/sci/ - Science & Math

>>12122790
whats a tensor

>>12122793
an array of floats

>>12122793

>>12122793
>>12122793
tensor is a [math]GL(n)[/math]-equivariant map from the set of frames into (suitable cartesian power of) the set of coordinates

Why do autistic trannies love category theory so much?

For even n, an infinite sequence of (fair, independent) n-sided die rolls can be interpreted as a random walk on [math]\mathbb{Z}^{n/2}[/math] by identifying the n sides with the n/2 copies of {-1,+1}.
Is it possible to extend this interpretation to the odd n as well, so that rolls of odd-sided dice can be interpreted as random walks in some exotic kind of space?

>>12123068
>coordinates
yikes

>>12123110
Why are you so obsessed with trannies?

Reminder:

Fuck niggers, kikes, spics, chinks, faggots, women, brainlets, normalfags, trannies, and janny

>>12123128
t. bluepilled undergrad category theorist

Which notation for open intervals is best? [math](a, b)[/math] or [math]]a,b[[/math]?

>>12122790
Anybody remember securityoverride? It was like Project Euler but for Hacking.

Great times

>>12123147
Autistic tranny detected. Join the 41%.

>>12123174
It should be in any graph theory textbook. You can see it by induction on the length of the walk. I'm sure it's in Stanley's undergraduate book.

>>12123335
¿Are you French or something?

>>12123335
I've not seen the second one before, but I'm already sold.

>>12123562
Does this matter?

>>12123112
this is not really a great representation, since its not injective first of all. And its fairly arbitrary which die sides 'cancel' each other. Is this for something?

>>12123609
Well you're asking whether to use the French notation rather than the standard one, so there is a question as to whether this will be considered unusual by your peers.

>>12123631
Ok, that explains things. I'll just go with the standard. We have too much French influence over here already and it's about time someone draws a line somewhere

>>12123650
Mdr

Is this equation solvable for positive x?
[eqn]\frac{x}{2} = \tanh(2x)[/eqn]

Mathematica can do this numerically but not analytically.

Do I need to learn trigonometry to learn calculus?

>>12123741
absolutely not. just know what sin and cosine are. That is, how they relate to the circle. Then all the trig identities you will encounter will be justified by basic geometry

>>12123736
You can get a series representation pretty easily, but idk if any positive answer is going to be expressible in terms of elementary functions.

>>12123335
first
>inb4 hurr durr it's the same notation as ordered pairs
if you're working with real intervals you're not going to see that much ordered pairs

>>12123758
Thanks you just saved me.

>>12123741
You don't need it to understand the theory, but you're definitely going to need it for examples and to grasp intuition. Also, many counterexamples involve trigonometric functions.

>>12123335
While the second one is the first one I learned, (a,b) is just so common that it's become my immediate reflex. (It's also less ugly and takes up less space)

Who invented the "mathematical maturity" meme?

>>12123802
Mathematical maturity is literally just having the basic skills to understand modern mathematics.

>>12123820
And what is modern mathematics?

>>12123149
based
>>12123736
nod possible
>>12123835
gay and pozzed, few people left that can do it properly

How many papers should you publish per year?

>>12123135
i dont get it

can u make an imaginary number that is simultaneously prime and not prime and use it to prove theorems about primes numbers?

>>12123777
AHHHHH

>>12123865
the fuck are you smoking
>>12123875
sorry f a m

>>12123865
all maths are imaginary
any rationalism is imaginary
that's because rationalists, like scientists, are not empiricists

>>12123650
french maths is better
99% of old maths and modern maths is french maths from Bourbaki and grothendiek

>>12123620
>this is not really a great representation, since its not injective first of all. And its fairly arbitrary which die sides 'cancel' each other.
I should probably clarify, my object of study isn't the presentation of what is essentially white noise, but random walks and their mathematical properties (including the possibility of cycles, recurrent/transient behavior, etc.).
More precisely, the trajectory of the random walk, represented as a [math]\mathbb{Z}^{n/2}[/math]-valued stochastic process, satisfies many nice properties (e.g. Markov, martingale), but this nice interpretation only works for even n. Can the odd-n processes be similarly interpreted as trajectories in some kind of space [math]Z^{(n+1)/2}[/math] such that these nice properties are preserved? The tradeoff would be that this space would have to be more complicated (in a topological sense?) than [math]\mathbb{Z}^{n/2}[/math] which is simply a free abelian group.
>Is this for something?
Not in the sense of leading to any publications, I'm just playing with the idea for recreation.

>>12123913
France was actually woefully behind until Bourbaki. The decades around 1900 were the German era of mathematics.

>>12123846
365
t. Chinese mathematician

>>12123865
can you make an imaginary number that is 1 and not 1 and use it to prove RH

>>12124052
You can prove anything from a contradiction.

>>12123110
>reading a textbook written by a woman
>reading a textbook written by emily (is)riehl
N.G.M.I !!!!!

My new chair is coming today mg. I think it will motivate me to start studying again! Thoughts?

>>12123846
0
t. Montenegrin mathematician
>>12123865
u may be interested in "field extension"

>>12124135
Is Emily Riehl a tranny or just a lesbian?

>>12124176
she's an emo femboy

>>12123110
>Half of the category theory books are written by women
Why I'm I supposed to take category theory seriously?

>>12124188
Because the men who invented it were so alpha they still have tons of chicks sucking their cocks even though they're all dead or close to it.

You can only respond to this post with 10/10 theorems.
https://en.wikipedia.org/wiki/Residue_theorem

>>12124209
https://en.m.wikipedia.org/wiki/Stokes'_theorem

>>12122793
A tensor is to a multilinear map what a matrix is to a bilinear map
Also it may behave slightly differently under symmetry operations such as reflection

>>12124188
Algebra: Chapter 0

>>12124188
>>12124235
this or just the first bit of vakil's "rising sea" concerning categori theory

>>12124209
the classification of finite simple groups

>>12124209
Gauss's theorem

>>12123110
>>12124197
>>12124246
set theory is better than category theory.

