/sci/ - Science & Math

>>11975242
Aren't maths just pen and paper RPG's mixed with hieroglyphs?

>>11975253

>>11975258
Are big equations and proofs just writen down encounters?

>>11975266
>Are big equations and proofs just writen down encounters?
Yes, and thesis's/papers are campaigns.

>>11975274
What you are saying is, Tolkien was secretly a physicist

>>11975242
Quartic curves

>>11975286
>What you are saying is, Tolkien was secretly a physicist
And Einstein larped in his free time

>>11975242
>Elliptic curves

Maybe algebraic geometry isn't as gay as I thought.

>>11975242
Fuck physicists, prove the area under a curve is the integral from first principles right now no internet.

>>11975635
>no internet.
Then how am I supposed to post the proof?

>>11975643
well you weren't suppose to so i could then reply to myself saying how 0 physicists could do it because they are all dumb numerical method tards

>>11975635
What first principles

>>11975659
cringe

>tangent vectors to a smooth manifold in the abstract are treated as point-derivations
>this strict treatment is thrown away in the case of smooth submanifolds of R^n
I can't take this anymore. My brain is malfunctioning because of this convention and I feel like I'm going to have a breakdown

>>11975673
Yeah that's what I expected

explain to this brainlet how 6+(-1)=7
If I have 6 dollarydoos and spend 1 I have 5 bucks left not 7

>>11975752
[math]-6-1=-7 \neq 6-1=5[/math]

>>11975755
but it's -3x - 6 not -3x - (-6)
Fuck this shit mayne, I guess i'm taking that remedial class

>>11975752
Anon you have -6-1 there, not 6-1

>>11975778
wack

Is quaternionic analysis an active field of research?

/gmmg/

Somebody was asking about bundles a few threads ago, so:
>Notes on equivariant bundles
https://arxiv.org/pdf/2008.01268.pdf

>>11975730
>>11975730
if you're just starting with diff geo, don't expect to understand everything at full rigour with all technical details. not possible, the structure is just too complicated. don't overthink it, you will get it eventually.

on my way to become the world's strongest mathematician
yesterday i learned about gamma and zeta functions, and why [math]\zeta(0) = -1/2[/math] and [math]\zeta(-1) = -1/12[/math]
today i'm starting elliptic shit

>>11976621
Prove firstly that the Riemann-zeta function diverges for all n less than or equal to 1. Explain then what do we mean by zeta(-1) and zeta(0)

>>11976936
>prove firstly that infinite sums larger poinwise than the harmonic series diverge
>explain analytic continuation
Why are some people so absolutely horrendous at showing off?

>>11976949
Whatever showoff.

>>11976936
did you mean: prove that [math]\sum_{n=1}^{\infty} 1/n^s [/math] diverges for [math]s[/math] with real part <= 1? because that's the only way i can make sense of your question

>>11976949
He's right though. Analytic continuation isn't real math but rather a handwavy physicist trick. "This function diverges but if I take a totally different function and pretend they are the same only because they share the same values on a small set, then I can make it converge to whatever I want." Cool story, bro. f(x) = x and g(x) = x^2 share the same values at x=0 and x=1, now I can pretend they are the same, too.

My brain is very smooth, can someone tell me what I need to look at on Khan Academy to be able to answer this type of question? ty

>>11976994
You don't understand analytic continuation.

[math]G=(\{2,5\},+,5) \\ G \cong \mathbb{Z}_2 \Rightarrow 2+2=5[/math]

>>11977025
That's just, I don't even know how to describe it anon. It's just commutativity of a sum isn't it? If x + y = y + z then necessarily x = z. At least on what you're working with

>>11977049
That's shameful. Those names are totally unmotivated.

>>11977049
>G=({2,5},+,5)
you didn't say how + is defined

>>11977049
I hate it.

>>11977055
Why not x = y?

>>11977159
Subtract y from both sides of the equation.

>>11977162

Thank you, that makes sense too me cheers

>>11977182
Cheers, lad.

why do retards not understand that the only reason we made analytic continuation was to get around their shithead trash definition for convergence
if we just accepted a more general definition we wouldnt need to continue a function, since it would already be defined

>>11977049
Retard

> TL;DR Bound on expected number of moves of a search algorithm

Consider an [math]n \times n[/math] undirected lattice, where one of the nodes [math]t[/math] contains a treasure. Each node in the grid points to one of its adjacent nodes, and following all such pointers forms a shortest path from the starting node to [math]t[/math] (i.e. pointers provide guidance to the treasure). However, due to noise, each node is faulty with probability [math]p[/math] (a faulty node points in some random direction). Starting from the middle of the grid, the goal is to reach [math]t[/math]. The complexity measure is w.r.t. [math]d[/math], the (unknown) distance from the starting point to [math]t[/math].

My heuristic in a nutshell: assign scores - based on the pointers seen so far - to all nodes that are adjacent to the nodes we've already visited, and jump to the node with the highest score until [math]t[/math] is found. I'd like to analyze the expected number of moves. How do I even begin?

>>11975242
Does anybody actually read textbooks word for word? If so, doesn't it take a very, very long time? Reading one Rudin proof sometimes would take me 10-30 minutes. Does this mean I'm a brainlet or is it normal?

>>11977703
>If so, doesn't it take a very, very long time?
it does, several month easily

>>11977703
>If so, doesn't it take a very, very long time?
Yes.

