/mg/

[Anonymous No.12424218]

Math, generally
>demiurge edition
Talk maths

Alte: >>12410456

[Anonymous No.12424233]

>>12424218
HOLY SHIT I HATE GERMANS SO MUCH

[Anonymous No.12424243]

Alternate OP image for the less artistically inclined

[Anonymous No.12424262]

How do I into geometry and stats?
I only know calc, lin a, and Diffies from college.
also I want into graph theory.

[Anonymous No.12424271]

>>12424262
>stats
Buy a book
>geometry
Just look at shapes

[Anonymous No.12424276]

>the stock market Chads are in my department again for their symposium

'ate em simple as

[Anonymous No.12424314]

For a linear differential equation of order n, we know that there are n solutions. Is there a connection between this and the fundamental theorem of algebra and if so is there a good paper on it? It's trivially easy to see for equations with constant coefficients but I'm curious about the other cases.

[Anonymous No.12424323]

>>12424271
any meme picture guides?

[Anonymous No.12424339]

>>12424276
How's life as an indentured servant treating you?

[Anonymous No.12424355]

>>12424314
Look for connections between gaussian elimination and theory of algebra, is my guess

>>12424323
If you need memes to tell you what to do youre NGMI.

[Anonymous No.12424368]

>>12424355
>Look for connections between gaussian elimination and theory of algebra, is my guess
Is it an unanswered question?

[Anonymous No.12424373]

>>12424355
>If you need memes to tell you what to do youre NGMI.
don't care.

[Anonymous No.12424376]

>>12424368
I don't know, I haven't looked into it, Im just replying to inspire you to find a solution yourself. But when you say "the connection", that can be diverse, there are infinite relations between infinite sets. So in a sense it can never be answered, at least not fully.

[Anonymous No.12424382]

>>12424373
Not saying that you're not gonna make it, bro. Just saying that your attitude needs to be changed. Look for books yourself rather than blindly following the advice of some halfway braindead anime trannies on 4chan.

[Anonymous No.12424384]

>>12424243
>for the less artistically inclined
AKA unimaginative code monkey drones

[Anonymous No.12424405]

>>12424382
going to google and typing "how to learn statistics" one results in meme articles and websites wanting my email
searching "how to learn geometry" brings up high school tier retardation.

[Anonymous No.12424429]

>>12424405
1. Go to any half-way respected university math department website.
2. Read the course outline for math majors.
3. Go to course webpages and find book titles.
4. Obtain books from interweb.
5. ????
6. Profit.

[Anonymous No.12424433]

Sorry if I'm being a cunt, since I've already posted a thread, but does anyone have any additions or critiques on this undergrad reading list? So far I've got
Basic Algebra I and II - Jacobson
Baby, Papa and Grandpa Rudin
Introduction to Topology - Borisovich et al.
A Concise Course in Algebraic Topology - May
Introduction to Smooth Manifolds - Lee
Partial Differential Equations - Evans
A Course in Arithmetic - Serre
Geometry, Topology and Physics - Nakahara

[Anonymous No.12424442]

>>12424218

> my university has "woman in mathematics" meetings for undergrads
> last year I went to one
> no undergrad girls showed up
> me and 4 other undergrad boys showed up
> grad student ladies were really hot
> it was kinda funny

[Anonymous No.12424451]

>>12424433
Can't go wrong with Munkres for point-set topology.

[Anonymous No.12424460]

>>12424442
We always get tons of undergrad women at ours. No guys ever come (they're not allowed) but the undergrads all want to be pampered data science meme professionals and think I'm a nerd for actually liking math. :/

[Anonymous No.12424482]

>>12424451
Is there much point in adding it in with what I already have though? I like Borisovich for topology, because he is fairly concise, while still giving some nice exposition, whereas Munkres seems like a much longer and more tedious slog. Would I be missing out on much point-set topology by foregoing Munkres?

[Anonymous No.12424492]

>>12424433
>A Concise Course in Algebraic Topology - May
no, it's far too advanced for a first reading in algebraic topology

[Anonymous No.12424500]

>>12424492
First reading would be in Borisovich, no? He goes over homotopy and homology, and introduces category theory.

[Anonymous No.12424533]

>>12424262
>also I want into graph theory.
Here's a good book which will help you get into one of those fancy european universities who learn how to solve the Riemann hypothesis on the first day of orientation week -> https://www.amazon.ca/lagrapheur-intrigues-policières-saveur-mathématique/dp/B006W9Q1Y2/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=&sr=

or if you're boring, this one looks pretty nice (especially for the price) https://www.amazon.com/Introduction-Graph-Theory-Richard-Trudeau/dp/0486678709/ref=sr_1_1?dchild=1&keywords=graph+theory&qid=1607211188&sr=8-1

[Anonymous No.12424653]

>>12424482
I can't compare because I've never heard of the book you're thinking of using. My department uses it as the standard text for undergrad topology and I found it quite thorough and enjoyable. It's a tome though, not meant to be read cover to cover when first seeing topology.

[Anonymous No.12424719]

>>12424653
How would you read Munkres then? Currently I'm working through it from the beginning, not skipping anything but maybe skimming some obvious proofs

[Anonymous No.12424746]

>>12424719
Well I didn't originally read it by itself since I was in a class, but I'd say up to section 37 on the Tychonoff Theorem, skipping starred sections, is basically a solid undergraduate course.

[Anonymous No.12425020]

i'm reading Euclid's Elements for a class, and i don't quite understand what is meant in Book X Proposition 10 "the ratio which a square number has to a square number." what does this mean?

also i apologize to the anon who said that it is important to pick a good translation for Elements, when i doubted him/her/they by saying "isn't Elements just a bunch of lines on a paper." it becomes a lot more complicated than that....

[Anonymous No.12425161]

Does anyone know where I can find an english translation of Galois' original papers?

>>12425020
My assumption is he just means the ratio A/B where A and B are both equal to some integer . So 36 25 49 100 all work, but 17 and 12 don't

[Anonymous No.12425175]

>>12425161
*Some integer squared

[Anonymous No.12425184]

>>12424460
Dinask tranny

[Anonymous No.12425249]

>>12425161
yeah i thought so too, but then what is the relation between them?

also, the link doesn't quite work but hopefully its better than nothing

http://www.galois-group.net/g/EN/intro.html

[Anonymous No.12425292]

>>12424442
uoft?

[Anonymous No.12425309]

Is Gorodontsev's Algebra book a good intro?

