/sci/ - Science & Math

>>9733160
1+2+3+4+... = -1/1/

>>9733160
1+2+3+4+... = -1/12

>>9733160
Most efficient way to calculate the amount of intersecting content of two high dimensional polytopes? Specifically an n-simplex and a hypercube.

Numerical solutions are also fair game.

Hi guys, what is a good beginners book on differential equations. I am not a mathematician but I need the book for my course on derivatives pricing. Thanks.

>>9733332
evans

>>9733266
>Numerical solutions
Refer to >>>/sci/engi/.
>>9733332
>differential equations
Refer to the above message.

>>9733160
Sick bro.

>>9733266
Ch*ck*d.

>>9733160
People who find this shit interesting must have mind like insects, truly autistic in the literal sense of the word.

>>9733552
*tips fedora*

>>9733332
Arnold

>>9733912
Neurotypicals are a waste product of humanity. They should be annihilated and replaced by Pureblood Autismos.

>>9733160
Hello /mg/

I haven't done any real math in about 3 years. I want to review Calculus and Linear Algebra for a different topic that I want to gain experience in.

What, in you guys' opinions, is the MOST effective way to quickly review Calculus and Linear Algebra, assuming I did well in school when taking those classes many years ago? Textbooks, online tutorials, or video lectures?

>>9734797
Start from the first chapter of a textbook and attempt a few problems. If you were able to complete and understand most of them, move on. If not, read the chapter. Repeat for a few chapters each day.

>>9734797
Just try to learn whatever new topic you're doing. When you see something from calculus you don't remember, go back and review it.
Re-doing an entire course is a waste of a couple weeks.

>>9734866
>>9734885
Thanks guys, I'll try out both of these approaches

When doing an x if and only if y proof, does the if direction correspond to assuming x and proving y, or assuming y and proving x?

>>9735165
It means "x if y and only if y", which can be divided into "if y, then x" and "x only if y". The latter means "if x, then y", so the "if" is y -> x.

>>9733160
>need n-dimensional vision to visualize n-valent graphs for n > 3
Talk about useless constructions.

What are some good books on differential forms and integration on manifolds? I've got Spivak's Calculus on Manifolds and Milnor's Topology from the differentiable viewpoint to get some insight into manifolds, but I don't think they (especially Spivak) explain it well enough

>>9733171
i know u r just meming but this https://en.wikipedia.org/wiki/Abel%27s_summation_formula

>>9733160
Recommendations on books about manifolds from topological pov?
Please don't mention Spivak, i can hardly understand a single sentence in that book.

>>9735741
Do carmo's differential forms and applications
>>9735799
milnor topology from the differentiable viewpoint is GOAT (bit short though)

>>9735741
Do Carmo
> no gun at the back
You could also try Lee's Introduction to smo

http://www.topologywithouttears.net/topbook.pdf
is this a meme or is it good for learning topology?

>>9736219
>is this a meme or is it good for learning topology?
Why don't you read it and find out?

Is there any benefit to learning physics alongside math, even if you're only interested in pure mathematics?

>>9736306
Try asking in a physics thread. >>>/sci/pg/ or >>>/sci/sqt/.

>>9736306
>Is there any benefit to learning physics alongside math, even if you're only interested in pure mathematics?
Yes, they complement each other. Try some string theory.

Can homotopy type theory be used to do the same things as model theory?

>>9736306
Actually, math can really be seen as a branch of physics. More specifically, it can be seen as a branch of empirical ATQFT. I recommend Landau-Lifshitz - An Introduction to Quantum Field Theory as a basic introduction and Sakurai - Algebraic Geometry of Black Holes to get a better physical feel for the deeper empiricism involved. After that you can easily read Shankar, Lurie - An Introdution to Non-Perturbative Algebraic Quantum Field Theories which contains a deep empirical proof of the cobordism hypothesis using nothing but our innate child-like physical intuitions. While these so called "mathematicians" still can't provide a proof of their homotopy "hypothesis". It's laughable.
>>9736315
Modern string theory is just a branch of empirical TQFT anyway. It's better to start with that so he can get to the physical proof of the cobordism hypothesis as soon as possible.

>>9736337
>homotopy
But the homotopy "hypothesis" still hasn't been proved. As my adviser likes to say: "Of what use is "homotopy" if it can't even prove the homotopy "hypothesis?"".

>>9736351
I don't really know the first thing about HoTT actually. I just read that it's more closer to the foundations of "actual" mathematics than Set Theory, so I wanted to know if it could also prove things like the completeness theorems and so on

>>9736361
>completeness theorems
And how would that be related to actual mathematics?

>>9736383
define "mathematics"

>>9736361
Empirically speaking, I would not trust anything connected to so-called "homotopy" as the homotopy "hypothesis" hasn't been proved. So it is empirically meaningless to speak of "homotopy" and everything defined in terms of it. I am currently working on purely empirical foundations as a small part of my thesis on TQFT. There I use the cobordism hypothesis in a crucial way to show that every internally definable property is invariant under empirical equivalence.
>>9736392
Me and my adviser defined mathematics as a particular branch of AQFT and string theory. This is the only currently known definition which is empirically sensible.

>>9736403
>Me and my adviser defined mathematics as a particular branch of AQFT and string theory.
Which particular branch?

>>9736404
The branch uniquely defined by its study of AQFT in the sense of Sakurai, Kontsevich and Lurie.

Why are old books so much comfier than modern ones?

>>9736219
Page 172 for compactness, page 200 for Tychonoff for *finite products*. And they put Reader's Compliments in the Introduction.
It is meme and it is very slow.

>>9736631
>compactness
>finite products
As opposed to?

>>9736679
As opposed to any normal book on topology it takes a lot of pages to get to those simple facts. It makes you cry.