>>11977703
It means you're a brainlet. I'm by far no genius but reading 100 pages of a textbook per day is normal for me. And I'm not talking about babby's first calculus.

>>11977958
>And I'm not talking about babby's first calculus.
post example of such textbook

>>11977958
I don't think you're reading with pen and paper in hand anon. That's no way to learn math, although you might pick up some buzzwords.

>>11977958
*tips fedora*

>>11977958
So if you can supposedly demolish the first 8 chapters of Rudin in one day, how is it that most people including people who eventually won fields medals usually take two semesters to do that?

>>11976621
truly glad mathematicians are /a/ guys

>>11978053
They're lazy retards who can't into Zimbabwean electro-pop fan forums.

>>11975286
Wasn't Wojak the Knight?

>>11978087
No.

>>11978095
:(

[math]\textrm{Let} \ G = H \rtimes_{\phi} K, K \rightarrow \textrm{Aut} (H) \ \textrm{and be homomorphic. There is only one} \ p \textrm{-Sylow subgroup of} \ G, \ \textrm{denoted by} \ P. p \ \textrm{divides} \ |\phi (K)|. \textrm{Prove} \ p \ \textrm{divides} \ |H|.[/math]

>>11978173
¿Are you having a stroke bud?

>>11978199

>>11977958
Based and learning-and-forgetting-and-relearning pilled.

>>11975242
If only there were a discipline that combined mathematics and physics--- a "physical mathematics", if you will.

>>11978255
Mathematics is physics where experiments are cheap.

>>11978326
Blasphemy.

>>11978326
>experiments are cheap
>no one can figure out whether some graph with like 50 vertices exists
>legendary figures got btfo by simple searches on vacuum tube machines in the 1950s

>>11975362
What are the pre reqs?

>>11978669
I wouldn't even attempt it without a PhD in septuple integrals, but of course then you can have any job you want.

>>11978669
Abstract algebra, number theory, and complex analysis. Check out Elliptic Curves, Modular Forms, and Their L-functions by Lozano-Robledo for a good intro.

Is tensor calculus just a generalization of vector calculus?

Are tensors just a generalization of vectors?

>>11978925
>>11979119
Every tensor is a vector.
Additionally, every vector is a tensor.
From this we conclude that tensors and vectors are the same thing, q.e.d.

>>11979119
Are tensors just a generalization?

>>11979140
Generally speaking they're a special case.

im a brainlet and want to start a math degree in 1.5 years.
I've just started learning about quadratics by completing the square. Is it possible to go from algebra 1, geometry, algebra 2, pre-calculus and then introductory calculus in 1.5 years assuming 2-3 hours of study a day?

I'm also currently completing a non-STEM degree part time.

>>11979278
A normal adult can literally go from no math to calculus in one year if they're motivated. I've had multiple friends do this.

>>11979284
Thats good to hear. did your friends have other commitments or did they dedicate themselves full time 6+ hours a day?

>>11979293
No, they were working full time and studied in their free time, but they were dedicated.

>>11979301
giga chads with good delay gratification and good motivations. fuck how do i do this

>>11979304
One of my friends who did this was a cis girl and another was a gay guy. Just git gud, it doesn't matter who you are.

>>11979293
stop looking for excuses faggot

>>11979293
Anon, calculus isn't some sort of super high level math. Millions of highschool students learn it every year.

>>11979312
>>11979349
>>11979363

>>11979363
>>11979349
>>11979312
ive spend 6 hours a day shitposting a day for years. ive destroyed my dopaminergic system. My concentration span in 10 minutes. Then I shitpost for 2 hours, do another 10 minutes. After the whole day im lucky if i studied 1 hours How do i motivationmaxx?

>>11977368
Isn't that more of a definition than a proof? Definition as in, it defines the sum to converge for imaginary powers. We wouldnt know if it converges otherwise, and there's nothing to check against because e^ix is as yet undefined

>>11979384
see i dont even have the attention span to properly respond

>>11979384
spend a few days detoxing on a camping trip, then dont shitpost when you get back

The ridiculous insistence on convergence has had devastating consequences for the field of mathematics and all fields that use it.

>>11979265
Hehehe

>>11979386
Yes, same with negative powers. You can think of it as being “defined” as a reciprocal to be consistent with the rules of exponentiation.

>>11979416
Based matharchoprimitivist schizo

>>11979416
BASTE

>>11979416
blow it out your ass fagatron

>>11977992

>>11979386
there are several definitions of exp, sin and cos. when you pick one the others become theorems.

>>11978173
please post solution

If I have x'=Ax and I know solutions x1 and x2 and I know A is a 2x2 matrix, how do I go about finding A?

I can give x1 and x2 I'm dealing with, but I'd like to understand the method generally speaking.

>>11976994
Who the fuck explained analytic continuation to you? That's one of the dumbest takes I've heard on the topic.

>>11977958
>I'm by far no genius but reading 100 pages of a textbook per day is normal for me.
lol

Am I too dumb for math? I read the first chapter of the Book of Proof and I can't seem to wrap my head around it. Should I just give up?

>>11980383
set up x' as a linear combination of the columns of a by x1, x2, solve

>>11980466
¿Do you have enough experience to start higher math? Maybe you need to do something more basic first.

>>11980483
I've only done Calc I/II and just a bit of Linear Algebra. Since I'll be starting discrete maths this fall, I decided to take a look at BoP. Right now, I get most of it but I just don't know how to solve any practice problems since they are a tad bit difficult. I know I'm self-studying with no help, but I really don't know where to start and I'm scared I might fail the class.