[Anonymous No.12425342]

>>12425249
>Set out two numbers B and C which do not have to one another the ratio which a square number has to a square number, that is, which are not similar plane numbers
So the ratio of lengths between B and C can't be anything of the form B/C = Q/Z where Q and Z are squares of some int. So B/C can be 2/11, but not 36/64

Ty for the link

[Anonymous No.12425429]

I found this.

[Anonymous No.12425442]

>>12425429
What the fuck is with adding integers and minus 1/12?

[Anonymous No.12425641]

Wtf does she has to do with the axiom of choice?

[Anonymous No.12425650]

>>12425641
She's a politician trying to change the age of consent, nothing to do with set theory.

[Anonymous No.12426074]

I know the answer but I still don't understand how you solve it

[Anonymous No.12426085]

have you guys learned rational trigonometry yet?
https://www.youtube.com/playlist?list=PL3C58498718451C47

[Anonymous No.12426177]

>>12426085
>studying a contrived variant of planar geo/trigonometry tailored to satisfy one person's whims
i'm pretty sure everyone has better things to do

[Anonymous No.12426201]

>>12426177
>anime posting
opinion discarded

[Anonymous No.12426268]

>>12424433
This reads more like a list of books to refresh your knowledge with rather than learn for the first time. No one should be learning out of Serre in undergrad.

[Anonymous No.12426274]

>>12426201
this

[Anonymous No.12426447]

>>12425429
>>12425442
Since the function is mirrored between -1 and 0 at x=-0.5, you'd always get a sum of -1, no matter what y-value in the value range you plug in there.

[Anonymous No.12426464]

>>12425442
I see, so [math]\sum_{n=1}^{-0.2} n = \frac{-1}{12}[/math]

[Anonymous No.12426574]

ok guys let's settle this once and for all
[math]\phi[/math] or [math]\varphi[/math]?

[Anonymous No.12426583]

>>12426574

both

[Anonymous No.12426628]

>>12424218
n/2
3n+1

n/2 for 1/3 domain space of n
(every other even)
3n+1 for 2/3 domain space of n
(Odd and every other even)

Inversely related wrt n
(n/2)(2/3n)=1/3
(3n)(1/3n)+1(1/3n)=
1+1/3n=
1+lim(1/3n)=
1

Functions converge to 1

[Anonymous No.12426678]

i need a source, not a book, but a good sum up of a field in math. the field is practically just proving limits and other qualities of groups and sets. any clue anyone?

[Anonymous No.12426713]

got my algebra exam tomorrow bros, never been so unprepared ;-;

[Anonymous No.12426827]

>>12426713
its ok i just got 5 out of 100 in a classical physics midterm

[Anonymous No.12426849]

Why are Russian math books so good bros

[Anonymous No.12426987]

>>12426268
I prefer learning from books like that though. I get bored if the book holds your hand, or if I'm not struggling to understand the concepts.

[Anonymous No.12427022]

>>12426574
[math]\phi[/math] because it's faster to write.
My choices go [math]\phi \ \psi \ \varphi[/math] and then random available greek letters.

[Anonymous No.12427141]

>>12426849
Which ones so you particularly like

[Anonymous No.12427194]

How can I prove a real function can't have infinite derivative on the whole of a nontrivial interval?
(reason I'm not asking on /sqt/ is I've had bad luck there with Qs like this)

[Anonymous No.12427315]

>>12426849
years of the Soviet Union actually investing in education and I&D while the West just jerked off over the market or some shit

[Anonymous No.12427804]

>>12426074

I have no idea what I’m talking about, but:

you have 101 elements, so the maximum checks is log(101)/log(2), which rounds up to 7

[Anonymous No.12427810]

They say love is a choice. Does the axiom of choice have anything to say on this subject? Checkmate atheists. (I have come to the conclusion that math is a waste of time. Disregard learning, acquire pussy.)

[Anonymous No.12427836]

>>12426177
based. fuck finitists

[Anonymous No.12427955]

>>12427810
sir you made me sad

>>12427194
can you make your question more precise
what's infinite derivative

[Anonymous No.12428007]

>>12427955
I make myself sad too. I was sexually abused by my babysitter when I was 6. Does that mean I am no longer a virgin?

[Anonymous No.12428046]

>>12427955
>can you make your question more precise
>what's infinite derivative
Meant simply f'(x) = inf for every x in the interval. As in sgn(0) = inf

[Anonymous No.12428110]

>>12427810
What if I'm gay, but genuinly not interested in acquiring the male equivalent?
If I don't kms, what I do is irrelevant, I better invest the time in something interesting. My life will be impotent anyway.

[Anonymous No.12428124]

>>12428110
Are you interested in being a househusband?

[Anonymous No.12428143]

>>12428124
I'm not aware of that word. (ESL)

[Anonymous No.12428144]

>>12427194
>>12427955
>>12428046
I guess you're asking why can't a function satisfy
[eqn]\lim_{h\to x}\left| \frac{f(x+h)-f(x)}{h} \right| = \infty[/eqn]
on some interval

[Anonymous No.12428151]

>>12428144
h->0 obviously

[Anonymous No.12428178]

>>12428144
>>12428151
It's not exactly equivalent as I had in mind a constant signed infinity, but still would be interested in such function or a proof it doesn't exist.

[Anonymous No.12428209]

>>12428178
right, just delete the absolute value then
do you know for sure that the statement is true?

[Anonymous No.12428234]

>>12428209
>do you know for sure that the statement is true?
It might very well be untrue..

[Anonymous No.12428255]

you should be able to solve this

[Anonymous No.12428262]

>>12424218
I assume someone here has taken a course in complex analysis. am I missing out by not taking this? I can only fit so many classes into my schedule.

[Anonymous No.12428263]

>>12428255
>>>/g/

[Anonymous No.12428308]

>>12428144
>>12428209
solution for the statement without abs value:
assume the condition is true for a function f on an interval [a, b], choose some n
then for each x in the interval, you can pick [math]l_x < x < r_x[/math] such that [math]\frac{f(p) - f(x)}{p - x} > n[/math] for all [math]p \in (l_x, r_x)\setminus \{ x \}[/math]
so this implies [math]\frac{f(p) - f(q)}{p - q} > n[/math] for all [math]p, q \in (l_x, r_x)\setminus \{ x \}[/math]
now by compactness choose a finite covering of [a, b] by the intervals [math](l_x, r_x)[/math] and fuck around to get [math]f(b) - f(a) \geq n(b-a)[/math]
but obviously that can't be true for all n, so there's a contradiction

[Anonymous No.12428361]

>>12428308
I'm not 100% convinced, but seems legit. Thanks

[Anonymous No.12428368]

>>12428124
I'm curious, why did you ask?