>>12124188
you aren't

>>12124263
>Gauss's theorem
the virgin gauss's theorem vs the chad gauss's lemma

>>12124268
Category theorem is a tool used to see links when they're not obvious at the first glance. I like its applications to differential geometry with covariant and contravariant functors.

>>12124268
undergrad detected

what would it take for me to actually understand the proof of Green's Theorem? Stokes Theorem? (only in 3, maybe 4 dimensions)

just a full year course in real analysis? or nah?

>>12124284
me: 10 minutes
you: a lifetime

>>12124284
fundamental theorem of calculus, geometric intuition

>>12124284
some complex analysis and some differential calculus. You can prove it with not much tools and do calculations for 4 pages, or advanced tools and not that much derivations.

>>12124291

>>12124309
stop projecting tranny. this is a selfie i took just now...i may not be a looker but at least i'm not fat or a faggot

>>12124319
>not fat

>>12124319 timestamp? I found that image on google so you like to lie 2 pathetic.

>>12124278
>I like its applications to differential geometry with covariant and contravariant functors.
are you talking about something more profound than >pushforward and pullback are functors?

>>12124365
here's a picture of me posting on this thread taken by my girlfriend (emmy noether)

>>12124282
correct
>>12124278
yeah i was just joshin ya. no fucking idea what i'm talking about.

>>12124392
>joshin ya

>>12123110
I was actually talking to this CS trans girl on 4chin a few days ago, and I asked her why is it that computers attract transgender people so much. She replied:
>My theory is that we seek a safe place where we can fit in, and because we cannot fit in with others or are intentionally excluded, we drift towards computers because they can't judge us or exclude us or poke at our insecurities. Like, I probably would've been a horse girl if I was AFAB (I love horses and took riding lessons for years until my classmates found out and bullied me relentlessly for it), and could have been a programmer but probably would've found a different major (like english, which I really wish I had double majored instead of just minoring in).
Then you combine this with the connections of CS and CT to get a partial explanation of why there are so many trans girls interested in CT.

>>12124371
lol later you still look like the pics I posted above

>>12124408
Why wouldn't they just do something that asocial women tend to do, like care for animals or do tedious administrative bullshit?

>>12124413
leave this thred and do not come back

>>12124421
>why don't men act like women

>>12124209
https://en.wikipedia.org/wiki/Floer_homology

>>12124209
Atiyah-Guillemin-Sternberg convexity.
Cartan-Hadamard.
Soul theorem and soul conjecture.
Virtually nilpotent implies polynomial growth.
Gromov non-squeezing.
Kondrakov.
>>12124496
>Floer homology is a theorem

>>12124268
>set theory is better than category theory.
kys

Sorry this is a bit retarded to be posting here, but my head has completely gone and needed to check I am looking at this the right way.

Two people are inheriting money, A and B.

A has lent B some money (X). A has also paid for some joint expenses of both A and B (Y).

When dividing the money received, C, is the correct way to do it:

1. A receives Y from C
2. (C-Y) is split equally between A and B
3. B pays A X

>>12124268
>set theory is better than category theory
Yes.

>>12124513
Debts shouldn't affect the splitting of an inheritance. Don't let your siblings scam you out of what's yours.

There is nothing wrong with set theory.

>>12124209
https://en.wikipedia.org/wiki/Superrigidity

>>12124516
'everything is a set of sets or something lol'

because that is perfectly what i imagine when i think of x mathematical object

>>12124605
What if ZF proves that for some particular Turing machine [math]t[/math] there exists ("exists") an [math]n\in {\mathbb N}[/math], such that [math]t[/math] halts at execution step [math]n[/math], while [math]PA[/math] can't decide it and it actually not ever happening in reality (i.e. in reality the Turing machine can never actually halt, i.e. what if ZF is self-consistent but wrong)?

>>12124652
Nope, IIRC by a variant of Friedman's trick ZF is conservative over [math]\Sigma_0^1[/math] formula.

>>12124646

>>12124646
all kidding aside, i am still very very new to reading about set theory and i struggle with the semantics frequently. i have already learned a little logic so coming from the sublimity of FOL's translation into natural language it is comparatively harder to appreciate what exactly set theory's language is trying to intuit. I could really use some resources that go into the semantics a little bit more in depth than some beginner texts.

>>12124674
w.r.t PA statements?
I stole the idea from

https://arxiv.org/pdf/0905.1680.pdf

where he talks about [math]\Sigma_1[\math]-validity

>>12124747
Over IZF. I am probably wrong judging from this paper, but I thought that by applying Friedman's trick to ZF you could extract a constructive content of any proof of a [math]\Sigma^0_1[/math] statement.

>>12124770
This paper seems to imply conservativity of [math]\Sigma^0_1[/math] formula for [math]\mathsf{PA}_2[/math] over [math]\mathsf{HA}_2[/math].

https://arxiv.org/pdf/1101.4364.pdf

So it's not quite ZFC but [math]\mathsf{PA}_2[/math] is definitely impredicative, and already quite potent logically.

>>12124745
Roitman "Introduction to Modern Set theory" chapters 3,4 and 6 has the best introduction the semantics of set theory. She presents quite a few theorems as to when certain sets are models of axioms. You might also look at Enderton's last chapter.

>>12124770
I know that IZF has the same proof theoretical strength in the sense of ordinal analysis, but can a classical theory really be conservative over an intuitionistic one if it makes literally ever "P or not P" (theorems not all provable in IZF) true?
I have no good intuition for which problems sit where on the hierarchy, so I can't judge for Sigma zero or one, so I wouldn't know. But yeah it seems the author would know. Here's he on MO btw.
https://mathoverflow.net/users/23141/nik-weaver

But in any case, I was sort of memeing about caring whether ZF makes statements nobody uses true. If I have an issue with set theory (assuming the weak set of axioms I do like), then it's indeed with its ontology:

>>12124799
It's not completely clear, but I read his statement not to be about model theory, but about "real world semantical interpretations" of the theory of sets. "What is a set"?

>>12124745
This is from the same guy on the topic
https://arxiv.org/pdf/0905.1677.pdf

I don't advocate it or him, it's just a coincidence you also ask a question about a topic that he wrote about.