>>9736716
His method of teaching gives you the concrete intuition needed for true understanding instead of jumping straight into unmotivated formalism and walls of notation/abstraction. I guess that's a bad thing for those people who like to hide behind walls of abstraction to prevent deeper empirical understand of the subjects involved. Some would even argue that he introduces these incomprehensibly abstract concepts too early. Also, I am willing to bet that you can't even draw a picture of a cofibration. We usually discourage this kind of learning in the TQFT community and obviously for good reasons. I guess the "mathematicians" still have a lot of catching up to do. This is especially clear if you look at the whole homotopy "hypothesis" situation.
>>9736729
I was recently reading his correspondence with Grothendieck regarding the whole homotopy "hypothesis" fiasco. He had pretty profound views on the subject, as expected of a true empirical mind. Maybe we can hope that "mathematicians" will catch up if we give them about 50-60 years. Maybe then they would actually be able to present a proper proof of this homotopy "hypothesis". Me and my adviser think not, because the cobordism hypothesis is the cornerstone of something concrete (i.e. TQFT, string theory) while this homotopy "hypothesis" is the cornerstone of absolute algebraic wank.
Basically, I suggest reading Von Neumann - Mathematical Foudnations of Quantum Mechanics to develop the necessary prerequisites for looking into these things at a deeper level. Then read Lurie's and Sakurai's elementary proof of the cobordism hypothesis, it should be somewhere on arxiv. If you have any questions about the physical intuitions involved in understanding the proof, feel free to post them here. You need to develop proper intuitions so you don't mistakenly believe that the results they obtain are "spooky" and "unphysical".

>>9736762
>concrete intuition
>jumping straight into unmotivated formalism
Most books do give concrete examples. I don't see this hard formalism. Maybe to you even sets are an obscure formalism (in this case, thumbs up).
>I am willing to bet that you can't even draw a picture of a cofibration
>the TQFT community
What are you even going on? Trolling?
>I guess the "mathematicians" still have a lot of catching up to do. This is especially clear if you look at the whole homotopy "hypothesis" situation.
Pic related.

>>9736490
I don't like when I have to read non-modern books. Old fonts are incredibly shit compared to properly implemented LaTeX and the trend of actually explaining your fucking math is a pretty modern one. Anything written before like the 80s almost always takes a very formal didactic approach, which is painful to learn from.

I've been studying Galois theory, and things haven't really been clicking for me at all, I feel like I'm missing the "big picture" or something. Everything feels pretty unmotivated. Broadly, I don't really know why do we care about field extensions. Some of the more specific definitions are difficult for me to swallow, not technically, but really I don't feel like I know what exactly they're supposed to capture. For example, in ring theory, being a Euclidean domain means you can do division like in the integers, being a UFD means you have something analogous to prime factorization, being a PID means that ideals end up always having a simple form. I don't feel I have this sort of understanding for splitting fields, normal extensions, the automorphism group, etc.

If anyone has some thoughts on this, it would be appreciated. I'd also be open to book suggestions. I've been using Dummit and Foote, which seems ok but not great on this topic to me at least.

>>9735741
>>9735799
See Atiyah - Geometry and Physics of Knots. It's a great read and will prepare you for TQFT.

>>9736955
>Maybe to you even sets are an obscure formalism
They are and most Mathematical Physicists would agree with me on this. Sets are a clunky way of describing our world, a world which is inherently invariant under physical equivalence, while sets are not. Physicists of the older generation are more likely to reject fancy mathematical constructs, but I'm sure this is about to change. So sets are merely seen as an obscure formalism with no real physical meaning backing it. You can sit around and compare large cardinals all day, but that would have no empirical meaning and so we choose to disregard such activities. I would suggest you do the same if you care about pesky little things like meaning and coherence.
>What are you even going on?
I am talking about the community of people who have dedicated their lives to studying TQFT, i.e. Mathematical Physicists. And I am talking about people who advocate for inane formalism and notation usually being unable to even draw a picture of a cofibration or a weak equivalence. Which just shows how the unnecessary abstraction has a tendency to limit creative and spatial thinking which is so inherent to human beings. Surely even you and other "mathematicians" must see that as a negative. It's never too late to develop your physical intuitions so I see no reason for this kind of behavior from you guys.
>Pic related.
I'm not stupid enough to believe that the homotopy "hypothesis" has any chance of being true. Clearly you disagree, so where are the proofs of this homotopy "hypothesis"? Answer: there are none, because as I have said previously, cobordism hypothesis is the cornerstone of something concrete while this homotopy "hypothesis" is the cornerstone of absolute abstract algebraic wank.

>>9737002
>the trend of actually explaining your fucking math is a pretty modern one
Read better old books.

>>9737003
Your situation is pretty common among "mathematicians". It's not surprising that a subject which tries really hard to push out any semblance of physical intuition from entering it is facing such difficulties with motivation. I would recommend stepping back from the insane level of abstraction and gaining a basic fundamental understanding of the use of fields in physics (this is useless formally and only useful for gaining intuition). That way you can develop your base physical intuitions so that you are better prepared to tackle stuff like QFT and TQFT in the future. Then and only then should you return to your walls of notations and unnecessary abstractions. Recommended reading - Jackson - Classical Electrodynamics, Landau-Lifshitz - Classical Mechanics.

>>9737005
>sets are merely seen as an obscure formalism with no real physical meaning
>I would suggest you do the same
No thanks.
>I am talking about the community of people who have dedicated their lives to studying TQFT, i.e. Mathematical Physicists.
I mean, what was the relevance to the previous post? You are just forcing your 'muh cobordism hypo much better than homotopy hypo' which has nothing to do with the post.
>Clearly you disagree, so where are the proofs of this homotopy "hypothesis"
Actually I don't know this stuff and I'm not interested.
>>9737016
This trolling is going beyond control, calm down now.

