands you like I can give better suggestions probably!
 
@meow $\int_{0}^{\pi / 2}\frac{dx}{1+\tan^{\sqrt{2}}(x)}$
 
what
 
Ted is a big fan of this one so you might have already seen it on here
 
11:26 PM
what is this
 
how does one take a tangent to the square root of 2-th power
 
ugh... I hate that notation; $f^n(x)$ is the $n$-fold composition of $f$, applied to $x$, i.e. $\underbrace{f\circ f \circ \dotsb f}_{n}(x)$.
 
The integral actually has some pretty simple solutions @meow
 
double yu tee eff is $\tan^{\sqrt{2}}$ supposed to mean then? huH?
 
You just need to think waaaaaaay outside the box
 
11:27 PM
stupid bad physicists with stupid bad notation
 
lol @xander
 
@XanderHenderson compose it with itself sqrt(2) times, come on!
 
BUT HOW?
 
some notation like generalized binomial theorem I'm guessing
 
you'd think $\tan^{-1} = \cot$
but youd be WRONG
 
11:27 PM
@Alessandro Thanks a lot! I shall check them out. I have heard some isolated Russian Circles from here and there
 
@MeowMix y would u think that
 
$\cos^2(x)$ = $\cos(x)^2$
 
Ah @meow think of it as $(\tan x)^{\sqrt{2}}$
 
@MeowMix yet another reason that $\operatorname{trig}^n$ is bad notation
 
Using powers for inverse vs 1/x
 
11:28 PM
OH
 
ahh i see
 
lol you guys
 
@XanderHenderson do it once, then do it .4 times, then .01 times, etc :P
 
$\operatorname{trig}^n(x) = \operatorname{trig}(x)^n$ for $n\in\mathbb{N}$, but $\operatorname{arctrig}(x)$ for $n=-1$, and is nonsense otherwise
 
i dont really write $\sin^{n}$ for $n \neq 2$. I do it for $2$ out of habit
 
11:30 PM
Actually someone once told me that she was asked to prove sin^2 + cos^2 = 1 but thought it meant composition and was confused
 
and now I am trying to figure out what fractional composition should look like :\
 
notation is difficult
 
and now I have to go play Barbies with my daughter :\
later
 
wtf
didnt even invite me
 
Meow is hurt and heartbroken
 
11:34 PM
If we work with real numbers, we have that $|x| < c \implies -c < x < c$.
 
yes
 
Does a similar thing hold with complex numbers? I don't see how it can
 
we do
well if you have $|z| < c$ you have an open disk
if you have $0 < |z| < c$ you have a punctured open disk
 
Yeah. Can I put a lower bound on $|z|$?
 
You don't really order the complex numbers
 
11:35 PM
if you have $c_1 < |z| < c_2$ for non-zero constants, you have an open annulus
2d donut basically
 
I'm doing an $\epsilon - \delta$ proof, and I have $|z| < c$, but what I need now is a lower bound, so when I invert the inequality, I have an upper bound for $\frac{1}{|z|}$.
I'm trying to see if $|z| < c \implies |z| > b$ for some $b$. But it doesn't seem like that's right.
 
to be honest my complx analysis is a bit rusty
and i dont remember doing $\epsilon-\delta$ with $\Bbb C$
someone else here should know, though. sorry
 
No worries
 
@MeowMix did you have to graph and find out if the graph is open or closed in complex analysis? ugh I'm facing the same nightmare from last semester
I could've sworn epsilon and delta proofs are in real analysis???
 
they are
 
11:50 PM
Okay, if $|z-z_0| < c$, then it must mean that $|z|$ is bounded below by some non-zero constant, otherwise it would not be inside the disc.
Is $|z_0 - \frac{z_0}{c}|$ a tight lower bound?
 
If I have $f(x)dx = f(y)dy$ then $x = y$ right? This isnt a trick question? Cause my prof has been throwing tricks at me left and right
 
@Semiclassical You there?
 
topology question, let $A= \{ 1,\frac{1}{2},\frac{1}{3},\frac{1}{4} ,.....\} $ what is the closure of A and what is the interior of A?
 
Closure would be $A$ with 0.
only limit point is 0.
 
as
an interval?
like [1,0] ?
 
11:56 PM
$\bar A = A \cup \{0\}$
 
wierd
 
Not an interval.
Why is it weird?
How did you define closure?
 
it is the intersection of all closed sets containing A
 
So, the smallest closed set containing A
 
yeah
 
11:58 PM
What does it mean for a set to be closed
 
syntax error
its complement is open
 
syntax error?
Okay, have you learned about limit points?
 
lol my way of saying i didnt know
 
@Faust That depends on the topology on $A$
 
i know what they are but we havent learned about them
 

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