>>11980489
Repeated exposures over time are the best way to learn, cramming won't help. Just give it time.

>>11976994
I don't think you know what analytic continuation means

When was the last time you solved a quadratic equation?

>>11980515
Yesterday when I was writing an exam.

Does someone have a reference for the mod p cohomology ring of [math]K(\mathbb{Z}, 3) [/math]?

>>11980236

>>11980900
thank you

>>11980489
What >>11980490 said. Your discrete class will help with a lot of the stuff, getting regular homework problems always helped.

>>11975242
Hey guys, I'm interested in learning Gauge theory. I know that the subject involves principal bundles, connections on principal bundles, and some other type of stuff like that. I've already taken grad courses in algebraic topology (Hatcher up to homology) and differential topology (up to de Rham cohomology and some stuff about characteristic classes). Which books would you guys recommend for someone with my background? I was thinking of getting Loring Tu's differential geometry book but idk if it's a good book. Any help is appreciated, thank you!

>>11978925
Tensor calculus is not actually a real subject.

>>11975242
Is Rudin still based analysis guy? I am enjoying the books but idk what I might be missing because I have never used any other.

>>11981198
Abbott, Pugh, Tao, Spivak, Zorich, Amann, the list uniroincally goes on and on.

>>11978925
Alright watch this shit. You got vectors right. Then covectors are linear maps that map vectors to a real number(technically they map to whatever field your vector is over but I’m assuming it’s the reals). A tensor is then a map that will take a shit ton of covectors and a shit ton of vectors and map all that to a real number.
Also watch eigenchris on YouTube he has a good video series about it.

>>11981281
Can you map from a tensor to a tensor?

>>11981351
Yeah it’s called tensor contraction. Also technically speaking, covectors, vectors, and even scalars are all types of tensors so almost any map linear algebra is going to be a map from a tensor to another tensor.

>>11980882
It's gross and doesn't have a nice description.

>>11981198
I would look at the textbooks in >>11981210 to give you another perspective on topics, they're all quite good.

I'm studying Pinter's abstract algebra book over the summer. I'm currently struggling with some of the exercises. Would it be better to keep working at every exercise no matter how much time it takes or should I just strive to finish the book before the summer ends?

I'm debating on whether it would be better to be vaguely familiar with more material or be somewhat proficient in less. I'll be taking a proper course in abstract algebra down the line too if that changes anything.p

>>11981481
What exercises in particular are giving you trouble? While Pinter is very introductory, some of the exercises are well above a first undergraduate algebra course.

>>11981508
That makes me feel a little bit better about skipping around. I guess since I'm bound to take a more detailed look at the topic anyway,I might be able to just get away with doing the mechanical exercises and the easy proofs.

I started struggling with some of the proofs on chapter 8. I'd post them, but I'm on my phone right now. I think E.1 and F.1

>>11979119
no one (in math) even cares about "tensors". All you are interested in is the tensor product as an operation between vector spaces/modules/rings etc.

>>11981536
https://www.math.wisc.edu/~mstemper2/Math/Pinter/Chapter08
Here are solutions for chapter 8.

>>11981538
>metric tensor
>curvature tensor
They're quite important in diff geo

>>11981481
>>11981536
In general, I think it's fine to skip around a little bit. If you're having trouble with some of the exercises, try moving on and see if some distance from the problems and future concepts can help make it easier.

>>11981596
Thanks.

In my book, the inductive proof for the existence of jordan bases assumes that "you can pick v such that Nv = u" where u is a vector in the jordan basis from range N. It doesn't require that the v you pick is not in range N, and the proof seems to imply that you'll arrive at a linearly independent list of dim V regardless what v you pick, as long as Nv = u. But, if v were in range N, the jordan basis would be linearly dependent. What gives?

Book is axler btw, i cant post a pic of it cause my pc camera broke but its page 183-186 on edition 2

Can anyone point me to some good resources, I'm looking for a good way to learn about pde. I'm about to finish my degree and taking a formal course in it isn't going to fit into my plans, but would like to be exposed and have some competency around them for the future.

>>11982251
Here you go https://www.maplesoft.com/support/help/Maple/view.aspx?path=Task/SolvePDENumerically

Formally speaking, is the empty set qua subset of the cartesian product empty set times empty set a bijective function from the empty set to the empty set?

>>11982382
It's called the empty function, and yes, it is.

Give me an interesting example of a lattice.

>>11982408
Post's lattice of clones on a two-element set.

>>11982417
You might not like it, but this is peak performance.

>>11982384
Nice, thanks.
>>11982408
The lattice of open sets on the unit circle.

Is Strang's book on differential equations and linear algebra good?

>>11982420
but I do like it

so what is the intuition in the continuous case?
sure the proof is black and white, but I still do not trust it intuitively.
In the discrete case, it is very straightforward (as there is no inverse scaling by abs(a)).

In the continuous case, though, one deals with the cdf instead. I do not really intuitively understand where the 1/abs(a) comes from.
I guess it has something to do with the area from the integral in the cdf being scaled down by 1/abs(a), so the derivative pdf would need to too, but I do not really know.

>>11983055
It's in essence the change of variable theorem not sure what other intuition you can take from it.