The main issue, maybe, is the aspect of non-typed set theory to reify the notion of collection.
I'm really not a fan of the empty set notion, at least when it can become a member of another set.

>>12124798
... and it actually mentions conservativity of [math]\mathsf{ZF}[/math] over [math]\mathsf{IZF}_C[/math]. Since AFAIK the latter enjoys the witness property, this implies that the paper you're citing above is wrong?

>>12124821
>can a classical theory really be conservative over an intuitionistic one
That's not what the conservativity result say. It's only true for a specific class of formulae, which are decidable. If you're given an integer n and a Turing machine, you can definitely check in a constructive way whether it halts in at most n steps.

Conservativity is not true for formulae of higher logical complexity, for a good reason.

Not math, but an arithmetical recreation and therefore slightly related. It occurred to me that [math]31^2 - 19^2 = 600[/math],* a nice difference. I then recalled that the nth square is the sum of the first n odd numbers, and as I expected, the (later) partial sum was 12 terms, or six pairs of 100. The recreation is generalized in the picture, and amounts to

(49 + 51) = 100,
(47 + 53) + (49 + 51) = 200, etc.

Of course, the importance of 25, 50 and pairs here reminded me of the old saw where Gauss quickly summed 1-100.

*Initially, this came to mind via some cultural references. 361 is the number of spaces on a Go board, while 961 is the largest square of a prime (or: prime power with non-unit exponent) which is yet under 1000, objects which come up in the flick Cube.

>>12124831
>>12124823
>this implies that the paper you're citing above is wrong?
Mhm, I don't think it rules out the ZFC<->HA issue at hand. In fact I'm pretty sure that the author of the paper doesn't even care for intuitionistic logic, he's just concerned with ZFC<->PA.

Namely, the idea is that you have an undecidable PA statement and the set theoretical axioms are just so strong that they may imply this and that. I think if he have no ZFC<->PA claim, we can't rule it out.
What's HA_2? If both HA_2 and PA_2 deal with subsets, then it might not be an apt comparison.

>If you're given an integer n and a Turing machine, you can definitely check in a constructive way whether it halts in at most n steps.
I don't follow just yet.
The idea is that ZF proves the existence result with strong axioms while PA is agnostic about it. In that case you'd have a ZF integer but can't compute it's properties (e.g. whether it's a multiple of 7).
This is maybe in analogy to how ZF proves existence of the well-ordering of R, while not allowing us to compute with it.

The only question is whether this can already happen with just arithmetic-like statements..

I hope you guys don't mind me asking here instead of sqtddtot.

I'm working on the mathematics of collision detection and item selection. Is there a way to encode two numbers as one number, unique to those two numbers? For example, I have two segments on the x-axis (10,13) and (155,158). Just using their distance isn't unique.

>>12124908
I tried to dig a bit but didn't really find anything hard. The idea is discussed, but e.g. that paper in pic related doesn't tell us too much about the hierarchy level

>>12124908
If you're just interested in natural numbers, you can find the quickest one here
https://en.wikipedia.org/wiki/Pairing_function

If you want to map between lists, you can iterate it.
You can also move between N and the set of all finite length lists of elements of N, see
https://youtu.be/soHBNEJlzL0

In fact, if you want to be invertible and implement it, I'd use that modular arithmetic instead of some root taking

>>12124908
I should maybe add that with disregard to bijectivity or runtime, the fact that you can do the encoding is evident as follows:

If n and m are two naturals, then
k = 2^n · 3^m
is injective and the inversion is given by prime decomposition.

E.g.
192 = 2^5 · 3^2
and no other two numbers can make for 192 since factors are unique.

But of course that's an exponential encoding and not onto, you can do the Cantor one for quadratic.

>>12124513
>>12124539
That’s not really an issue, we’re just trying to work out numerically what is fair

>>12124408
Why is it that women and the like always have social reasons to do shit?

>Be talking with (female) girl math major
>The topic of reasons of going into math comes up
>"Well I went into math because it felt like a place for misfits and I'm a total misfit"
>She's actually struggled BAD the whole major, insane work ethic is why she's gotten by

Is that the most contrived reason to go into a field? It literally has nothing to do with the field itself or you talent with it. It doesn't make any sense.

I went into math because I find it fascinating, not because I thought I belonged to some arbitrary social group.

>>12124408
I think you are a tranny too

New book just came, lads.

>>12125215
which one?

>>12125223
Taylor - Measure Theory and Integration

>>12124224
>A tensor is to a multilinear map what a matrix is to a bilinear map
this so wrong wtf

is it appropriate to write [math]x<0[/math] or [math]x>0[/math] if the set in question is something like[math]\mathbb{R}\setminus\{0\}[/math]?

>>12125391
Low IQ post

>>12125395
yes

>>12125397
Based.

>>12124224
this a riddle? hmmm... think bone hurts some. wheres that faggot alex trebek when you need him . he has answers after the buzzer

>>12124988
Just divide the inheritance in half. You each take half. Settle up whatever debts you have afterwards. That's what's fair.

>>12123835
a language game

>>12125541
do you require games to be fun?

>>12125543
no but they have to be compelling

>>12125549
'Compelling' is completely arbitrary.

>>12125395
no, it's actually a good question

>>12125577
It is. If you're being careful you'll notice that < is commonly used to indicate many different binary relations.

>>12125590
i see [math]\le[/math] more often.

>>12125921
[math]\leq[/math] is just a poor man's [math]<[/math].

So I just failed another math test. How do you anons do it? I think math is interesting but there's so much specific shit to remember

>>12126017
>remember
There's your problem.
Understanding > memorization

>>12125968
<'ed and ordinalpilled

>p or q
>not p
> p->r
>therefore not r

is this valid?
Not homework anymore.

>>12126244
what are you asking?

>>12126257
it is basic prop logic.
>>12126244
dude. denying the antecedent. look it up.

Are lattices actually an important structure or are they just a coping mechanism?

>>12126278
Have something against algebraic number theory?