[Anonymous No.12428386]

>>12428308
So for the courageous we've still got the absolute value version. This is a nice and simple proof, but the impossibility must follow from uncountability, I believe. Simply can't act discretely with it.

[Anonymous No.12428564]

>>12428143
It means you stay at my house and do all the chores and provide definitely not gay comfort for me when I get back from work.

[Anonymous No.12428566]

>>12428368
Probably because I’m interested in having a househusband?

[Anonymous No.12428573]

>>12428262
It is something you will definitely see in grad school again if you plan to go down that route. What would you rather take instead?

[Anonymous No.12428639]

Brainlet moment here. If [math](f^{-1})'(x) = \frac{1}{f'(x)}[/math], when [math]f'(x) \neq 0[/math], why isn't [math]log'(x)=\frac{1}{exp~x}[/math] (on positive reals obviously)?

[Anonymous No.12428644]

>>12428639
>If (f−1)(x)=1f(x)(f−1)(x)=1f(x)
this is not true. think about it.

[Anonymous No.12428649]

>>12428644
Brainlet moment cured, thanks.
It's just that for f(a)=b, the inverse derives to 1/f'(a) at b.

[Anonymous No.12428700]

>>12428573
combinatorics and graph theory.

[Anonymous No.12428765]

>>12424233
ok chang

[Anonymous No.12428864]

What is the use behind Euler's theorem on homogenous functions? I've had the proof as an exercise but even after googling I have no idea what's the point.

[Anonymous No.12428937]

What's a good book on analytical geometry for someone who pursues a math major?

[Anonymous No.12428973]

>>12428700
If you are willing to learn complex analysis on your own during your winter or summer break, then it is not as essential as a solid foundation in analysis, algebra, and topology, but it is good to have.

[Anonymous No.12428977]

You ever be like "wait i'm not in high school anymore"

>>12428937
>analytical geometry
any trig book probably

[Anonymous No.12429161]

hello Senpai
who wan self-learn ? join our autists
we have all manner of degenerates, including but not limited to
rust programmers, category theorists, one (1) communists, and of course a tranner
wide ranging interests Guaranteed to Enrich and Officially Debunk ur worldview
we're doing Advent of Code 2020, join for the wild ride
previous projects include SICP, overcoming dekinai, forallx, GEB groupread, attempt x to kill slack, and some dude doing combinatorics really loudly

Unironically, we're a group of people interested in all kinds of intellectual subjects related to STEM and the humanities. Looking for more maff people.
https://desu-cartes.org/signup.html
dog bless

[Anonymous No.12429256]

>>12429161
>dekinai
what is this

[Anonymous No.12429594]

Post trivia about 18

[Anonymous No.12429616]

>>12429594
18 is the smallest difference between an emirp and its reverse

[Anonymous No.12429871]

>>12429594
17 = 17 * 0.99...

[Anonymous No.12429914]

>First time doing Analysis
>Professor gives a 5 exercise list about the first chapter of the book
>Can't solve a single one of those questions
>Several people in class managed to solve them all
Sighs, I'm so tired of this. I'm starting to believe I chose the wrong field. I love mathematics, but I can't solve basic analysis shit, I'm completely worthless, this is so sad, if I fail this semester I'll quit maths even if it will pain me very much.

[Anonymous No.12429917]

>>12429914
Is that baby rudin?

[Anonymous No.12429919]

>>12429914
me too bro, got an analysis exam later today

can someone help me please, im trying to solve a singular perturbation problem but I'm having trouble rescaling

i have Ex^3 + x -1 = 0, where E is epsilon

what should i rescale x = y/(E^v) as?

[Anonymous No.12429920]

>>12429914
is this your first proof-based course?

[Anonymous No.12429923]

>>12429917
Yeah.
>>12429920
It is, I read a little of How to Prove It at the beginning of the year to get ready for proof based courses, but I already forgot most things of the book and the ones I remember didn't help me much.

[Anonymous No.12429945]

>>12429923
transition from computational math to pure math is insanely hard. the people from your class who completed all problems are simply ahead of you, but they probably sucked hard at first. you need to grind proofs at least for a few months before you can decide that you "chose the wrong field". I recommend linear algebra or basic group theory instead of analysis to become acquainted with proofs though.

[Anonymous No.12430074]

I'm looking for a seminorm whose unit ball has this kind of shape.

Any ideas?

[Anonymous No.12430283]

How do I find the number of un-ordered subsets of size N in a set of size M? I intuit that it relates to the nature of euclidean space. It's also interesting that the solution for N=2 is generally (M-1)M/2 which is the sum of the first M-1 integers!

[Anonymous No.12430308]

>>12430074
The length of a vector is the amount you need to dilate that shape by in order for your vector to lie in the dilate.

[Anonymous No.12430310]

>>12430283
I think I have found a recursive form! The degree of N corresponds to the power of the polynomial of growth!

[Anonymous No.12430325]

>>12430283
>unordered subsets

[Anonymous No.12430331]

>>12430325
Pedantry gets degree
Autism speak sets you free

I think I discovered a general explicit form too using integrals on the recursion, I'm too lazy to compute it though. Can you check my work that this is the correct series for N=3?
1
4
10
20
35
56...

Interesting that 3 starts off more constrained than 2 but then blows up faster. Groups of size 3 don't fit as well in small space but allow for more creativity - they have more options when there is room in larger space

[Bill Dubuque No.12430343]

Bill Dubuque

[Anonymous No.12430558]

>cumtrapz
wtf i love numerical analysis now

[Anonymous No.12430560]

>>12430558
Also:
>trapz(x, y)

[Anonymous No.12430588]

Would the category of isomorphism classes finitely generated modules be small in general? Do you know any conditions for the ring that would be needed?

[Anonymous No.12430603]

>>12430343
>Bill Dubuque
https://en.wikipedia.org/wiki/Macsyma
Bill Dubuque

[Anonymous No.12430655]

>>12429914
Seconding >>12429945 the first proof based course often tears people up, especially if it's Rudin style analysis. Give yourself more time before deciding you're stupid

[Anonymous No.12430660]

>>12430603
(indefinite integration, limits, power series, number theory, special functions, functional equations, pattern matching, sign queries, Gröbner, TriangSys)

[Anonymous No.12430665]

>>12430588
Well every finitely generated module over a commutative ring R is isomorphic to a quotient R^n/Q with Q a submodule of R^n.
Using that, it looks pretty easy to prove that it's a small category.