>>11983060
>not sure what other intuition you can take from it.
In the multivariate case it becomes the Jacobian determinant of the transformation matrix, so you can import any intuition from linear algebra that you might happen to have.

>>11981538
>>11981538
>no one (in math) even cares about "tensors".
??? if you're not joking, you're either an undergrad, a category theorist, or an undergrad category theorist

>Just enrolled to a calculus course
>I don't even know high school-level algebra


How fucked am I?

>>11983120
you are fucked BIG TIME mate

>>11983152

What do

/gmmg/

>>11983099
A real category theorist would most likely not say that. They have their monoidal categories and the tensor product functor can be used to provide both an archetypal example of a left adjoint and an example of three functors giving two adjunctions, namely the induction-restriction-coinduction triple in which induction is done using the tensor product. An undergrad category theorist, on the other hand, could very well think like that.

>>11980882
I refer to it as "something nasty", or "Kurwa(Z, 3)".

>>11983163
learn math. it's honestly not that hard.

Could somebody please recommend a good online particle physics or cosmology course? I want to make sure my passion is actually for physics, rather than popsci books about physics, before ditching my bio degree.

>>11983120
im in the same position. course starts end of september. i've just been using khan academy it seems to be okay-ish for math up to calculus

>>11983238

I've got the 5th edition of K.A Stroud's Math of Engineers, gonna do the F2 - Algebra and hope it'll give me enough to be prepared for the course

Hows your progress so far?

>>11975730
If you want something that lends itself to nice geometric interpretation, one way to think about tangent vectors on abstract manifolds is actually as equivalence classes of smooth curves (under what specific equivalence relation, I can't remember). You recover the point-derivation definition of a tangent vector by taking the directional derivative of a function through the curve at the point. (The equivalence relation should be defined so that this makes sense.)

Then if you're working on a submanifold of R^n, you should be able to identify the "equivalence class of curves" with the actual tangent vector equal to the derivative of the curve.

I'm not sure if this helps, but it's one way I like to think about this. As the other anon said, differential geometry can be messy with all the different things going on. You'll eventually understand it anon.

Is p-adic geometry a meme?

>>11975242
Lads, let's say I have a 2D polynomial with every odd coefficient from y^1 to y^7 and x^1 to x^15, how do I minimise the number of operations to evaluate it? Do I compute each line for a given degree of y using Horner's scheme then apply Horner's scheme to each sum of each line, or do the same in the other axis, or start from the highest degree corner and go down diagonally multiplying the sum by x^2*y^2 everytime?

applied math chad here

>>11981712
Reply

>>11983540
Unicorn here.

I have a class in Discrete Math over the fall. topics to be covered are:
>Logic and Proofs
>Sets, Functions, Sequences, Series, Sums and Matrices
>Algorithms
>Number Theory and Cryptography
>Relations
>Graphs
Book to be used is Rosen's Discrete Math 8th Edition. It seems bloated. Any other good supplementary texts that are more direct?

>>11983653
>>Logic and Proofs
Introduction to higher mathematics
>>Sets
Naive set theory
>>Functions, Sequences, Series, Sums
Understanding analysis
>> and Matrices
Linear algebra done right
>>Algorithms
lol kys
>>Number Theory
Rosen
>> and Cryptography
that pink book
>>Relations
Introduction to higher mathematics
>>Graphs
Graph theory: an introduction

>maths today is just spewing terms that confuse the layman to stroke your own ego while all those simple matters could be explained in a simple way
Ah....

>>11983722
That's right, it's a trick to confuse the normals. Math is just making up silly words for everyday things.

>>11983181
Morning, lad.

Shouldn't the answer be negative infinity, meaning the limit exists?

>>11983783
In this context, negative infinity means the limit doesn't exist.

>>11983783
a limit 'exists' if it's equal to a number

>>11983804
>>11983793
Thank you

>>11983793
>>11983804
But negative infinity is a number in the extended real number system.

a*b=0
a= 0/b
b=0/a
a=0, b=0

but these are not the only solutions to a*b=0, what gives?

Assume V=R2
Operator T maps R2 onto v1
Operator Q maps R2 onto v2
Operator R maps R2 onto v3
all 3 vectors nonzero

now assume w=av1+bv2+cv3 = 0
Tw = av1=0, so a equals 0
Qw = bv2=0, so b equals 0
Rw = cv3=0, so c equals 0

so for w, a list of 3 vectors, the solution is all coefficients are 0 if they sum to 0. meaning this list is linear independent. but it is 3 vectors in r2, so it mustn't be

obviously 0 is a solution, but other solutions exist. yet the equations i used above lost those solutions. Why?

Do you guys have any advice on how to learn faster? It takes me way too many times reading something and messing up on exercises before something finally clicks.

>>11983987
learn programming

>>11983934
>Operator T maps R2 onto v1
>Operator Q maps R2 onto v2
>Operator R maps R2 onto v3
not linear maps

>>11983997
have you heard of projections anon?

>>11983997
i meant v3's space if that wasnt clear, so v3*some coefficient not neceasrily v3 *1

but yeah just imagine TQR are orthogonal projections

>>11983988
But programming is not fun

>>11983987
Don't. Focus on understanding over speed.

>>11984018
then
>Tw = av1
>Qw = bv2
>Rw = cv3
is not true in general

>>11984014
retard

>>11984030
oh
thats true

>>11984029
I'll try to do this. I just get frustrated sometimes because I really want to move onto new stuff, but I'd get stuck on something for a couple of days instead and I don't even know what it is I'm having trouble understanding until finally it just starts to make sense and I feel stupid for stumbling over things that, at that point, seems pretty obvious.