>>9737003
It's because those definitions, normal separable, are cooked up so the fundamental theorem of Galois theory goes through. The commutative algebra stuff you mentioned arise naturally since those are generalizations of properties of integers. If you have access to a library there is a UTX book on Galois theory that is pretty cute. Otherwise you just have to work problems. The best big picture I can think of would be to understand the Galois correspondence as it pertains to other fields, like covering spaces (alg top/geo), definablably closed saturated models (model theory), among others.

>>9737062
You mean this book? If so I'll give it a look, thanks for the input!

>>9737062
>definablably closed saturated models (model theory)
Do you have a reference?

>>9737157
Poizat, Hodges, Marker

>>9737157
https://arxiv.org/abs/0909.4340

Guys there are two classes named Abstract Vector Spaces and Complex Variables that are both 300 level courses. What are they about? I'm definitely taking real analysis next semester but I don't know what those two classes are about. Does your school have them?

>>9737378
>What are they about?
Does your school not offer course descriptions?

>>9737378
>Abstract Vector Spaces
As opposed to what?

>>9737378
Complex variables is like calculus, but using complex numbers instead of real. Depending on your department it might be a bad idea to take this before real analysis, maybe not.
I've never heard of anyone offering an "abstract vector spaces" course but my guess is that what your university calls "linear algebra" is a crappy course that only deals with R^n and this course does linear algebra properly.

>>9737395
>>9737378
Abstract Vector Spaces Description: Axiomatic treatment of vector spaces, inner product spaces, minimal polynomials, Cayley Hamilton theorem, Jordan form, and bilinear forms.
Complex Variables: Analytic functions, elementary functions, integrals, power series, residues, and conformal mapping.

I just don't understand what to expect.

>>9737403
My school has upper div real analysis as a prereq for complex analysis, which is a 5000 level class.

>>9737409
There are two types of complex analysis courses, those with reals as a prereq and those without. Those without are could be called complex calculus. Since he said it is a 300 level class I'm guessing it is one without.

What's up nerds. Please forgive me selfishly using /mg/ for a recommendation.

My calculus teacher is specific about the kinds of calculators he will allow on exams. Scientific, non-programmable, no derivative/integral functionality or equation solving.

I survived this semester well enough with only a TI-30, but trying to build both sides of a multi-term fraction in an intelligible manner on a 2-line calculator is piss.

The TI-30XS seems to be a nice four-line calculator that won't calculate a derivative, but it lacks complex number support. And any calculator I can find with complex number support also includes higher functionality that I'm not allowed. So I guess what I'm asking is: Do you know of any calculator that can handle imaginary numbers, but is still more or less crippled in terms of basic calculus functions?

(If you make the apples, skip the tarragon. Ruined it.)

>>9737540
I don't know your situation so I can't guarantee but I'm going to suggest you probably won't ever need to do complex arithmetic on your calculator. All the basic operations are just chained operations on real numbers and most problems you'll be given are simple enough to do by hand anyway.
The TI-30XS is a very nice calculator. Second-nicest I've ever used next to the casio fx-991 but that can integrate and solve systems of equations.

>>9737556
So I've heard. Well, thanks. I'm sure I won't need the functionality, but I thought it might be foolish to not reach for as much as I can.

>>9737556

>>9737540
lol fag. Learn to do complex airthmetic by hand and you won't need some fag calculator. Can't believe some of you people.

>>9737630
>fag
Why the homophobia?

>>9737630
I like to work backwards from an answer to reproduce the given problem, when possible, to ensure no mistakes were made. While I can do this by hand with little margin of error, a calculator does it faster with no margin of error. Especially handy with complex roots etc

>>9733332
Tenenbaum and Pollard.

help a brainlet out here
how am I supposed to solve this
am I supposed to just assume they all pass through (0,0)?
or is that irrelevant and I'm just that much of a brainlet

>>9738183
Please post this in the /sqt/ instead
Thanks

>>9738186
fine :(

>>9737633
It's not a phobia. There is no fear. It is pure disgust. I'm sad that even my country has legalized sodomite marriage.

>>9738183
Just use the dot product.
a•b=0
Then solve for t
Retard

ok listen up faggots
answer this if you are smart:
let M := {U element P(R): U is countable}
let c: M -> M
c(x):= {cluster points of x}
Does there exist an x element M such that for all k, c^k(x) is infinite?

>>9738241
>{cluster points of x}
What do you mean by that exactly?

>>9738254
p is cluster point of x
<=> for all e > 0 exist infinitely many q element (B_q(p) cut with x)
<=> for all e > 0 exist infinitely many q element x such that |p - q| < e
<=> for any bijection a_n: N -> x there exists a subsequence of a_n which converges to p.

Is there a solutions manual for Bishop/Goldberg Tensor Analysis on Manifolds?

>>9738241
Not doing your homework. Post it in /sqt/

>>9738278
It's just something genuinely curious about it but I cant prove it myself.
I suspect that such a set would have to be uncountable but I cant prove it.

>>9738296
Hint: consider the set of rationals

>>9738305
ah shit well that was easy
Still I think thats pretty cool.
You can have a sequence where the cluster points of cluster points of cluster points... are still infinitely many

√(x^2) = |x| while (√x)^2 = x;
so does
x^(a/b) = bth√(x^a) or (bth√x)^a ?

>>9737003
Most "modern" expositions of Galois theory are pretty contrived. Like the other poster said, everything is just cooked up so the main theorem holds. The main point is usually just to prepare you for seeing other deeper Galois-like correspondences later on. Think of it as like Precalculus. There are a few books out there that explain the historical context, but ever since Grothendieck got his grubby hands on the subject, all the life and wonder has been sucked out of it, leaving only a dessicated shell of an adjoint functor.