I haven't worked that much with non-commutative rings, so I don't want to say something wrong, but the argument can probably be adapted to that case.

[Anonymous No.12430671]

>>12430665
Thank you very much my friend. I was expecting all the replies to be something like:
>algebra
dilate tranny
You have a nice day or evening or whatever it is in your country right now!

[Anonymous No.12430675]

>notation for a set to be normal in a group is the same for the notation for a set to be an ideal in a ring

[Anonymous No.12430684]

>>12430675
Being a normal subgroup = being the kernel of a homomorphism of groups. Being an ideal = being the kernel of a homomorphism of rings.

[Anonymous No.12430698]

is [math] (R,-) [/math] a non-associative magma?

[Anonymous No.12430704]

>>12430684
still dumb.

[Anonymous No.12430728]

>professor won't let us download the video lectures

why

literally MIT and so many other universities just let all of their stuff be open source, commons license, whatever the fuck it is. why does my mid tier little uni object to me having video lectures a personal reference. It's obviously not me in the lectures so why does anybody care

[Anonymous No.12430735]

>>12430728
they hate you and want you to suffer

[Anonymous No.12430738]

>>12430704
if you feel like there's anything weird about this, you should probably stop pursuing math

[Anonymous No.12430739]

>>12430728
>tfw yoinked the entire course shell and webex uploads before it went down

[Anonymous No.12430742]

>>12430738
i get the similarity, i just don't like sharing the notation. same with principal ideals being like (a). just makes shit involving polynomials a pain to read. use angle brackets and now it looks like cyclic groups.

[Anonymous No.12430783]

>>12430698
If it makes sense to ask if it's a magma and it isn't a category then it's a magma.

[Anonymous No.12430894]

>>12430698
Yes. More specifically, it's a loop.

[Anonymous No.12431234]

its over, scholzecels
http://www.kurims.kyoto-u.ac.jp/~motizuki/Explicit%20estimates%20in%20IUTeich.pdf

[Anonymous No.12431263]

>>12424218
I do not understand fourier transforms bruh

[Anonymous No.12431282]

>>12430728
If nothing else works:
Use ffmpeg with screen and system audio as the device to record from, and go full screen.
You are probably breaking implicit or even explicit rules of your uni there, though, so better watch out!

[Fourier Transform No.12431289]

I don't understand idiots bruh, what part don't they get, the swapping frequency and time part, the differentials, the convolutions?

[Anonymous No.12431291]

>>12430742
>i get the similarity, i just don't like sharing the notation
They're both (essentially) congruences in the sense of universal algebra. Now your autism can rest easy because they're literally the same thing.

[Anonymous No.12431295]

>teacher keeps talking and talking but in the end says nothing

[Anonymous No.12431305]

>>12431295
In what maths course does that happen?

[Anonymous No.12431309]

>>12431305
videocall class

[Anonymous No.12431315]

>>12430665
You can do the same thing with any variety of algebras, including modules over noncommutative rings. A finitely generated algebra is going to be (isomorphic to) a quotient of the free algebra F in your variety on countably infinitely many generators. There are typically quotients of F which are not finitely generated, but in any case F can only have a set worth of congruences and hence there are only a set worth of finitely-generated guys, up to isomorphism.

[Anonymous No.12431329]

We are given an ODE z''(x) - z(x) + 2g(x) = 0

I end up through rearranging and using fourier transforms with F[z]/F[x] = 2/(1 + s^2)

But there is no equivalent to multiplication and convolution, for division right? Or I might just be setting this up wrong

[Anonymous No.12431332]

>>12431329
F[z]/F[g] i meant

[Anonymous No.12431365]

>>12431234
Mochizuki's "proof" of ABC is a failure.

[Anonymous No.12431396]

>>12431291
are there other notions that share this congruency?

[Anonymous No.12431412]

>>12431295
i'm a teacher and i do that

[Anonymous No.12431471]

Can i ask career questions here?

[Anonymous No.12431476]

>>12431396
Basically anything you quotient out by in algebra corresponds to a congruence. So in addition to normal subgroups and ideals you also have submodules and subalgebras (in the sense of an algebra over a ring). Congruences of semigroups come up in the theory of string rewriting systems.

[Anonymous No.12431498]

>>12431476
neat, thanks.

[Anonymous No.12431562]

What are some books that (at least try to) answer what mathematics are?

[Anonymous No.12431574]

>>12431562
you don't know what is mathematics? i'm pretty sure 7-year-olds know that, you don't need a whole book to understand that word

[Anonymous No.12431590]

>>12431471
You can, but no one here's an actual tenured professor, so you won't be getting any advice from people who have actually made it.

[Anonymous No.12431604]

>>12431590
>no one here's an actual tenured professor
This, I'm a postdoc and that's about the max level on /sci/ right now. Check back in 2030.

[Anonymous No.12431675]

>>12428386
Anyone dares to try? I'm curious.

[Anonymous No.12431723]

>>12431234
ok here's what i don't understand:
here's a paper of mochizuki and 4 of his friends, and supposedly there are more people working on iutt, let's say at least 20 of them
did those people also read the entirety of iutt and they believe it all works out?
or do they try to prove results using iutt as a black box?
or did they just jump on the bandwagon of a new memetheory and believe that mochizuki has his shit worked out?

[Anonymous No.12431754]

>>12431675
It's probably false, but a counter-example would probably be non-measurable (solid intuition), and constructing non-measurable functions is a tremendous pain in the ass.

[Anonymous No.12431923]

[Anonymous No.12431973]

>it's another french visiting professor

GOD I hate europeans, specially the French, so much. They ALWAYS have fucked up hair, stench of milk and have an insufferable personality.

[Anonymous No.12432115]

If all you need for linear algebra is high school level algebra why is it usually taken after Calculus

[Anonymous No.12432227]

>>12431973
>stench of milk
what does that mean

[Anonymous No.12432229]

>>12432115
in my undergrad we took linear algebra in first and second semester

[Anonymous No.12432244]

>>12431604
>I'm a postdoc
What's your field?