>>11975635
What are you defining the area under a curve to be? What do you think an integral is?

https://www.mathtrainer.org/
https://youtu.be/AdBoybKnzZw

how do you claim to be a mathematician if you can't pump basic operations like a mad lad?

>>11977958
I highly doubt you're actually learning much that way.

>>11977703
I'm midway through my PhD and, if I do say so myself, am doing pretty well. I dont read text books, and when I do, I dont read entire proofs. I make sure I understand how things fit together and ignore the technical bits (although that works because I'm very good with the technical side by now). I dont have to attention span to read, but I do love solving problems and coming up with my own narrative for things. With this method, I eventually learn everything I need to know, though I have a bad habit of reinventing the wheel. I was a meh student early on because of this. Find what works for you, but ya, dont read a book cover to cover. Learn to find the important ideas and focus on those. The rest follows naturally over time.

>>11975882
I have heard of analysts considering quaternions in their work as an extra level of abstraction. I've also heard of it from the algebra camp in the context of Hopf algebra stuff, but I dont think its terribly active. Like some people like them but I doubt anyone specializes in quaternions. I could be wrong. I like quaternions but I haven't thought about them since my undergrad.

>>11983783
[eqn] \lim_{x\rightarrow 0^-} \frac{2}{x}=-\widehat{\infty}[/eqn]
Thank you for coming to my Tooker talk.

[math]x \in \mathbb{S} \mathbb{N} \mathbb{E} \mathbb{E} \mathbb{D}[/math]

>>11984141
He most likely meant the relationship between the antiderivative (indefinite integral) and the area, aka the fundamental theorem of calc

So doubling the unit cube with ruler and compass
I got to v= 2.12 is that good enough?

>>11984322
Not happening bro

Honestly, the hardest part about algebraic topology is all the combinatorics

>>11984563
Combinatorics is a great field with numerous applications.

>>11984563
Is there actually a lot of combinatorics in AT?

>>11984620
It'd be easier to ask what field doesn't have a lot of combinatorics in it.

>>11984620
kind of. One of the main objects of study, CW complex are fairly combinatorial objects. Also some non-elementary calculations need many convoluted and confusing integer calculations

>>11984694
>>11984735
fuck

>>11984816
Do you dislike combinatorics or something?

>>11984816
Chill out, it's literally just counting lol.

>>11984987
Just a bit of a weak spot for me

I'm struggling with manipulating cubic equations and trigonometric equations in calculus, especially polynomial factoring and division. Can anybody recommend a text that I can use to get good at this shit?

>calculus

Are there any texts that provide an introductory overview of Clifford algebra?

>>11985049
The correct answer is to learn abstract algebra.
The practical answer is to take a look at Lange's 'Basic Mathematics'.

I come from a social science background and I can crunch numbers in SPSS, but I want to pull back a bit and get a solid understanding of statistics. Theorems, proofs and all that. You guys have a chart or something?

>>11985227

>>11985227
SOCIAL
SCIENCES

REEVALUATE

>>11985263
I recognize some of these words. I obviously meant some "so you want to learn statistics like a man" chart with a series of books.

>>11983673
>>11983653
Does Axler even touch much on matrices?

>>11985326

>>11985569
I already worked through statistics for social science students, so a "highly accessible" book for business majors won't cut it. I want Spivak for statistics. I want to grow statistical hair on my statistical balls.

Thanks for your photoshop efforts though anon, really appreciate it.

>>11985608
>Spivak for statistics
https://math.stackexchange.com/questions/2635903/a-book-on-statistics-similar-to-michael-spivaks-calculus

>>11985608
>Applications
Sheldon Ross "First Course in Probability"
>Theory if you never need to talk to mathematicians
Larry Wasserman "All of Statistics"
>Theory if you do
Kai Lai Chung "A Course in Probability Theory"

>>11982251
Undergrad mathlet who took an intro pde course here:

-Semilinear eqns, some 1st-order eqns, can all be solved with the "Method of Characteristics". eg Ux + x*Uy = 0 -> any U will be constant along the curve defined by dy/dx = x/1, so y = .5x^2+C, and so any individual U can be uniquely found by C, so U=f(y-.5x^2); solve given initial condition.
- Heat eqn: Ut + kUxx=0. Solving analytically is a bitch that took us like 3 class periods: U = integral dy over all R of S(x-y)(y), where is the IC, S is the Green's function (1/sqrt(4kt pi) )e^((x-y)^2/4kt).
-Wave eqn: Uxx - c^2 Uyy = 0 -> (d/dx - cd/dy) (d/dx + cd/dy) U = 0. Letting this be Vx - cVy, use characteristics to solve for V, then find an additional homogeneous solution; end result is (1/2)(f(x+ct)+f(x-ct)) + int(x-ct to x+ct) g(y)dy. (Or something like that)
- 2nd order 2var pdes can, by change of variables, be roughly heat, wave, or laplace's eqn, plus some terms. Don't actually solve stuff that way, anything but simple edge conditions make the above gross (you need eg function extensions that flip at the boundary axis so as to match on the half-plane of actual definition).
- Instead, try separation of variables for homogeneous problems: assume U=X(x)*Y(y), plug it into the PDE, separate into f(X(x))=g(Y(y)), which must equal some eigenvalue C. This becomes ODEs for X and Y; the infinite weighted sum of possible Uc is a full soln of U. To match to ICs, look up Fourier series; we mostly did PDEs which had sin or cos in the eigenfunctions, so it was "easy" to match to a Fourier sin or cos series.
- The real takeaway was that basically no PDEs can actually be solved analytically, and the solvable ones are still disgusting. Look up numerical methods for PDEs, I can't summarize it because it was really finicky and my professor had an accent and we went online at that point and other excuses, etc
tl;dr cry