How should I prepare for the gre?

what tf is a manifold

>>9737003
>I've been studying Galois theory, and things haven't really been clicking for me at all, I feel like I'm missing the "big picture" or something. Everything feels pretty unmotivated. Broadly, I don't really know why do we care about field extensions. Some of the more specific definitions are difficult for me to swallow, not technically, but really I don't feel like I know what exactly they're supposed to capture. For example, in ring theory, being a Euclidean domain means you can do division like in the integers, being a UFD means you have something analogous to prime factorization, being a PID means that ideals end up always having a simple form. I don't feel I have this sort of understanding for splitting fields, normal extensions, the automorphism group, etc.
>If anyone has some thoughts on this, it would be appreciated. I'd also be open to book suggestions. I've been using Dummit and Foote, which seems ok but not great on this topic to me at least.

>>9738610
>what tf is a manifold
https://simple.wikipedia.org/wiki/Manifold

>>9738305
Proper class of rationals*

>>9738277
Pls.

so much math to learn, so little time. how do I budget my time appropriately, /mg/ ?

>>9739370
Use math.

>>9739388
How would he use math for something non-mathematical?

>>9739395
>he
excuse me?

>>9738277
>Is there a solutions manual for Bishop/Goldberg Tensor Analysis on Manifolds?
It exists in your mind.

>>9739489
"No"

Is there an english version of Grothendieck's EGA?
What about FGA and SGA?

test

>>9739704
>EGA, SGA
No. Just learn French, mathematical French isn't hard at all if you know English.
>FGA
See FGA Explained - https://ncatlab.org/nlab/show/FGA+explained

>>9739704
>>9739730
>Just learn French

This. It is a lot easier than you would think.

Literally anyone that does math without listening to kpop is a pleb and shouln't be allowed to post here

Literally anyone listens to kpop is a pleb and shouldn't be allowed to post here

>>9740008
fight me you ungifted tard

>>9739956
That just makes you a second class citizen. The particians of these threads listen to J-pop while doing math.

>>9740028
I can't think of a single metric where J-pop is better than k-pop, K-pop is the ultimate tool to focus on my problem sets

>>9740056
It might be fine if you're still at a level where you think of doing "problem sets" as being worthy of your time.

>>9739430
It

>>9739956
the only purpose of k-pop is for you to stroke your weewee to asian girls in short shorts
listening to k-pop is like ordering a hamburger and only eating the bun

>>9740085
I bet you ain't even out of HS

>>9740213
He's probably a drop out.

>>9740213
They have "problem sets" in HS so your post doesn't make sense.

>>9739730
>Just learn french
You're right, i'm trying, but in the meantime i wanted to read it

>>9735803
>>9735834
Carmo reads like ass

>>9737396
Vector calculus in [math] \mathbb{R}^n[/math]?

>>9740352
You can learn by reading it. You'd be surprised at how much you can actually understand without any real knowledge of the language. Also see http://people.brandeis.edu/~jbellaic/French.pdf for some math vocabulary and so on.

Woah. He found a flaw in the foundations of maths. Phi isn't (1+root(5))/2 it's (root(5)+1)/2. How will sci recover?

Hey bros, does anyone know any good literature on the four color theorem and graph coloring, I need to write a paper for my algorithm complexity class.

>>9738617
>Don't understand the motivation behind basic field theory ? Here, take this book that rushes through the basics in 10 pages then follows it with 300 pages worth of content 5 layers of abstraction deeper

>>9741599
>>Don't understand the motivation behind basic field theory ? Here, take this book that rushes through the basics in 10 pages then follows it with 300 pages worth of content 5 layers of abstraction deeper
Who are you quoting?

>>9740471
>this man does addition without commutativity and basically you are fucking stupid
>How? Just click here!

If I have two means of independent distributions, lets say E[D] (distance) and E[T] (time), can I calculate the expectation of velocity as simply the division of those two?

>>9741648
Nevermind I'm a brainlet

All math is based on induction.

>>9740471
How many times do you reckon this retard has told (bragged) others about his love of math and science?

>>9734797
if you need a handle on the mechanical aspects, just run through some problems, if you need a conceptual refresher, hit up 3blue1brown's youtube series on calculus and linear algebra

>>9741826
is induction based on induction

>>9742602
Suppose induction works for all k<n. Then, by induction, it works for k=n, so we can induct inductively.

Where can I get an overview of what modern geometry is all about? Something like a book summarizing it would be great!
>inb4 arxiv

>>9742674
Your physical intuitions should tell you everything you need to know.

>>9742662
><
So induction working assumes the axiom of choice?

>>9742674
coxeter geometry revisited is a classic

>>9742674
These days it is mostly about finding good models of TQFTs in nature (of which there are plenty since TQFT is a very empirical field). Usually these TQFTs are modeled by locally existent black holes, these are black holes which might not "fully" exist in our universe, but they exist infinitesimal-locally in various (non-classical) topologies on some chosen exotic space-time. A good reference for this kind of stuff is Shankar - Principles of Quantum Mechanics and the masterpiece Sakurai - A Modern Approach to AQFT. Then you should take a look at the deep paper titled "The cobordism hypothesis" - https://arxiv.org/abs/1705.02240.. Many people consider it to be the cornerstone of empirical TQFT since it uses concrete physical intuitions in quite deep and subtle ways, so you need to master the art of thinking physically before even attempting to read it. You can develop these intuitions by thinking extensively about empirical problems. So exercises you might find in Sakurai - An Approach to String Theory via AQFT should do the job. I have to warn you though: if you are a "mathematician", then your brain isn't well-suited to empirical thinking so it would be very hard for you (but nevertheless still possible). My adviser was actually a "mathematician" himself before he discovered Mathematical Physics and TQFT.

>>9742867
From a cursory glance, I think the book does a very poor job of conveying the TQFT-perspective on the subject. And as most Mathematical Physicists and even some "mathematicians" these days know, the TQFT-perspective is incredibly valuable to the field as shown by works of Witten, Sakurai and Kontsevich.

>>9742981
This stopped being funny months ago.

>>9742986
I don't see how this is supposed to be funny. If you dislike Mathematical Physics, then what are you even doing on a science board?