>>12124964
>not onto
Onto? Otherwise, killer post man, you rock. I think that might be just what I need. Cantor's is so simple holy hell I wish I'd thought of it. We shall yet see though whether my simple mind will also simply implement it.

>>12124209
Brouwer's fixed point theorem
>>12124408
Aww that's kinda cute

>>12125112
Unsure to what degree of bait this post represents, but the psychology you are describing is not by any means specific or maybe even particularly highly correlated with women. Most people I have met do most of the things they do because they feel socially obligated to do it.

any (maybe analysis?) textbooks that cover the asymptotic behaviour of functions (big oh etc)? I'm looking for a mathematical approach (with limits, convergence etc) rather than a computer science one such as in CLRS

idk if this is going to post twice captcha is buggy for me

>>12125391
That depends a lot on the question. If the question is "for which x is this the case" then sure. If the question is "what is the set such that this is the case" then no, write a set.

>>12126445
Maybe you want something like de Bruijn's Asymptotic Methods in Analysis. It's a good book with lots of nice techniques. The first chapter really gets through the silly "what is big oh" stuff but the really juicy stuff comes after when it's applied to a very wide variety of problems and situations.

>>12126488
>Asymptotic Methods in Analysis
The first chapter looks really relevant to my course, the others not so much. I think the topic is used as a primer on inequalities and proof, and then later on the distribution of primes. This proof seems very similar to the stuff in thats shown in our notes, anything else more like this? It's probably pretty elementary but that's all we really need for this class I guess.

>>12126527
I should also mention the class is an introductory analysis class

>>12123741
I would memorize trig identities and the unit circle, as for the rest...meh.

>>12123741
calc 1 not really, at least at my uni in burgerland. calc 2 has an entire segment on trigonometric substitution so i would touch up on some material tangent (hah) to it:

1. Understand how cos+sin relate to unit circle
2. Understand arc* functions
3. Memorize a lot of trig identities (you might get a reference, but it is faster if you know them). Trig proof exercises help with this.

>>12126555
as a footnote, 2+3 can be satisfactorily understood from probably less than five khan academy videos, if you don't have the attention span for a real book.

>>12126558
>khan academy

I can't grasp Mathematics passed Algebra, I've only had success in Geometry. I can't learn Engineering, I can't learn Chemistry, I can't learn shit all about anything relient upon Math. I need help, can any Anon help me.

>>12126583
yeah i know. i hate that site. but if you are a spastic zoomer who just needs a little trig to get him past his calc 1 engineering prereq or something it is applicable.

>>12126660
Sure. What do you need help with?

>>12126660
Ok...but have you tried sitting down and doing the same problems over and over until you memorized each step and then applying those steps to a different problem? i.e. studying.

Any recs on a good proof book for beginners?

https://johncarlosbaez.wordpress.com/2018/09/16/the-5-8-theorem/

>>12126782

>>12126386
Dude I've seen most stuff like that with women. The first thing they consider for everything is the social aspect. I'm not immediately judging it tho, maybe it's beneficial, but it's definitely contrived.

>>12123741
You don't even need trigonometry to understand trigonometry

>>12124857
You can also simply say [math]a^2 - b^2 = (a+b)(a-b)[/math]
And now, of course, if you arrange to take a nd b symmetrical around 25, a+b = 50 and a-b is even.

>>12127507
Well memed, my friend.

This is not valid.
p implies r, but r does not necessarily imply p.
Consider the statement p(x) "x is prime"
and the statement r(x) "x is odd or 2",

now, x = 9 means that p(x) is false, but r(x) is true.

>>12124284
>be you
>anonymous
>the year of our lord 2020
>have a function
>have a curve
>you notice...
>you can sum up the function on the edge of the curve
>but wait! it's in 2D, this function has partials
>you can look at the change in the partials across area enclosed
>this tells you the net change of each piece as you go along the loop
>voila

>>12126660
Just the way you type is a clear indication that there is no hope for you in mathematics. It's highly disorganized, it has little to no flow or structure. If your typing style is at all reflective of how you reason, then you shouldn't bother.

Some anon the other day said trace relates to graph theory. What was he talking about? I'm referring to linear operator trace, is there another form? I tried to visualize his idea but to no avail.

>>12126349
>onto
another word for surjective. I think Americans use it. I started using it sometimes because I kinda like it.

It's easy to implement. As I said, don't take the square root inverse as given on Wikipedia, the video I linked implements it with modular arithmetic, I remember.

>>12127646
Sounds like he meant w.r.t. adjacency matrix

>>12126790
I don't read blogs of white supremacists

>>12124209
https://en.wikipedia.org/wiki/Pythagorean_theorem
A timeless classic, easy to prove and has multiple applications

>>12127644
Shut up condescending fat retarded gay poop eating piece of poop with pee in your butt retard

>>12126660
Hello dear anon. The way to succeed is to understand that math is symbols. Symbols are like legos, they can move in various ways via "actions" like addition, subtraction, multiplication. When two terms are equal, when x=y, that means x and y ARE the same (in the math realm, where the actual letter is irrelevant). The equal sign is the most holy symbol in math, it means mathematical Sameness. The other thing is that math has patterns. You know that legos need to be moved, but how to place them properly? This comes with practice. With time, 8 looks instantly like 2^3 = 2*2*2. 1 and 0 identities are other common patterns, x*1=x, x-y=0 means x=y. Just patterns. Btw, x*1 can also be x*(x-1/x-1)=x, and this dividing out a factor can be useful in some radicals. Anyway, keep at it, do practice, and remember to BE ABSTRACT!

>>12127656
>The way to succeed is to understand that math is symbols.
That's as retarded as saying "the way to succeed in maths is to understand arrays of numbers". Sure, symbols can simplify and encode a lot of stuff but that doesn't warrant saying that all maths boils down to symbol manipulation.
For example, when I think about mathematics I rarely think about symbols at all and only latter encode my reasoning through symbols.
>remember to BE ABSTRACT
Neophytes do not need to be told this. Their natural inclination is to be as abstract as possible, hence so much attraction towards category theory and other abstract nonsense.
I don't remember where I've read this, but your rule of thumb should be something like for every new definition/abstraction that you learn, you should know at least 5 concrete examples which justify the introduction of the said abstraction.