>>11985762
Woops, I made some mistakes. Oh well, not like it matters. Again, tl;dr is to learn Matlab and forget about solving analytically. Course, my school only really makes contributions to mathematics (to the extent it does, natch) in the realms of computational pde solving, numerical methods, etcetera, so maybe there's bias on my professor's part

WIll learning abstract algebra make elementary algebra easier?

>>11984563
>he doesn't know

>>11985803
Elementary algebra just boils down to applying inverse functions, or taking advantage of factorization properties of certain rings.

>>11985803
Not easier per se, but you'll have a lot better understanding of things that you previously took for granted.

how can we tap into holy energy to understand math better?

>>11977703
It might help you to read over the section multiple times paying increasing attention to detail each time. the first time skip over proofs and just focus on an understanding and intuition / the big idea. the second, skim over proofs to get an idea of how they're tackling the problem and what is actually being proved (some books don't explicitly state this at times), skipping tedious parts and the third going over it again.
>>11979278
It's definitely possible. Are you sure you're even interested in math in the first place and not falling for /sci/ memes?
>>11979384
I was this way not long ago, the secret is to remove the distractions instead of trying to gain the willpower to overcome them. Study away from any electronics, leave your house and go to the library if you have to.

>>11985914
Ask Ramanujan

>>11985118
Any text on spin geometry starts with Clifford algebras

>>11985824
k-know what?

>>11981041
just take a physics book before a math book

also i think ischam has a book on dg

>>11982408
>Give me an interesting example of a lattice.
http://www.paultaylor.eu/ASD/foufct/sierpinski.html

7.1. We have already made use of the analogy between topology and set theory, in the form of the Lindenbaum–Tarski–Paré theorem in §§4.5 & 5.3, as part of the motivation of the monadic property as an abstract formulation of Stone duality. Similarly, we begin by looking at the subobject classifier (lattice of truth-values) Ω in a topos, in order to identify the relevant properties of its analogue in topology.

>>11983224
the best book for gr is dirac

you can read sean caroll lecture notes too for gr

or just go to arxiv and search for introduction to gr and cosmology

>>11985118
Traubenberg_2009_Clifford Algebras in Physics.pdf

Alder_2009_Geometric Algebra An Introduction to Clifford Algebras.pdf


Boi_2009_Clifford Geometric Algebras, Spin Manifolds, and Group Actions in Mathematics and Physics.pdf


Hiley, Callaghan_2010_The Clifford Algebra approach to Quantum Mechanics A The Schroedinger and Pauli Particles.pdf


Hiley, Callaghan_2010_The Clifford Algebra Approach to Quantum Mechanics B The Dirac Particle and its relation to the Bohm Approach.pdf
Hiley, Callaghan_2011_Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation.pdf


Hestenes is autistic about clifford algebras and want to rewrite all geometry with that

Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics
Book by David Hestenes and Garret Sobc

>>11986310
>Hestenes

start with Hestenes_2003_Oersted Medal Lecture 2002 Reforming the mathematical language of physics.pdf


here is some more autism
Hestenes_2010_New Tools for Computational Geometry and Rejuvenation of Screw Theory.pdf
Hestenes_2010_Modeling Theory for Math and Science Education.pdf
Hestenes_2009_Zitterbewegung in Quantum Mechanics.pdf
Hestenes_2008_Reading the Electron Clock.pdf
Hestenes_2008_Gauge Gravity and Electroweak Theory.pdf
Hestenes_2005_Gauge Theory Gravity with Geometric Calculus.pdf
Hestenes_2003_Spacetime physics with geometric algebra.pdf
Hestenes_2003_Oersted Medal Lecture 2002 Reforming the mathematical language of physics.pdf
Hestenes_2003_Mysteries and insights of Dirac theory.pdf
Hestenes_1991_The design of linear algebra and geometry.pdf
Hestenes_1986_Clifford Algebra and the interpretation of quantum mechanics.pdf
Hestenes_1981_Reply to ’’Comment on ’Spin and uncertainty in the interpretation of quantum mechanics’ ’’.pdf
Hestenes_1979_Spin and uncertainty in the interpretation of quantum mechanics.pdf
Hestenes_1975_Observables, operators, and complex numbers in the Dirac theory.pdf
Hestenes_1974_Proper particle mechanics.pdf
Hestenes_1974_Proper dynamics of a rigid point particle.pdf
Hestenes_1973_Local observables in the Dirac theory.pdf
Hestenes_1968_Multivector functions.pdf
Hestenes_1968_Multivector calculus.pdf
Hestenes_1967_Spinor Fields as Distortions of Space-Time.pdf
Hestenes_1967_Spin and Isospin.pdf
Hestenes_1967_Real Spinor Fields.pdf
Hestenes, Ziegler_1991_Projective geometry with Clifford algebra.pdf
Hestenes, Holt_2007_Crystallographic space groups in geometric algebra.pdf

>>11986295
>>11986310
>Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics
I believe I've seen most, if not all, of the limited number of books on geometric algebra and its calculus, and this book (the first) remains the most comprehensive, but not the most comprehensible. A tremendous amount of insightful thinking went into this book, and a diligent and thoughtful reader will learn a lot. But the book cannot be considered the place to begin one's study of geometric algebra. Understanding what Hestenes and Sobczyk are saying and trying to see its significance is maddening. There's a quotation, from whom I've forgotten, to the effect that "Pioneering work is clumsy." _5 stars_ for the content, but only_2 stars_ for readability.