>>9742981
>putting this much effort into "trolling"

>>9743011
It's a sapiosexual biologist or one of the people he thinks are his online friends. Just ignore him.

>>9742981
My question is more like, what are the things that people do in differential geometry/topology that are not physics?

Why wasn't I taught category theory as a child?

>>9743204
Why weren't you aborted before birth?

>>9743013
>It's a sapiosexual biologist or one of the people he thinks are his online friends.
I'm not a "he".

>>9743079
>what are the things that people do in differential geometry/topology that are not physics?
https://arxiv.org/list/math.DG/recent

>>9742674
There is a very cool book by Marcel Berger called Geometry Revealed which is basically a not-too-technical survey of a bunch of different faces of modern geometry. Sounds like it fits what you want, although it's like 800 pages.

How would one go about computing this integral? It appeared on my exam today as the second part of a question. The first part was to state Cauchy's formula, but since the poles of the function lie inside the disc I really couldn't see how you'd make use of it.

Given that this was only worth 3 marks out of 100 and on the intentionally really easy question on the paper, I can't imagine there's too much to do

>>9743487
I forgot to mention, B_2(0) is a disc of radius 2 centred at the origin.

>>9743487
[math]z \in \partial B_2(0) \Leftrightarrow z = 2e^{i2\pi\theta}, \theta\in [0, 1][/math]

How can I prove the Monotone Convergence Theorem assuming the Bolzano-Weierstrass property? (in real numbers)

damn I wish I weren't dumb ...

>>9743487
Durham lol?

also, just find the partial fraction expansion and use cauchy formula on each integrand

>>9743511
they're both true statements, so they're logically equivalent

QED

Can someone please explain how to use quaternions like I'm five?

>>9743860
>Can someone please explain how to use quaternions like I'm five?
Do you understand complex numbers?

>>9743867
Yes.

>>9743871
What do you not understand about quaternions?

>spent 5-6 hours on pages 5-7 of apostol's calculus volume I trying to understand one proof
>finally worked through each step and didn't even understand the proof
>practice problems are still out of reach
I never asked to be born retarded.

>>9743874
I was just looking for a general overview of how to apply them to rotate a set of coordinates. Mainly interested in the intuition behind them.

>>9733160
Can a coherent sheaf with no global sections have non-trivial higher cohomology?

>>9743750
No, Warwick. This year's paper was incredibly hard.

Also, Cauchy doesn't apply because 1/(z+/-1) isn't analytic in disc, no?

Pure mathsss is the ultimate incel subject.

https://arxiv.org/pdf/1805.05501
>Revisiting the de Rham-Witt complex
>Bhargav Bhatt, Jacob Lurie, Akhil Mathew
>(Submitted on 15 May 2018)
>The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect fields of characteristic p.
>We introduce a category of cochain complexes equipped with an endomorphism F (of underlying graded abelian groups) satisfying dF=pFd, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham-Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Berthelot-Ogus operator Lηp on the p-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison in AΩ-cohomology theory.

join the cool math discord server
https://discord.me/math

>>9743933
You are mixing up Cauchy's theorem and the Cauchy integral formula. The latter applies after partial fractions.

>>9744036
Mathematics really needs to stop this fucking meme of just naming objects after all the people who invented them.

Am i retarded? wtf is this? What happened to A=BH? Is this some common core bullshit?

>>9744060
Holy shit. It does work. So there was a reason for us stating it after all...

Now if only I could have known that 12 hours ago...

What am I missing here?

>>9744089
what's the base and what's the height? (rhetorical question for your learning benefit)
(you are a retard btw)

if you don't want to think about it, i'll explain it later in the post.

.

imagine taking each of those triangles you see in that rhombus, and constructing parallelograms out of them. what happens?
you get 2 rectangles with base p/2 and height q/2.
so that means you have a total area of 2 * (p/2) * (q/2) = p q / 2

>>9744115
>What am I missing here?
What have you tried?

>>9744121
Various things I’m to embarrassed to show. Truthfully I’m just lost

>>9744121
6x^3-30x^2= L x 2x(x-5)
6x^3-30x^2= L x 2x^2-10x
(6x^3-30x^2)/2x^2-10x=L

>>9744115
What's the formula for area of rectangle?

A=LxW. So you have the area and width. Write those in to the equation, and try to work out the length from that.

>>9744142
>>9744138
That’s what I did here but I’m stuck

>>9744147
Oh, didn't realise that was you. I'm not too hot on my algebra but try google "dividing polynomials" and that should give you what you are looking for.

Any recommendations on an introductory differential geometry book for someone who just finished multivariable calc and linear algebra?

>>9744117
Brainlet here again. Is there a way to convert base and height measurements into the diagonals? Even my cheat sheet study guide thing doesn't mention this PQ/2

>>9744147
try multiplying by (2x^2+10x)/(2x^2+10x) and then simplifying

>>9744253
Some anon helped. Thnx though

>>9744240
Do Carmo's "Differential Geometry of Curves and Surfaces" would be appropriate for your level. Shifrin has notes on diff geo that are also pretty good (google them). Both have plenty of examples - it is essential you work through them, it helps develop your geometric intuition

>>9744113
lad... this year's paper wasn't hard, you just didn't know the material

>>9743892
No help as usual. Excellent work /mg/, keep it up.

> 3/4 way through my stat BS
> all of the non-"do algebra plox" posts are still gibberish to me

Am I just retarded?

>>9733160
I studied fractal geometry in college. It was cool. My professor had brain cancer and he was a student of Mandelbrot's.

https://users.math.yale.edu/public_html/People/frame/Fractals/

>AMA fractal geometry

>>9744724
>>AMA
you have to go back

>>9744727
I donno.. fractal geometry is cool I thought some nerds might have a question or two

I'm looking for an introductory resource on graph grammars.