>>12127671
I don't mean it like that. It's a few things: one he said he struggles in Algebra, which is a symbolic thing mostly. Sure, the other day I did some algebra using visualizing rectangles. But symbols help. The other thing is, I consider geometry and other things to be symbols too - sensations encoded with meaning. Also, knowing how operators fuse with geometric intuition is a good way of "manuevering" through thought

As far as me saying to be abstract, I mean that also maybe in a different than usual sense. Specifically, I struggled a lot with the "equal sign" in high school, what it truly meant. I mean being abstracrt as in understanding each concept for what it is truly, and not letting assumptions cloud action (although anon if you are reading - assumptions are useful in getting to the next step, an assumption is an idea to check out and verify with logic)

>>12127650
Anon said it's the number of closed loops iirc. But picrel says the trace is 0 but there are closed loops in the right 2 matrices.

>>12122790
Project euler is a programming challenge - not a mathematic one.

Reminder to ignore all math advice.

>>12124209
Why is this a 10/10 theorem? Very interesting results but the known proof is with series. I can't see a good intuition for series in general so I can't understand the proof's deep meaning.

>>12127689
You can get this sort of info from it.

But at the end of the day I doubt that this should be viewed as the trace being related to graph theory - it's just that in the graph theory case, the trace is this and that.
In the same vain, I wouldn't say that the function [math] x\mapsto x^{-2} [/math] can be understood in terms of the gravitational potential, simply because it pops up there.

Maybe it's better to view it in the guise of


[math] \mathrm{tr}\ A = \log\mathrm{det}\left(\mathrm{exp}(A)\right) [/math]

or

[math] \operatorname{tr} \left(A(t)^{-1}\cdot A'(t)\right) = \det A(t)^{-1} \cdot \left( \det A(t) \right)' [/math]

>>12127671
>so much attraction towards category theory and other abstract nonsense.
lol the seethe is ridiculous. 99% of mathematicians do not care about CT and 100%of grads and undergrads never heard of it.

>>12127792
You can't say scheme without category theory and you can't read much AG without schemes.
So.. questionable.

I didn't wanna have to do this, right.
But I'm bored, so I'm posting my all new meme shill list that I'm not actually qualified to make.

>>12127792
you have no idea what you're talking about.
>>12127671
>your rule of thumb should be something like for every new definition/abstraction that you learn, you should know at least 5 concrete examples which justify the introduction of the said abstraction.
This is a pretty reductionist view. What exactly do you propose, that a proof of a theorem isn't sufficient without "concrete examples"?

>>12126244
No it's not valid. Since p isn't true, the implication p=>r is true regardless of what r is. So we can't use that fact to make any conclusions about r.

How long does it take to become fast at LaTeX? I've only ever had to use it sporadically and am painfully slow as a result. I'm typesetting my homework this semester to practice, but the first week's has taken so long I might just give up ;(

I know most of the codes at this point, but I am so used to doing math on paper.

>>12128014
Don't give up, just keep practicing no matter what. Try to do as much math as you can straight in latex without writing anything down. You'll get fast soon enough. Plus all the time you save not having to erase/rewrite when you change something is a huge boon.

Did KhanAcademy remove the part of the site with the math dashboard? That was the only part of the site I used.

>>12123110
The fact that the cakes, custard and category theory book exists is completely infuriating.

>>12124209
Baires category theorem.
Cantor's theorem.
Fundamental theorem of calculus (and by extension stokes theorem)
Kantorovich theorem
Banach's contraction mapping theorem
Heine-borel theorem
Euler's theorem (topology)
Inverse mapping theorem
Cauchy construction of the reals (dedekinds construction is cringe)

>>12124023
Are you the famous Zhang et al ?

>>12126272
>it is basic prop logic.
then it shouldve been written in TeX
and not p shouldnt have been written under p or q as if it followed from it

>>12128014
Really fast. At this point I even write my general electives (usually social science or humanities related) homeworks on latex

>>12128059
Chill out, it's just our version of the mystics droning on about quantum energy and the God particle.

What kind of infinite sums, or perhaps even finite sums, of irrational numbers, sum to integers?

>>12128059
Not as infuriating as the omission of CWM.
Also why is Borceux there?
>>12128128
[math]\sqrt{2} + (-\sqrt{2}) = 0[/math] for starters.

>>12128149
>the omission of CWM
Using a book written by some w*ite m*le oppressor instead of the standard text written by a transgender lesbian.

>>12127944
Have you read all of those books?
I found Fomenko&Fuchs to be skipping too many steps to be a good book pedagogically.

>>12128171
Other than Hall, yeah.
Most of my knowledge of Lie groups comes from Bump, but Bump sucks.

>>12128080
Based.
https://www.youtube.com/watch?v=LSWIFXP2r14

>>12128200
And you worked through all the exercises in Fomenko&Fuchs?

>>12128128
Just off the top of my head
[eqn]\sum_{n=0}^{\infty}(-1)^n a_n[/eqn]
Where [math]a_n \in \mathbb{I}\forall n \in \mathbb{N}[/math]

>>12128218
Not really.

>>12128096
>then it shouldve been written in TeX
yes.
>and not p shouldnt have been written under p or q as if it followed from it
no. it is standard to write propositions in a list, regardless of whether they actually are necessary to prove the conclusion.

>>12128217
This video is positively retarded.

>>12128123
Chill is ebonics. Do not use ebonics in /mg/ threads.

>>12128240
Not an argument.

Not enough math on the math general.
Prove this as an exercise (no cheating!)

>>12128253
Explain in your own words what is retarded about the following definition:
A Dedekind cut is a subset S of the rationals Q such that:
- S is nonempty
- if s is in S and rational q is s.t. q<s, then q is in S.
- There is a rational r such that q<r for all q in S.
- S does not have a greatest element.