[Anonymous No.12432255]

>>12432115
...but calculus is a high school subject.

[Anonymous No.12432278]

>>12432255
That doesnt change my question m8
Why is linear algebra often taught after Calculus

[Anonymous No.12432282]

>>12432244
the one with a brown cow and some grass in

[Anonymous No.12432285]

>>12432282
ah, that's the one with the naked faggots sucking each other off.

[Anonymous No.12432298]

>>12432285
yeah I dont know why your dad and your brother come to my field to do that

[Anonymous No.12432310]

>>12432298
Why to respect the local custom, of course.

[Anonymous No.12432373]

>>12432310
guess ill see you there too tomorrow then

[Anonymous No.12432395]

>>12432373
Cool. I will bring my faggot stomping boots.

[Anonymous No.12432531]

Wtf is happening in this thread.

[Anonymous No.12432862]

is there a "canonical" definition of pi that is grounded in number theory? i.e doesn't rely on the "geometric circle"

>>12432115
because (one can say that) everything you've learned in highschool math thus far was to build up towards the result of finding derivatives of polynomials.

lin alg as a continuation offers no real major results, whereas calculus, the definition of the limit, and the fundamental theorem of calculus would probably be among the first major results of anyone's math education, and allows one to begin a serious physics course

[Anonymous No.12432929]

>>12432862
>lin alg as a continuation offers no real major results

Omfg lol.

[Anonymous No.12432934]

>>12432862
i'd argue vectors and matricies are far more useful to the 'normal' persona than calculus is.

[Anonymous No.12432948]

>>12425020
the dover version has extensive commentary on phrases like this. are you reading that version?

[Anonymous No.12432956]

>>12432929
nothing comparable to derivatives and integrals in a first semester imho

>>12432934
the most obvious usage of vectors is vector calculus in physics; i do not see any immediate and interesting results, asides from maybe neural nets, which require the chain rule anyways/...

[Anonymous No.12432984]

>>>/bant/11771766
Tooker's having an episode

[Anonymous No.12433070]

>>12432984
nice

[El Arcón No.12433082]

>>12432984
One of these days, I'm going to make you know that this post post was far less satisfying to you than you thought it might be.

[Anonymous No.12433093]

>>12433082
Are you saying the bant thread poster is an impostor

[Anonymous No.12433143]

Guys I'm really scared I'm gonna fail my metric spaces and topology exam. How do I learn this shit in a week.

[Anonymous No.12433354]

>>12431305
Random Signals and Noise

[Anonymous No.12433390]

>>12433143
Just play with ur balls haha

[Anonymous No.12433422]

>>12433390
Kek. But I'm still gonna fail lmao

[Anonymous No.12433837]

is Lie groups and representation theory a good course for the last year of my yundergrad? it is a course of the graduate course in theoretical physics but seeing its syllabus it's purely a maths course, also i should have all the requirements

[Anonymous No.12433845]

>>12433837
i should point out that i have taken abstract algebra (up to galois theory), some algebraic topology (singular homology, homotopy groups) and basic differential geometry (mostly of curves and surfaces but also some general results)

[Anonymous No.12433868]

>>12433422
you deserve to fail for not having paid attention, retake the course

[Anonymous No.12434033]

Is there ever anything good on the ArXiv math.GM tag? Most of what I'm seeing are retards who think they can prove the Riemann with elementary complex analysis or articles on pedagogy.

[Anonymous No.12434041]

>>12433837
take the course, Lie groups are general knowledge

[Anonymous No.12434160]

I'm into arithmetic geometry and a friend of mine is into logic, so I thought it would be fun to study topos theory together, and see the interaction between our respective fields.

Is pic related a good textbook for such an endeavour?

[Anonymous No.12434180]

>>12434160
I don't know the first thing about arithmetic geometry but there's certainly a lot of things in that book that have little to do with it.

[Anonymous No.12434187]

>>12434160
I don't know the first thing about arithmetic geometry but there's surely a lot of things in that book that have little to do with it.

>>12433837
There's a /sci/ reading group on matrix Lie groups using Hall's book

[Anonymous No.12434219]

>>12434187
I don't mind learning some logic for now, and maybe later I'll read more about the cohomology theories that use topos theory on my own.
That being said, do you know a topos theory book that has a better balance between algebraic geometry applications and logic ones?

[Anonymous No.12434261]

>>12434219
No it's probably fine and reasonably balanced, try it.
Most of the book I'm aware of are much more logic oriented than it.

[Anonymous No.12434763]

>>12433868
Bro I attended all the lectures and I understand the concepts. I'm just too pea brain to grasp writing my own proofs from scratch.

[Anonymous No.12434922]

Is the Touhou paper worth reading?

[Anonymous No.12435041]

>>12434922
Link to that?

[Anonymous No.12435047]

>>12435041
https://en.wikipedia.org/wiki/Grothendieck%27s_T%C3%B4hoku_paper

[Anonymous No.12435059]

>>12435047
Oh, I thought it was a paper made by the Touhou poster.

[Anonymous No.12435274]

I want to learn combinatorics but I'm also new to formal proof-based math (hardest math I've taken was cookbook linear algebra for college credit, senior year of high school).
Can anybody recommend a combinatorics book that starts from the basics, the way I've seen algebra and analysis books start with the definitions of sets and unions etc.

[Anonymous No.12435279]

>>12435274
why do you want to learn combinatorics?

[Anonymous No.12435283]

>>12435279
it looks fun and helps people solve games like poker or puzzles

[Anonymous No.12435319]

>>12435279
also, I want to make my way to graph theory

[Anonymous No.12435552]

>>12435274
Stanley's Algebraic Combinatorics

[Anonymous No.12435579]

>>12435319
For intro graph theory use West.

[Anonymous No.12435619]

>>12434922
>>12435047
I think it is for historical purposes, but if you're trying to learn homological algebra, go for Weibel.

[Anonymous No.12435681]

>>12434922
>>12435047
>the Touhou paper
I'll admit I laughed.

[Anonymous No.12435805]

Is there any relation between probability and measure theory?

[Anonymous No.12435853]

>>12435805
Probability is about probability measures on measurable spaces. Random variables are measurable functions with respect to such a measure.

[Anonymous No.12435855]

>>12435805
probability theory is literally a subfield of measure theory. measure theory is about assigning a number (an abstract volume) to subsets of some space. in probability theory, this space is the set of all possible outcomes, an event is some collection of outcomes, and probability of the event is simply its measure.