I have a background in pure math, comp sci, and formal logic, including a course on category theory so I'm not afraid of a formal treatment (in fact I find formal resources easier to understand, so that would be preferred).

pic not related,

>>9744724
Some time ago I read the intro of a text on Fractal Geometry. It said that 'fractal dimension' is a generalization of the ordinary notion of dimension in the same way that a 'metric' is a generalization of the notion of 'distance'. Could you elaborate on this a little?

Is there a set of lecture notes on the topic that you would recommend? I find that lecture notes get right to the point without all the verbose exposition you find in textbooks.

>>9744866
>formal treatment
As opposed to?

>>9744880
Sorry, I should have elaborated on what I meant by that.

In my experience there seem to be two different approaches to teaching math. In the most common approach the text will go from special cases and real world examples and little by little build up towards more abstract cases with more general definitions along the way (linear algebra texts are the worst offenders of this). In the less common appraoch the text will start off with a barrage of very general/abstract definitions in formal logic and primarily use examples to highlight possible pitfalls and degenerate cases.

I much prefer texts written taking the latter approach.

>>9744943
>In the most common approach the text will go from special cases and real world examples and little by little build up towards more abstract cases with more general definitions along the way
I haven't seen this approach taken in any mathematics text. What makes you think it's really the most common one? And I don't quite get the meaning behind the phrase "real world examples".
>in formal logic
Are you sure you're talking about mathematics? Fields outside of logic don't tend to write anything in formal logic.

>>9744664
Nope. I didn't take the paper out with me so I can hardly prove it, but just nope

>>9744999
>doesnt even know cauchy's integral formula
>can't prove a babby tier integral
>no, i was not unprepared, the exam was hard!!

>>9743933
how's warwick for maths? or life in general? I'm going there next year for a phd

>>9744966
>I haven't seen this approach taken in any mathematics text. What makes you think it's really the most common one?
It's my main annoyance with most textbooks I read.

I specifically mentioned linear algebra as an example since many introductory linear algebra textbooks have a progression along the lines of: systems of equations, some visual stuff on [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math], norms, dot/cross products, [math]\mathbb{R}^n[/math] for finite [math]n[/math] and linear transformations as matrices, real vector spaces, etc... and they never cover vector spaces over a field. Of course, there are many great linear algebra texts that don't do this (like Hoffman and Kunze). I am trying to say that given the choice I would prefer Hoffman and Kunze any day.

>And I don't quite get the meaning behind the phrase "real world examples".
As in applications to science and other fields. Though I guess my annoyance extends texts that focus too much about 'nice' special cases when the general theory has more potential (eg. intro algebra texts).

>Are you sure you're talking about mathematics? Fields outside of logic don't tend to write anything in formal logic.
I don't mean it has to be in formal logic but the more formal it is the better. To be honest, many of the texts I've liked reading aren't that formal but they are always very clear, concise, and move swiftly.

See pic related, the formalism makes it super easy to understand what is and is not being said as well as how to prove such a statement (or it's negation). It also makes it trivial to see the relationship to topology. Most importantly, it does this in half a page without monologuing about some uninteresting example that takes just as much work to understand as the definition yet is nice enough to create potential for misconceptions.

>>9745122
>how's warwick for maths?
Clearly it's pretty bad if they study computations of "integrals".

>>9745144
>systems of equations, some visual stuff on R2R2 and R3R3, norms, dot/cross products, RnRn for finite nn and linear transformations as matrices, real vector spaces, etc... and they never cover vector spaces over a field
You're using some weird non-standard terminology. Nobody considers that to be linear algebra.
>As in applications to science and other fields.
Why would you be mentioning that in a thread titled "math general"?
>See pic related, the formalism
I don't really see any formalism there. It just seems to be an engineering text which defines continuity for real valued functions separately for some unknown reason.

>>9745120
>>9745147
Somebody's rather upset

>>9745122
The PhD support seems quite good. I've a few friends on the MMath who know people doing their PhD here and I've not heard any complaints.

The maths dept itself is really nice. Lecturing is average. Reputation obviously very good, but I've no idea how much that even matters for a PhD. Try and get accommodation in Leamington Spa rather than Coventry- for obvious reasons.

>>9745166
>You're using some weird non-standard terminology. Nobody considers that to be linear algebra.
Here is a popular undergrad linear algebra book. Compare with the table of contents:
https://warwick.ac.uk/fac/sci/maths/undergrad/ughandbook/content/ma106/elementary_linear_algebra_10th_edition.pdf

Either way, intro linear algebra texts were meant to be an example so that you could understand what I mean, not to debate the defintion of linear algebra. Another example would be any intro group theory text that wastes too much time developing modulo integers or a lambda calculus text that starts out by giving examples in Haskell.

>Why would you be mentioning that in a thread titled "math general"?
I'm only saying that I'd rather avoid textbooks that waste walls of text on that stuff instead of math.

>I don't really see any formalism there. It just seems to be an engineering text which defines continuity for real valued functions separately for some unknown reason.
I agree, the ideal thing to do would be to define it topologically. That said, many texts, including Spivak (who also does it over reals), takes multiple sections to define these. In Spivak's case continuity is defined in terms of limits,he never gives a definition for continuous on a set (just continuous at a point and some very cumbersome definitinos for continuous on intervals), the proofs are a mess, and uniform continuity is pushed into the chapter appendix.

>>9745203
>Try and get accommodation in Leamington Spa rather than Coventry- for obvious reasons.
no, i dont know the obvious reasons, i dont live there, nor have i ever been there

also, i mean like student life in general?

[math]\int \frac{dx}{x}[/math]
vs
[math]\int \frac{1}{x} dx[/math]

?

>>9745607
The first if you're the only one who will read it, otherwise the second.

>>9745607
>>>/sci/sqt/

>>9745607
[math]\log x[/math]

>>9745203
As a(nother) foreigner perhaps going to the UK for a PhD, how does one get accommodation anywhere? Is getting that guaranteed by being accepted in, or does it require waging war against the bureaucrats?