>>12128272
a+bi -> a-bi for a,b real is an automorphism of C. Thus if q is a root of a real polynomial, so is q* (the complex conjugate of q).
The polynomial (x-q)(x-q*) is a real polynomial for any nonreal q.
QED

>>12128251
Lol that's cray.

>>12128272
[math]A_5[/math] is the smallest simple, non-Abelian group.

The digits 7777777777 are coming. Everyone go to >>>/pol/ and tell them to read Serre.

>>12128305
Oh yeah baby state random facts with no exposition, that really advances the conversation.

> includes Fourier transform as a set of exercises

>>12123339
there are a ton of websites like these they're called "ctf challenge sites", i like rankk.org the best because it's multidisciplinary and has a nice aesthetic
>>12128123
>>12128059
i mean, eugenia cheng is a legitimate category theorist.

>>12122790
Just coomed to tomboys, and finished an integral.

>>12128478
It looks like her primary interest is music and popular math exposition. I saw a talk on categorical music theory at the joint meetings a few years back. I wouldn't be surprised if that was her or someone in her general circle. In any case, I think my mysticism comment mostly still stands.

How are integrals related to integers?
We say a number is integral if its an integral.
Then whats integral about integrals?

>>12128642
Having a stroke bud?

>>12128649
We say a number is integral if its an integer*.

>>12128649
>Having a stroke bud?

>>12128667
Integrity.

>>12128642
>We say a number is integral if its an integral.
Actually, the name is period.
https://en.wikipedia.org/wiki/Period_(algebraic_geometry)

The dot product is such a simple geometrical concept. Why is it so convoluted in linear algebra? The whole trickery with dual space of linear forms is because we cannot do a dot product of two column vectors so we have to transpose one of them so that is a row vector? And thats the purpose of the linear form <v,.>? To take w from V and that way we multiply a row vector by a column vector? This is mind boggling. Why can't we just assume v.w is possible for two contravariant column vectors.

>>12128691
Because not every vector space has row vectors as its elements.

>>12128691
Linear algebra is more general. Let V be the space of all continuous functions on [0,1]. Then we can take product if two vectors by integrating the product of functions.

Integrals are integers with integrity. Integrality is the quality of an integer to be integral. Integrals are not integers but integers are integral.

>>12128701
I see that there are linear functions on V that aren't dot products of two functions in V, such as integration against other integrable but non-continuous functions, or things like the Dirac delta "function".

>>12128766
Astute observation!

Why is complex analysis so much more patrician than real analysis?

>>12128642
In algebra, an integral element of an extension of rings is a root of a monic polynomial with coefficients in the base ring.
It has nothing to do with integrals from calculus.

The words are different in french, for instance. The integrals from calculus come from "somme intégrale", while and integral element is "un élément entier".
Same root, different words, but with english language being nothing but a french dialect gone wrong, we can't always expect to get a clear translation.

>>12128859
Don't ever reply to me again.

>>12128863
I'd be happy to oblige but you'll have to get yourself a trip code

>>12128863
This.

>>12128867
But you just replied to him...

[eqn]\sum_{n=0}^\infty \frac{1}{\sqrt{n!}}[/eqn]
is this something simple? not sure if I'm being retarded

>>12128901
yes it is super trivial. what have you tried

>>12128901
Yes on both accounts.

>>12128901
are you sure it is a square root not an n-th root?

>>12128918
Yes I can solve the nth root, but not this.

>>12128928
Try reducing n! to its factorization, maybe you can pull things out, eventually seeing a pattern? Bumping for interest

>>12128901
Mathematica returns no closed form solution so it's probably not simple, and don't trust smug faggots who reply that it's "super trivial".

>>12128944
The answer is zero right? if you drop the square root and evaluate 1/n! it is zero as n->inf since you get a huge number in the denominator. And the square root doesn't matter. or use sterling approximation.

>>12128969
It's 3.46951 apparently, but fuck knows why.

>>12128975
[math]\forall n \geq 7[/math] the terms are negligible to the overall sum.

>>12128975
Ah sorry my bad I thought you were taking a limit

If biliniear form is a matrix, it is a structure, an entity sandwiched between two vectors like in the picture and matrix multiplication is performed twice with each vector and the result is a scalar. Now imagine the dot product which is also a bilinear form but there is literally nothing in between, so the vectors touch each other directly. How should I think about this nothingness?? What is the "form" here, the dot product itself? But that is an operation, not an entity, not a "structure".

>>12128944
>using mathematica
fuckin loser. ask math stack exchange, some autist like joriki or cleo will have the answer within a day

>>12129350
btw no need to use mathematica i just realized octave also supports symbolic computations via the symbolic package that requires python/ sympy. pretty based. how can mathematica and especially matlab even compete.

>>12129342
identity matrix

>>12128901
I like this question.

You're not the first to ask, given I can find two cringe texts on it
https://mae.ufl.edu/~uhk/FUNCTION-F.pdf
and
https://pilitron.fandom.com/wiki/SQ_constant

It makes me ask the question whether [math] \sqrt{n!} [/math] can be natural, i.e. whether [math] n! [/math] can be a perfect square - which after another google search we find it can't.

Which leads to questions such as whether this sum is transcendental - which I'd expect it is and I'd also expect it may be hard to find out, simply because it's so hard for the Riemann zeta function.

Looking at
[math] \sum_{n=0}^\infty \dfrac{1}{\sqrt{n!}} [/math]
I'm also quickly led to write it as
[math] \sum_{n=0}^\infty \dfrac{\sqrt{n!}}{n!} [/math]
and compare it against
[math] \exp(c) [/math]
where we find c=1.244012.
I couldn't come up with anything more..

>>12129378
yeah I thought about it but it is kind of fake and contrived to think about "something" that takes up space between the two vectors. imagine if we did the same for simple arithmetic multiplication: instead of 5 * 2 we would write 5 * 1 * 2. whats the point. linear algebra is so contrived...

>>12127696
How could you possibly not have intuition for the fucking Residue theorem of all results? It's literally Cauchy's theorem + understanding how circles spin around poles. It's extremely intuitive and wonderfully beautiful from the geometric viewpoint.
Obviously it reduces to the claim integral z^n dz around the unit circle is 2pi if n = -1 and 0 otherwise (which should be obvious to you if you know how to take a fucking integral).