[Anonymous No.12435886]

>>12435855
this
and measure theory is literally a subfield of set theory, since measure spaces are defined in terms of sets and measures is a function and functions are relations which are also sets

[Anonymous No.12435928]

>>12435853
>>12435855
Thanks anons.
I was reading some elementary probability and Kolmogorov axioms reminded me of measure theory.
Is there any other application of measure theory outside of probability and geometry?

[Anonymous No.12435935]

>>12435928
Ergodic theory, lol.

[Anonymous No.12435936]

>>12435886
I usually like your posts okay, but this one doesn't really add much...

[Anonymous No.12435938]

You just find the initial and final energy and set them equal to each other to solve for velocity right? I keep getting an imaginary number in my answer.

[Anonymous No.12435986]

>>12435938
bro this is the maffs thread

[Anonymous No.12436000]

>>12435986
Physics is basically the same thing

[Anonymous No.12436010]

>>12435938
Idk i did that shit like 3 years ago. No fucking clue. That's highschool stuff mate.

[Anonymous No.12436044]

>>12435938
I got 3.13m/s might be regraded though ngl

[Anonymous No.12436075]

>>12425641
>>12425650
No, she works for Admiral Overseas Corporation and makes computer monitors for the poor using science she learned from /sci/

[Anonymous No.12436080]

>>12435936
He's shitposting, literally everything other than synthetic geometry and logic is a subfield of set theory, by this logic.

[Anonymous No.12436115]

>>12436044
What equation did you use?

I set E = .5iw^2+.5MaV^2+Ma*gh-(.5Mbv^2+Mb*gh)

Do you just subtract the energy of the B mass from the total mass since the system is going in the direction towards A?

[Anonymous No.12436205]

>>12424218
https://www.math.ucla.edu/~jerryluo8/teaching/Winter2020-33A/The%20Manga%20Guide%20to%20Linear%20Algebra.pdf
think you guys might be interested in this

[Anonymous No.12436250]

>>12436080
I think it was sarcasm aimed at making fun of the "probability is a subfield of measure theory" assertion.

[Anonymous No.12436374]

>>12436250
...that's what I said.

[Anonymous No.12436385]

>>12424218
wat do?

[Anonymous No.12436435]

>>12436385
WHat have you tried?

[Anonymous No.12436441]

>>12436385
It's immediately obvious from the definition of multiplication.

[Anonymous No.12436468]

>>12436385
It's not true
Let A=
[1 0
0 0 ]
B=
[0 0
0 1]
Then AB=0 but tr(A)=tr(B)=1

[Anonymous No.12436564]

>>12436435
>>12436441

[Anonymous No.12436579]

>>12436250
I don't think that's warranted at all. Maybe I'm retarded, but until I understood what a measure space was I always found finite combinatorial probability problems ill-posed since it seemed like you had to learn some implicit information about the problem to answer correctly. It turns out I was right and that most of the difficulty of basic probably is that you're not given the actual probability measure in question and have to intuit which one the author means. This is also why Monty Hall and other stupidity exists.

[Anonymous No.12436614]

>>12436385
Anonku, okłamali cię, weź macierz I, wtedy tr(I)=n, ale I=I^2 i tr(I)tr(I)=n^2.

[Anonymous No.12436724]

>>12436080
>by this logic
I disagree, measure theory is really native in set theory, wheras anything algebraic (while having models in a theory of sets) is not essentially set theoretical.

[Anonymous No.12436787]

I am trying to solve the ODE that's in the first line. I solve it as a Riccati's equation for y=1.

The second line is the end result.

How am I supposed to proceed? Have I done something wrong?

[Anonymous No.12436793]

>>12436787
ignore the first image, this is the correct one

[Anonymous No.12437469]

>>12431675
>>12428144
i googled around and found this
https://www.ams.org/journals/proc/1990-108-02/S0002-9939-1990-0987607-1/S0002-9939-1990-0987607-1.pdf
if you look at the beginning of the proof of theorem 2.4 there should be a theorem of kolmogorov and vercenko which gives you a negative answer
sadly the reference only says "chapter ix in a book lmao" and the chapter is 50 pages long http://matwbn.icm.edu.pl/ksiazki/mon/mon07/mon0709.pdf
on the positive side, it appears that the topic of "take arbitrary function or continuous function R^n -> R and look at slopes" has been extensively covered by various people about 80 years ago so you should be able to read this if you care enough
personally i give up

[Anonymous No.12438222]

Where's the Finnish algebra tranny?

[Anonymous No.12438400]

>open analysis textbook
>see this bullshit
wtf hahaahah

[Anonymous No.12438537]

>>12438400
What’s the problem?

[Anonymous No.12438545]

Is it true that the following function forms a bijection between the power set of Z+ and the infinite cartesion product of {0,1}:

For each element p of P(Z+), define f(p) to equal an infinite string of 0s, where the nth 0 is replaced with 1 if n is a member of p
>>12438400
Same

[Anonymous No.12438763]

>>12438545
What have you tried?

[Anonymous No.12438799]

I have finished with my study of irrational numbers and now I will be moving on to setless groups. These are groups that do not have an underlying set structure.

[Anonymous No.12438869]

>>12438763
The question was to find a bijection and the function is my attempt at answering it, I'm asking to check my work. The idea is that ti's obviously injective and it's surjective because you can generate any order of 0s and 1s in the string.

[Anonymous No.12438902]

>>12424218
Do people at your school fail out of the math major? How does it usually happen? I'm just in my first year but idk if I'm gonna make it

[Anonymous No.12438916]

>>12438902
In my school we have a math major with Down's syndrome. I think he's doing pretty well.