>>9743860
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
This is a simple discussion of geometric algebra. Basically in 2d it's the complex numbers, in 3d it's the quaternions. Instead of working directly with the quaternions you can work with a matrix representation of them called the Pauli matrices. They describe spin in quantum mechanics. One of the first exercises in Sakurai's book is about performing rotation with the Pauli matrices. You should look at the geometric algebra and clifford algebra for more about them.

>>9733160
Is the differential of right multiplication on a matrix lie group just right multiplication?

How do you people keep math fun?
I enjoy doing it as long as I can keep up the effort but even when I like something I can fall out of the habit. Inb4 someone thinks I'm just a lazy student: it's not about passing some exam, it's about that I want to enjoy mathematics.
What books/subcategories of math/video speeches or whatever have you strongly enjoyed?

>>9747091
>video speeches
https://www.youtube.com/watch?v=OFWFpVu5aQo

Guys I'm in my first year of college (econ) and I'm doing pretty bad in everything except for calculus - including linear algebra, like wtf I thought that I should be good at it too. Correct me if I'm wrong, but is calculus really that easy? Has it been dumbed down for brainlets like me? Sure, it helps that I'm kind of liking it, but I doubt that I'm that good. I feel like a fraud.

>>9747091

try to apply what math you know to real world problems :^)

Yesterday I played a bit with numbers and found a general test for divisibility using smallish numbers (each of the mods ensures we never get large numbers):
A number n in base B with digits d is divisible by k iff
[math]\sum_{i=0 }^\infty(d_i\cdot((B-k)mod\ k)^imod\ k)mod\ k=0[/math]

Calculation that gives you all of the standard divisibility formulas like alternating sum for 11, sum for 3 and 9, last digit for 2, last two digits for 4, last 3 digits for 8 etc.

*patiently waits*
https://totallydisconnected.wordpress.com/2018/05/09/the-latest-hot-abc-news/

>>9747092
kys

>it varies between fields whether poset means "partially ordered set" or "preordered set"
It's a small difference, but still.

>>9748335
Which fields use it to mean preordered set? Never seen it that way

>>9748216
>month(s)

>>9748338
The category theory book I'm reading uses it as preordered set, and claims that other category theorists do too, but I just checked another book and nlab and they both use it as partially ordered set. So I guess this book just sucks.

>>9748216
The end of the Mochizuki Meme.

>>9748216
>https://totallydisconnected.wordpress.com/2018/05/09/the-latest-hot-abc-news/
>Two prominent and very well-regarded mathematicians have isolated a specific and serious error in Mochizuki’s proof of the abc conjecture. They are preparing a detailed writeup explaining the issue, which should be available publicly in the next month(s).
>(I heard this from good friend X, who heard it directly from one of the two people involved.)
>Whether or not their writeup is widely read, I am certain that for most in the community, the verdict of these two persons will instantly settle the matter.
What could be the motivation behind not at least pointing out the specific conclusion that doesn't hold, and then doing a detailed write up later?

>>9748874
This is old news, stop posting this.
>What could be the motivation behind not at least pointing out the specific conclusion that doesn't hold, and then doing a detailed write up later?
Attention and clickbait. Builds up hype for the western brainlets with inferiority complex that finally the Japanese overlord will be defeated and the westerners will un-cuck. I am sure however that whatever they think is a mistake will just be clarified and debunked by Mochizuki himself, even if he did make a mistake it's no surprise since it's more than 500 pages of an entire new and arguably most advanced theory in mathematics. He won't be shunned or lose any respect since his work is far beyond anything most of them will ever do.

>>9748216
I feel genuine pity for "people" who don't have the patience/mental fortitude to read and properly understand Mochizuki-sensei's works.

https://arxiv.org/pdf/1805.06746.pdf
>A Proof of the Riemann Hypothesis Through the Nicolas Inequality
>Tom Milner-Gulland
>(Submitted on 15 May 2018)
>A work by Nicolas has shown that if it can be proven that a certain inequality holds for all n, the Riemann hypothesis is true. This inequality is associated with the Mertens theorem, and hence the Euler totient at [math] \prod_{k=1}^n p_k[/math], where n is any integer. We shall show that indeed the Nicolas inequality holds for all n.

Will he ever publish?

>>9741625
its called paraphrasing

>>9736762
literally the gayest post on this site
could only manage ~20 words of it

>>9748892
Are you a teenager?

>>9749446
>Are you a teenager?
Not anymore.

>>9748892
>someone wrote something really long so its right
just a big lol

Any good books on introductory graph theory for a CS student who just finished learning about proofs and set theory?

>>9749455
>graph theory
>CS
Consult >>>/sci/sqt/ and >>>/g/.

>>9749455
>Any good books on introductory graph theory for a CS student who just finished learning about proofs and set theory?
Modern Graph Theory

>>9733160

I know this isn't physics general, but I'm sure you guys can help me. If I have one region of high pressure and one region of low pressure, then how do I calculate the rate of gas flowing from high to low pressure? I am interested in this as I want to model how a hovercraft would work.

>>9749695
The flux depends on diffusion parameters and the power of your convection inducing motors, you'll need to simulate it in fluid mechanics software. This is not a question for mathematicians nor physicists.

A shortcut for hovercraft is that the thrust of an air bearing is usually specified by the manufacturer. So that and the mass of your hovercraft is all you need to do any simple calculations you want.

>>9737408
Sounds exactly like a generic upper division linear algebra class. Basically a generalization of your intro to LA class, where you really flesh out the direct correspondence between linear operators and the matrix algebra you've already learned. It's a fundamental class, and quite insightful too. At Berkeley it's called Math 110 and it's recommended as your first upper division math class because it's comparatively easier than say, abstract algebra or real analysis.

>>9749699

Thank you for your reply. However so if I had a 30 ton hovercraft with no cushion and a small skirt, how many watts would keeping it flowating consume?