There are several other books which one should go to first. Hestenes' own "New Foundations for Classical Mechanics", written on about a junior or senior level, is much more clearly written than was his older "Clifford Algebra to Geometric Calculus" (graduate level). Easiest to read by far (sophomore level) and offering lots of contact with traditional courses on linear algebra and vector calculus are Alan Macdonald's inexpensive "Linear and Geometric Algebra" and its follow-up, "Vector and Geometric Calculus"; Macdonald's books also introduce one to use of free geometric algebra software. Clear writing, geometric insight, and lots of informative figures characterize "Geometric Algebra for Computer Scientists", by Dorst, Fontijne, and Mann; be warned that the book has an extremely lengthy errata list at its website, so the book should only be read with a printout nearby of that list. Students wanting to see applications to physics other than classical mechanics should consider "Geometric Algebra for Physicists" (graduate level), by Doran and Lasenby.

>>11986339
As noted above, Dorst, Fontijne, & Mann maintain a website where errata for their book may be found, and so also do Doran & Lasenby and Macdonald for their books. That's a big help for the puzzled student who can't figure out why something doesn't make sense. But if there's an errata list for "Clifford Algebra to Geometric Calculus", I haven't been able to Google it.

A cautionary note when comparing the various books named: Hestenes and Sobczyk use a kludgily defined "inner product" between elements of the geometric algebra. Hestenes was the pioneer in the field, so many subsequent writers have adopted his inner product, most notably Doran and Lasenby. But Dorst has argued convincingly for replacement of the Hestenes inner product by more-cleanly-defined, easier-to-use, and easier-to-interpret, left and right "contraction products". Macdonald uses the left contraction product exclusively, although he calls it the inner product and denotes it by a centered dot instead of by Dorst's right floor bracket.

Why are pajeets so fucking based?

>regularity and existence of solutions to elliptic equations

>>11977958
zoomers mad at the responses. think they can watch youtube videos on every topic

What kind of math examines these shapes?

>>11986799
Haha top one looks like benis :DDDDDD
More seriously those looks like epicycloids.

>>11986816
It seems like if I took a circle and twisted it, I’d get these shapes
There’s a lot of ways to draw them if you insist on using a continuous line
Epicycloids was a nice wiki read, and it does work to produce these, but I’m interested in perhaps mapping points on a circular line to these points once the line has been manipulated/transformed

>when u read kants critique of pure reason and then watch modern day “scientists” try to prove kants transcendental idealism with their half baked conspiracy theories
It’s like you never made it to Hegel

>>11975647
but that is true in any case

>>11978076
always have been, always will be

>>11986975
Polar functions.

>>11987495

I just completed my undergrad but never had a chance to take topology. Can anyone recommend a good beginners book or a decent online university lecture series?

>>11987671
https://www.amazon.com/Introduction-Topology-Third-Dover-Mathematics/dp/0486663523/ref=sr_1_3?crid=2INHVP1D6X22&dchild=1&keywords=topology+dover&qid=1596917794&sprefix=topology+dov%2Caps%2C186&sr=8-3

>>11987671
Everyone recommends Munkres, so go with that. I would also recommend Janich and maybe Sidney as a supplement. Munkres himself has a schedule, lecture, and problem sets here: https://ocw.mit.edu/courses/mathematics/18-901-introduction-to-topology-fall-2004/index.htm

>>11987671

>never had a chance to take topology.

yeah you did, you should have enrolled into an elective insread of wasting time with having a social life

>>11987671
If you took analysis, you know all the topology you need to know.

>>11987731
I doubt they teach (co)homology in analysis

>>11987750
Exactly.

>>11987687
My university doesn't have topology as an option so I literally never had the chance

>>11987750
You don't need cohomology if you aren't staying in academia.

>>11987785
You literally did.

>>11987789
you don't need topology if you aren't staying in academia

>>11987802
But you do.

>>11986799
Those are algebraic curves, so you'll want algebraic geometry.

>>11987800
Really? How would I have done that?

What are some unusual group operations?

>>11987822
Commutative exponentiation

>>11987818
By taking advantage of the plethora of resources on the topic available for free at your fingertips.

>>11987830
So you mean... exactly what I'm doing right now?

>>11987834
Yes, so you admit you had a chance. Glad the discussion is done.

>>11987840
Cheers, thanks for the insight.

>>11987830
holy fucking shit stop embarrassing yourself

Did I forget anything?

>>11987848
What's that CA book in the bottom left?

>>11987848
Oh, right, Basic Maths.
I've got some other candidates in my head, but I don't think any of them make the cut. I could use a second opinion:
>Atiyah's Commutative Algebra
>Eisenbud
>Additive Combinatorics
>Conway's CA
>Kodaira's CA
>Spivak's Calc
>Apostol's Calc
>Hatcher's Algebraic Topology
>Jech
>>11987859
Alfohrs.