[Anonymous No.12438918]

>>12424233
Dubs confirm
>>12424355
Dubs. Anon speaks the truth. If you're learning maths for fun you should study that you want. Best way to do this is search "field I want to study pre reqs" and then find books on those to build up background. Happy studying! :)
>>12424433
No love for number theory or probability? Also dubsdubs :0
>>12424500
A lot of people start off with munkres first. It's a nice read. Dubs.
>>12424533
Thank you anon, have a good day! Dubs.
>>12426177
Dubs of truth.
>>12427022
Dubs. I also like \phi.
>>12427955
Please don't be sad, dub poster :(
>>12428144
Dubs.
>>12428255
My combinatorics isn't that great, but I'll give it a go! (dubs)
>>12428566
You're in luck, plenty of gay autismo who study maths! (dubs)
>>12428644
Should there be a space between the equations? (dubs)
>>12428700
Based computer scientist. Also dubs.
>>12428977
Last year of undergrad :( (dubs)
>>12430588
Dubs.
>>12430655
Agreed. Everyone in mine found it very difficult but after the first month everything became a lot easier! (dubs)
>>12432244
Dub dubs.
>>12432255
Dubdubs.
>>12433422
Sorry anon. Learn the bare minimum so that you can reproduced proofs on your own. Is it an online exam? Dubs.
>>12434033
Dub.
>>12434922
Dubs
>>12435855
I love measure theoretic probability! Also dubs.
>>12436000
Trips! You can't say physics is maths when the question you've posted doesn't involve proving any propositions.
>>12436044
Thank you anon for your hard work :) (dubs)
>>12438222
Trips! Who?
>>12438400
Dubs.
>>12438799
Dubs. What book are you planning to use for set less groups?

[Anonymous No.12438924]

>>12438918
>What book are you planning to use for set less groups?
By reading a book on setless groups I would be robbing myself of the joy of discovering the theory on my own.

[Anonymous No.12439147]

Lets say I have a Markov chain [math] X[/math] with state space [math] S=\{1,2,3,4,5 \}[/math] and transition matrix [math] P[/math],
how can I calculate [math]\mathbb{P}[X_k = j_k, X_l = j_l | X_m = j_m][/math] without using the law of total proability a lot of times? Doing that way seems rather awful?
For example, suppose that I have [math]\mathbb{P}[X_{10} = 1, X_2 = 2 | X_6 = 3][/math]. I'd have to use it 7 times which is truly a pain in the ass having to calculate

[Anonymous No.12439170]

>>12439147
Idk do like the P^10 thing

[Anonymous No.12439252]

>>12424218
What kinda undertale OC shit is that and what does it have to do with maths

[Anonymous No.12439254]

>>12439170
I thought about that too but the problem is that in that way I'm taking "all the paths" insteads of the paths that go through a certain state

[Anonymous No.12439439]

found a potentially useful image

[Anonymous No.12439484]

>>12431263
just a dot product except with infinite vectors
infinite sums become integration, so its an integral
your basis vector is [sin(x),cos(x),sin(2x),cos(2x)] etc, and that's strange, but it really just is a dot product.

[Anonymous No.12439548]

>>12438902
usually people just swap over to something easier
one girl swapped to comp sci

[Anonymous No.12439580]

>>12438222
Here. Why?

[Anonymous No.12439744]

>>12439580
Hi! Have you been busy? No arxiv links or things like that for ages.

[Anonymous No.12439757]

>>12439744
Hiya! Not really busy, just dead inside and stuff. If you people want me to, I can spam some papers every morning, but otherwise I'll let you be in peace.

[Anonymous No.12439853]

If |a-b| <= |b|/2 then a and b have the same sign.

That's a cool trick. I can see why but it wasn't immediate for me.

[Anonymous No.12439878]

Anyone's got an algebraic formula for both branches of complex square root?

[Anonymous No.12439939]

>>12438400
>wtf hahaahah
That looks completely standard.

[Anonymous No.12439943]

>>12438400
imagine being filtered this hard

[Anonymous No.12440213]

>>12430325
Is this from Disgaea?

[Anonymous No.12440232]

>>12431562
None
Maybe >>>/lit/
This board does not care about what mathematics are, is about doing mathematics

[Anonymous No.12440244]

Does Artin's Algebra require any LA background?

[Anonymous No.12440263]

What are the best Universities in regards to their research into the areas of algebra and representation theory?
Is it the usual suspects, MIT, Princeton, Oxford?
I'm thinking of where to apply for a postgraduate position

[Anonymous No.12440271]

>>12440244
No.

[Anonymous No.12440311]

>>12440263
You should delete this thread before it gets banned, because posts like these strictly go on /adv/

[Alimus No.12440313]

Someone help me

[Anonymous No.12440335]

>>12440313
What have you tried?

[Alimus No.12440354]

>>12440313
Nothing. Can't understand my teacher's discussion due to slow internet connection.

[Anonymous No.12440383]

Any ideas?

[math]y(x)=5+ \int_{0}^{x} (t^2-2y(t))\,dt[/math]

[Anonymous No.12440387]

>>12440383
Derivative & FTC

[Anonymous No.12440527]

>>12440232
What is your favourite mathematic?

[Anonymous No.12440528]

>>12430283
>number of (unordered is redundant) (sub)sets of size N
wouldn't it always be N!

[Alimus No.12440651]

>>12440313
Can you guys provide some video link for these lessons...please

[Anonymous No.12440717]

>>12440528
That would imply that the number of subsets of size 2 in a set of size 100 is 2!

[Anonymous No.12441071]

How do I show that if y solves the ODE y' = sin(x+y), that the limit of y is -\infty? It's obvious from the slope field, but that's not very rigorous. I know the solution is tan(x+y)-sec(x+y)=x+c but I'm not sure how that helps.

[Anonymous No.12441154]

Let a be a real number. Let f:R==>R and lim _x-->a f(x) = infinity. Then the limit lim_x-->a cos(f(x)) DNE. How do I prove this? I know to go by contradiction but can I use the sequential criterion?

[Anonymous No.12441184]

>>12440313
>(a)
It tells you what the rule is, just apply it. [math]R=\{(2,6),(3,6),\text{ etc. } \}[/math]
>(b)
again, it tells you the rule. just apply it with those values.
>(c)
Recall the definition
>(d)
Recall the definition.
>(e)
Refer to question 2 for an example.

[Alimus No.12441191]

>>12426628
Thanks man, i figured it out. Can you help with b part here?

[Alimus No.12441197]

>>12441184
Wrong reply above lol

[Anonymous No.12441208]

>>12441191
C'mon man, this is sad. Do you know what the definition is for domain? Do you know what [math]F(x)[/math] means? Have some initiative.

[Anonymous No.12441213]

Hi math gods. In another thread someone showed me the Michael Penn video of why analysis doesn't work on the rationals because for example the rational function f(x)= 1 if x <sqrt(2) and 0 if x>sqrt(2) is continuous at every rational number q but does not look continuous, i.e. has a jump at sqrt(2).
I pointed out that this could be fixed if you reformulate the condition to continuity by saying that f:Q->Q is continuous if for every rational Cauchy sequence (x_n), the sequence f(x_n) is also Cauchy. I think this definition fixes the issue and the function f becomes discontinuous, but people in that thread keep claiming I am wrong and I don't understand their reasoning.
Can anyone explain how I'm wrong?
Also can anyone explain why reals are necessary for analysis, like give examples which don't work if you only work over the rationals?