>>12127944
This is extremely based. Saved.
>>12128171
Filtered.

>>12128014
You absolutely 100% need to practice and homework is the best place to do it. If it helps, write out your answers roughly on paper / on a tablet and then put on comfy music or a podcast or something while you type them up. Just do yourself a fucking favor and stick to typing your homeworks, it will get much more efficient after 2 or 3 weeks and then you'll never want to go back.

>>12128080
Good lord. This is an extremely good post. I'd just add Gauss-Bonnet and the von Neumann and Birkhoff Ergodic Theorems.

>>12128838
It isn't. This is the opinion of a very, very immature child (or an algebraist I suppose). Pathology is a necessary condition for beauty.

Normally, I'd be negative, and tell everyone who posted any of these to fuck off and never come back.
However, I'm in an excellent mood, so I'll say: if you didn't make any of these posts, God bless you. If you didn't even reply to any of them, then I love you, brother, please post more.

>>12129485
wew, thank god I had nothing to do with any of those.

But I'm also sure you can find more than those 10 posts here if you search for unnecessary posts

>>12129430
Dude, you need to take a fucking nap or something. There's nothing "contrived" about it. The whole point is that every inner product on a finite dimensional vector space can be written as a matrix like this (which ends up being symmetric and positive definite). Like obviously you're not going to waste your time doing this all the time for the typical inner product. But the point is that you CAN, so it's a result.
The idea is, say you want to prove something about ALL inner products on a vector space. If it's finite dimensional, sometimes it is easiest to represent that inner product <x, y> as x^T B y for some matrix B. Then you do your proof with this arbitrary symmetric positive definite matrix. Later on, someone wants to know something about the standard inner product. Well you proved it for arbitrary B, so now shove in B = I and you've given them the answer. But someone else might come up and be curious about a different inner product, and now you've answered their problem too.
It's not writing 5 * 1 * 2 for 5 * 2, it's proving something about 5 * x * 2 for any number x and then noting that now you have the result in the case x = 1 (typical multiplication of 5 and 2) as well as in cases for any other x.

>>12129485
I've responded to many of those in the last 10 minutes, but purely to berate them.

>>12129489
>thank god
So you are seeking an approval of some anonymous faggot who selected some random shitty posts out of 99% of equally shitty ones and you are happy you didn't make the list? Sad.

>>12129489
>But I'm also sure you can find more than those 10 posts here if you search for unnecessary posts
Yes, but my basic quality standards are very low.

>>12129501
you are an insufferable faggot. kys

faggots all of you

>Trannies throwing tantrums itt

>>12129544
Are you angry because I corrected you about something?

>>12129604
Yes.

Is there a function to detect the signed-ness of a number?

>>12129629
|x|/x, and define to be 0 at 0. This is known as the signum function.

>>12129629
https://en.wikipedia.org/wiki/Sign_function

>>12129485
just don't reply to those, /mg/ has enough drama as it is
pretty sure the guy whose favorite theorem was Gauss' theorem was shitposting, the joke being that Gauss has a fuckton of theorems attributed to him in many different fields. My favorite theorem of Gauss would be Gauss-Bonnet on smooth surfaces

>>12129660
Doesn't Gauss' theorem usually refer to the divergence theorem?

>>12129668
Yes.

Why is it so slow? Is it literally counting to infinity? How does the algorithm even work? What is considered infinity? 10 trillion?
It's been 2 hours.

>>12129632
>>12129639
Thank you!

Ah! Here. I restarted octave and now it is returning immediately. Looks correct.

>>12129737
oh wait, it didn't do shit, it just dumped the sum in some shitty ascii art format. where is my result?!

>>12129741
That's how a computer says the problem is too hard.

>>12129766
Could be, yeah. I thought maybe mathematica could do it better, but it gives me total nonsense. It doesn't understand sqrt??

Ok anons, this one has been fucking with me for a while now. Consider the squares of integers 1 <= n <= 100. How many 50 element subsets have a unique sum?
I have no idea what the insight is supposed to be. You can't iterate over all subsets, too big

Turns out It converges really quickly, so no need to count to infinity. Mathematica is so shit btw. Octave+sympy all the way.

>>12127695
I trust it if it's posted alongside an anime pic

>>12129785
You didn't apply the sqrt function correctly. I thinks sqrt is a variable or something.

>>12129861
Yeah I think it requires square brackets.

Yay. So has to be [] and also capitalized Sqrt.
Quirky. But very fast when it works. Mathematica is so awesome. Octave is so shit. It can't go higher than 150, it will just hang forever.

how do i pronounce math and make it not sound like meth

>>12129878
Mathematica works well if you're willing to put in hours of computational time after about an hour of formatting and correctly syntax for something you could do in seconds computationally in a real coding language. It does make cool plots though, but you really have to have lines and lines of formatting to make them look professional, none of this gray axis bullshit

>>12129881
수학

>>12129881
Mathematics

>>12129906
>do in seconds computationally in a real coding language
lol thats exactly what I thought. I could have coded that sum in literally any language. Except I don't know if it would work as fast without using some clever numeric algos. Not in this case probably anyway.

>>12129881
Try ennunciating like a normal person?

>>12129917
I'm from Latvia.

>>12129660
>My favorite theorem of Gauss would be Gauss-Bonnet on smooth surfaces.
Holy fucking based.
>TFW the most important quantity in Riemannian Geometry is the same as one of the most important quantities in Algebraic Topology
how the fuck

>>12129924
You could say "maths"

>>12129422
root n! cant be natural as a corollary to bertrand postualate if you think for 2 seconds
>>12128901
Yeah the question is super easy, first year calc. Everyone saying its not is trolling you

>>12127944
I liked this one too, have a spurdo

>>12129501
uhmmm... someone on the phone wants you

>>12124209
no retract from n dimensional disk to it's boundary. (algebraic topology)

I'm entering a PhD program in a foreign country and while there is an algebraic number theory department, people's interrest (including my advisor's) are very different from mine (I'm more into arithmetic geometry).
My advisor told me he can help me with navigating academia and figuring out articles and journals, but not on research, or not even to find a subject. He just advised me to scout on mathscinet (I should get my access next month) until I find something cool in my area.