>>11987873
Pinter's 'Abstract Algebra'
Velleman's 'How to Prove It'

>>11987888
Proof books are a meme.

>>11987888
Done.

>>11987848
>>11987873
>>11987930
Don't see Grothendieck
Don't see Bourbaki

>>11983055
Its the absute value of the determinant of the derivative of the mapping of the variable. Intuition is that determinant measures volume. If you want to get familiar with this study multivariable calc, specifically chage of variables formula (integration)

>>11975242
>Reminder to work with physicists edition
I want to kill whoever faggot keeps replying this shit like a broken recorder. kill yourself

>>11988092
for good reason

>>11983289
yes

>>11988138
t. Smooth brain

>>11987930
Nothing for the Foote fetishists?

>>11987930
Kohn's 'Measure Theory' is also mentioned a lot.

>>11988092
I don't think I've ever seen anyone here seriously recommend Bourbaki other than as a reference book.
The EGAs are recommended every now and then, yes. But I'm pretty sure that every single introductory algebraic geometry has been recommended here at some point.
>>11988320
Second line, second entry.
>>11988354
Fair enough. Adding it in, brb.
Obviously the kino cover, miss me with that new edition shit.

>>11988364
>Second line, second entry.
Ah I appear to have become blind

why is math general exuding such harmful auras today?

reinigen
reinigen
reinigen
reinigen

>>11988364
In case anyone's concerned, I take 30 secs to add it in, pnggauntlet just takes long to optimize the image.

>>11988092
No one reads EGA anymore, thank God. Some brave autists already sacrificed their lifetimes picking out the useful parts and translating it to something readable.

>>11988320
Based and Footefetishpilled

>>11988555
Checked.

>>11975762
When you're dealing with equations of this sort in algebra the key is to treat all of it as addition by default. If you see a subtraction sign in front of one of the variables or coefficients, then don't think of it as subtracting a positive number, think of it as adding a negative number.

>>11987873
>>Spivak's Calc
>>Apostol's Calc
I remember seeing those way more often than C&J

>>11988805
Quick rundown on Apostol?

>>11988885
issa book

>>11988805
I don't really recall seeing them actually recommended here, but I'll trust you.

>>11988905
This is actually a pretty good chart, but it's lacking set theory.

>>11988885
I'd rather let another anon do that since I only ever worked with Vol I, but:
Rather dry at parts (I kinda like it that way), pretty rigurous, some historical bits throughout, some applications (shudder), a nice introduction to Linear Algebra which makes me wish he had done a full book on it, a lot of exercises, some of them neat, some rather laborious. It's a good book imo.
No mention on physical quality, because I don't have the hardbacks.
>>11988905
The thing is a lot of people spend a lot of time minmaxing their introduction books, so there's a lof of talk about Calc books, LA, and basic stuff. So watch out that the chart doesn't become cluttered with that shit. You already got 3 books on Calc (Intro to Analysis almost).
That said, yeah I used to see them a lot.

>>11988915

>>11988915
The chart contains books that are often recommended.
No one here recommends some set theory book, so there is no set theory on the list.

>>11985930
should i be able to prove any theorem from the book at will after learning it?

>>11988973
But set theory is a critical part of math and used in nearly every branch.

>>11988978
Yes, but that's irrelevant.
Shill for some set theory book three times here while pretending to be different people and I'll add it in in the morning.

If aleph_1 is the cardinality of R, then is aleph_2 the set of all subsets of R, and alpeh_3 the set of all subsets of subsets and so on ad infinitum?

>>11988982
Admittedly, I'm not really familiar with any good set theory texts, which was partially why I was asking.

>>11989004
Jech

Morikawa's hundred year unsolved problem was just solved, lads.
https://arxiv.org/abs/2008.00922

>>11989017
coolio

>>11988986
No that is the generalized continuum hypothesis.
>>11989012
Reference book

>>11989084
nah, jech intro

>>11989017
>https://arxiv.org/abs/2008.00922
what do you think they used for the pictures of circles

>>11975618
It's gayer than you ever thought.

what the fuck

>>11989649
Nothing surprising about this.

>>11988978
It is not. It is a language that is essential to know, but most people don’t need to know more set theory than what fits in a chapter of one of these books. Most mathematicians have no idea what set theorists are up to.
Same thing with category theory. It is a very powerful framework/pov, but the category theory that category theorists study is irrelevant to most people’s needs

>>11977692
If you revisit a faulty node can it take you to a different adjacent node than when you first visited it? If so then the score is misleading. Additionally it would be even more problematic if the probability itself can change upon revisiting.

I'm going back to school for applied math but it's been a while since I did some senior year math. What's a good little crash course I can do to refresh myself?

Either you're born trans or you're not.
Doesn't it follow that 50% of positions should be held by trans people?
I found this but even in ACT there's less than 50% trans as of now. How to solve this?

https://youtu.be/dn_whW1DIws

>>11988969
qt

>>11988978
>used in nearly every branch
not really
As in, the set theory that is used by other fields it the set theory that was established long before, say, 1925 when von Neumann tried to reformulate it.

>>11988969
qt

>>11988978
>set theory is used in nearly every branch
Not really, though.

As in, the set theory that is used by other fields it the set theory that was established long before, say, the twenties when von Neumann tried to reformulate it.

Can somebody explain motives to me?

>>11990339
New thread

bonp