[Anonymous No.12441222]

>>12441154
Without assuming f is continuous, this is incorrect. There are functions f such that lim_x->a f(x) = infinity but lim_x->a of cos(f(x)) exists.

[Alimus No.12441254]

>>12441208
Yeah it's sad. I didn't understand the lesson cause of our internet connection

[Anonymous No.12441262]

>>12441222
Can you give me an example of such a function?

[Anonymous No.12441264]

Is there any term fo a multiplicative group of matrices whose absolute value of determinant is 1? ([math]M \in GL(n, R) \wedge |\det M|=1[/math])

[Anonymous No.12441266]

>>12441262
I can.

[Anonymous No.12441267]

>>12441213
write a formal proof please.

[Alimus No.12441271]

>>12441208
Oh shiii. I figured it out. Thanks btw

[Anonymous No.12441275]

>>12441264
Yes. It's called the "shit group of irrelevant
matrices".

[Anonymous No.12441276]

>>12441213
https://math.stackexchange.com/questions/2855609/prove-or-disprove-if-x-n-is-a-cauchy-sequence-in-x-and-f-is-a-continuou?rq=1

[Anonymous No.12441280]

>>12441213
>Also can anyone explain why reals are necessary for analysis, like give examples which don't work if you only work over the rationals?
IVT, mean value theorems, bolzano-weierstrass etc. all of this is becomes false and it's just baby stuff

[Anonymous No.12441281]

>>12441267
Formal proof of what?

[Anonymous No.12441285]

>>12441264
the group of volume preserving linear transformations

[Anonymous No.12441287]

>>12441264
the group of prolapsed anus

[Anonymous No.12441302]

>>12441275
Is group that contains special linear and orthogonal groups really so irrelevant?

[Anonymous No.12441303]

>>12441276
Thanks but I don't see how this is relevant to what I said.
>>12441280
These theorems become false as usually stated because we're no longer dealing with real numbers, but I think there are simple modifications that are very similar that give you all you want in applications, no?
For example, instead of saying you can find a zero in the IVT, you say you can find a Cauchy sequence whose values approach 0, and so on.
It seems to me like the real numbers are used to "forget" important information about sequences to perhaps make the statements a bit simpler but it comes at the cost of making things uniform which should not be viewed as uniform. This comes at the cost of making the theory much more computationally obtuse and obscure.

[Anonymous No.12441358]

Say I have a rational number p/q. What's a good algorithm to find the best rational approximation a/b to p/q such that |a|, |b| < M for some natural number M?
For example, I would like to approximate 100000001/200000000 by 1/2, even though they're not equal, to save space.

[Anonymous No.12441369]

>>12441358
continue fractions solve the problem "what is the best rational approximation p/q to a real number x, under the constraint q<N" where N is a number of your choice
the algorithm takes log(N) steps on an average real number so it's pretty good

[Anonymous No.12441374]

>>12441266
Okay, what is it? I've already tried everything I can think of. I can't come up with something but I keep getting an essential discontinuity or oscillation.

[Anonymous No.12441375]

Do you study crystal tilings? I feel like they're planar permutation groups

[Anonymous No.12441379]

>>12441374
a=0
f(x)= 2pi * floor(1/|x| ) at x!=0
0 at x=0
Then lim_x->0 of f(x) = infinity
but lim_x->0 of cos(f(x)) = 1

[Anonymous No.12441389]

>>12441303
I think you should work through some analysis textbook, and then do it again but formulate everything via cauchy sequences. that's the only way you'll find your answer.

[Anonymous No.12441402]

>>12441379
Thank you anon. I feel stupid now.

[Anonymous No.12441407]

>>12441402
Mathematicians get to enjoy feeling stupid much more often than normal people. Get used to it.

[Anonymous No.12441413]

Found a very good article about exactly what I was wondering about
https://www.dpmms.cam.ac.uk/~wtg10/reals.html

[Anonymous No.12441450]

what series is this?

[math]1-x+x^2-x^3/3+x^4/24[/math]

[Anonymous No.12441467]

>>12424218
So I applied for a trading internship a few days ago and they have now asked me to do a quantitative reasoning test. Is this going to be similar to the GRE section do you think? How could I prepare a little bit?

Probably not too relevant to this thread but thought I'd ask.

[Anonymous No.12441483]

>>12441450
isn't the 3 supposed to be 6?

[Anonymous No.12441489]

>S. So a moral you could draw is that when most mathematicians talk about the real numbers, they don't know what they are talking about.

>L. That's a rather blunt way of putting it. I'd prefer to say that they leave unspecified what they mean by the idea of an arbitrary subset of a set, because they don't need to worry about it. They have the axioms for a complete ordered field, and if you believe in the external reality of mathematical objects then you will have to say that those axioms are ambiguous, because the completeness axiom relies on the notion of an arbitrary sequence, which is not fully explained. But if all you want from your axioms is a set of rules for when you are allowed to deduce one mathematical statement from another, then the axioms for a complete ordered field give this to you.

Tim Gowers is in agreement with Wildberger, so it seems.

[Anonymous No.12441504]

>>12441483
no

this is from a picard iteration for n=4

[Anonymous No.12441518]

>>12441450
And a /2 with the x^2 and an ellipsis? Otherwise it's just a quartic equation

[Anonymous No.12441543]

>>12441450
[math](1,-1,1,-1/3,1/24,0,0,0,0,\dots)[/math]

:)

>>12441489
That's overshooting, since Wildberger wants to pursue people not even use R.

>most mathematicians talk about the real numbers, they don't know what they are talking about.
Sure, that's almost a given. Whoever you are, if you study foundations proper, you end up with some itching conviction that most people who don't care don't want to hear. Logic people "go crazy", which is not to say anything bad about them. Crazy is just different than the rest who don't even know about it. Most mathematicians just want to find a topic they like, marry some woman and live to their death proving theorems in their subject, not worrying "what it is they do" beyond the subfield of math they are good at.

[Anonymous No.12441574]

>>12441413
Great read, loved the reference to Richard's paradox.

[Anonymous No.12441642]

>>12441264
Special Snowflakes

[Anonymous No.12441659]

>>12441656