Has anyone been in a situation of mathematical isolation? How did you manage it? I'm thinking that once I get contacts, I can work with people abroad through the Internet, but I can't really expect to attend international conferences for now, with everything being dead or postponed.
For now, I'm reteaching myself algebraic geometry and I'm gonna find myself a class field theory textbook as well (if you have any recommendation, btw...).
I'm also scouting grad student talks at online conferences to see if I can't find myself a low hanging fruit to get started with doing research, but I feel like some sort of vulture looking to steal research topics from my fellow grad students.

>>12130292
You know what's nice? A nice lecture, just sitting and watching somebody talk.

>>12128901
N cannot be negative and you must rationalise your surd.
N=1
1/surd(1)=surd(1)/1
N=4
1/surd(1*2*3*4)

>>12128901
[math] \pi + \log(\pi / 8) + \tan^{-1}(\pi)[/math]

>>12130331
Checks out.

I come from a strange world.

Teach me your ways oh math wizards, teach me of the legendary z, the third dimension, and even maybe the 4th.

>>12130342
Take [math]\mathbb{R} ^n[/math] and set [math]n = 4[/math].

>>12130316
I didn't quite catch your point.

Am I supposed to fully grasp analysis proofs that involve delta-epsilon and the like? Usually they are fine, however today we proved that the thomae function is continuous at rationals and the intuition behind the choice of delta was very foreign to me, and I probably wouldn't have been able to come up with it if I sat down for a week. I'm not sure if any other student has been able to do it themselves. Does it get easier bros?

>>12130405
continuous at irrationals*

>>12123335
We've learned the Bourbaki one in highschool, but everyone uses the other one after that

>>12130405
>>12130408

I don't know how it was proved in your lecture, but to be quite honest, looking at the function for a few seconds is enough to get an idea of proof :
We just need to prove that if x is irrational and n is a natural number, we can find a [math]\delta[/math] such that any rational number [math]q \in (x-\delta;x+\delta)[/math] has its minimal denominator greater than n.

Now, you divide the real line in interval [math](0;\frac{1}{k})[/math], [math](\frac{1}{k};\frac{2}{k})[/math] and so on.
For all natural number k lower than n, x falls inside one of those interval (and clearly not precisely on the boundary). Just take [math]\delta[/math] to be half of the minimal distance between x and the bound of one of those intervals and you're all set.

So I guess the answer is yes, it gets easier. A problem like that is not very hard to solve if you have good intuition and vision of the behavior of real numbers.
However, as you get better, problems you're dealing with usually get more complicated (at least if your goal is pure mathematics), since the easy problems have often already been solved.

>>12130292
your only bet is on conferences (ideally, paid from some stipend)


get to know the people in the topic you want to do a thesis, by browsing articles on arxiv, then ask them by email the openings for a phd

>>12130437
And of course I fucked up my tex...

So in the middle, it should have been
>Now, you divide the real line in interval [math](0;\frac{1}{k}), (\frac{1}{k};\frac{2}{k})[/math] and so on. For all natural number k lower than n, x falls inside one of those intervals (and clearly not precisely on the boundary). Just take [math]\delta[/math] to be half of the minimal distance between x and the bound of one of those intervals and you're all set.

>>12130331
lmao nice

>>12130437
>>12130445
>minimal denominator greater than n
This was the part that I thought I would struggle to come up with. I guess I haven't used the archimedian property enough yet, need more problems lol.

>>12124209
Feit-Thompson
Implicit function
Banach-Hahn
Zorn's lemma
Existence and Unicity of solutions of ODEs
Brouwer fix point theorem
(I like exotic spheres, and that we have a one-parameter family of differentiable structures on R^4, but these are only interesting, and not that useful)

>>12130464
fuck, i forgot the Wedderburn-Artin theorem

>>12126790
We just had this as an exercise lol

>>12128080
Great taste right here lads

>>12130462
Yeah, this comes to mind quickly because I've encountered the question of approximation of rational numbers by irrationals, in the form of a lecture about continued fractions I attended 6 years ago.

It's the algebrist in me talking, but quite often, you'll get insight on these sort of things by knowing intuitively the overarching structure of whatever objects you are working with.

So the best thing you can do to get more confident about these, aside of course from practicing on problems, is to widen your understanding of the material by getting interrested in many little things here and there.
It's hard to fully explain what I mean, especially as english isn't my first language, but you have to somehow get "intimate" with the material of your courses, if you want proofs to come naturally to you.

>>12130464
>(I like exotic spheres, and that we have a one-parameter family of differentiable structures on R^4, but these are only interesting, and not that useful)
When I first heard of it, I got very excited at the idea,and I remember finding this book :
https://www.worldscientific.com/worldscibooks/10.1142/4323
Now, a quick research let to that article :
https://www.arxiv-vanity.com/papers/hep-th/9411151/

I don't know enough physics (though the sentence "The analysis is based on the A. Connes’ construction of the standard model." from the article did give me a slight hard-on) to figure out how promising it all is, but the idea that they might use exotic differentiable structures in physics is quite exciting, and apparently not entirely dead.

>>12128272
If we have a a root of f, then we can use polynomial division, to extract a factor of x-a, the remainder has to be zero, as seen, if we input x=a into (x-a)*g(x)+r(x), and from analysis we know that every polynomial of odd degree has to have at least one real root.

>>12130479
Whaat, so they're useful after all.
Well, now I know something that i didn't before, thanks anon.

>>12123865
prime number are numbers that have only 2 factors: 1 and themselves. composite numbers can be represented as the the multiplication of two smaller numbers. so how the fuck i would go about doing a number that contradicts itself?

>>12130804

>>12124209
GRR

>>12130464
>Existence and Unicity of solutions of ODEs
underrated

>>12130445
>For all natural number k lower than n
personally I would rather fix k than leave it ambiguous like that, less room for error even if it isn't causing problems right now