>>9749695
>>9749699
>>9749702
>>9749703
Physics discussions belong in the physics threads over at >>>/toy/. Or alternatively try >>>/sci/engi/ and >>>/sci/pg/.

>when 90% of the discussion in mathematics threads is xirs telling people that what they're doing is not mathematics, or that it is not well-defined

I'm trying to educate myself at home for work and Geometry is a big part of my job. Is Euclid a good starting point for my studies? If not what should I be looking for?

I hate you all.

>>9750893
>I hate you all.
Why?

>>9750920
Not really, but I am envious and disappointed at the same time. Some of you do math stuff and enjoy it like I used to enjoy doing it, or maybe even more, and some of you just derail these threads so no actual discussion can happen. I'd like to talk about stuff, but I feel too weak and stupid to start the discussion. Really I just hate myself and try to project my negative qualities onto others, like the posters in this thread.

Does this upset you?

>>9750936
Why would it? dx, dy and dz are non-zero and linearly independent.

>>9750928
cringe

>>9748874
He probably does not know, considering it's thirdhand gossip.
And the people publishing certainly don't want to announce they've found an error until they've got an impenetrable proof that it's both correct and not repairable.

>>9750972
>not repairable.
Well they should still announce even if it is repairable

Am I the only one who hates exotic unpronouciable alphabets as notations?

>>9751020
I haven't encountered any of those, so I can't really say.

Anyone familiar with the book Riemann's Zeta Function (H.M. Edwards)

I've always wanted to learn more about this renowned function, and in particular how it relates to the Prime Number Theorem. The extent of my number theory knowledge is a few chapters of Ireland and Rosen, but I have taken a one semester course on Complex Analysis (and I have a copy of Ahlfors).

How should I go about learning the Prime Number Theorem?

>>9751194
>How should I go about learning the Prime Number Theorem?
Read a proof.

>>9751236
I'm wondering about prerequisites mainly.

>>9751245
Complex analysis and number theory?

I wish I had a math friends
It's just easier and more fun to put effort into stuff together than to always be alone.

>>9751194
Just start reading the proof on wikipedia and find out what they start talking about and start googling, what are you disabled?

Pic related, which class is most interesting, and how related are they. I go to Berkeley, 113 is abstract algebra, 104 is real analysis, I've taken both.

>>9751931
The first two are pretty similar, the last one is almost completely unrelated at this level, and probably going to be significantly harder.

The most interesting is up to you. They all seem pretty lackluster to be honest, especially the first two (I assume you haven't or don't require general topology? That would take a big chunk of the time). I would take the last one given that I'm more of an algebra person. I'd suggest the first one if you like calculus/differential geometry a lot more, the second if you're more interested in more flexible geometry and less about smooth structures, and the third if you're more interested in number theory and algebra

>>9751245
>>9751363
Honestly the proof the prime number theorem is not super high-octane. There's virtually zero number theory involved (as far as I remember the only thing you need is unique factorization). The only prerequisite you may possibly not have is knowing about analytic continuation from complex analysis. Other than that it's just a bunch of specialized approximation results.

>>9751893
im your friend anon

>>9750952
>linearly independent.
But dx, dy, dz aren't vectors. They are numbers. (well, they aren't really something, but they are supposed to intuitively represent numbers)

>>9751245
no prerequisites needed just read the elementary proof by selberg

>>9752512
That is stupid. The analytical proofs by Jacques Hadamard and Charles Jean de la Vallée-Poussin are much easier. You only have to know a little complex analysis.
The proof by Atle Selberg and Paul Erdős is elementary but pretty complicated.

>>9752508
>they aren't really something
What do you mean?

>>9752508
They are functions, stupid. They are called "differentials".

>>9752589
I mean that they are not defined.

>>9752597
I am pretty sure that the retarded physicist/engineer who wrote that text wasn't using differential forms.

>>9752603
Even if he wasn't intentionally using them, he was using them.

>>9752669
differential forms aren't infinitesimals

>>9752688
And?

>>9752508
Yes they are vectors you massive brainlet. They form the basis for the dual space of R^3. In the context of being a dual basis you can call them covectors if you like, but the point is that they're elements of a vector space, hence vectors.

Is there a method, book or theory which can help me setting prices? I mean, how to derive a price for a given product given some data (like surplus, expired and defective items, quatity of products I purchased at a given price and quantity of products I purchased at a new price, etc.)

>>9733160
science proves God

>>9752598
How are they not defined?

>>9753914
How are they defined?

Tools for looking at the determinant of two 3x3 non hermitian matrices? I'm aware of the Minkowski inequality and the Matrix Determinant Lemma but they don't quite work, unless I want to try the singular value decomposition of one of my matrices.

>>9739430
>"Did you just assume my gender?" the post. Faggot.

>>9753386
Google “economics 101”. I hope you like graphs.

>>9753941
I think microeconometrics is more applicable here.

>>9753386
>>9753941
>>9753959
Wrong thread. Use >>>/sci/sqt/ and >>>/biz/.

>>9750936
No, it doesn't.

>>9752508
>But dx, dy, dz aren't vectors. They are numbers.
Now THIS upsets me.

>>9753921
It's right here dude >>9752771

>>9754176
hyperreals numbers

>>9744062
Naming shit is hard.

Has Mochi been indulging in too much tempura?
http://www.kurims.kyoto-u.ac.jp/~motizuki/

Can someone explain this post: >>9752771 ?
I don't understand how dx,dy,dz are a dual base of R^3.

>>9754742
It's really a matter of definition, but basically you can think of dx, dy, dz at one point p as being the dual base to the tangent space of R^3 at p (which we can identify to R^3).
But now why would we set this strange definition ? Say you have a smooth path p(t) such that p(0) = p. Then what would be the "infinitesimal variation" of, say x, at 0 ? Probably the x-component of the tangent vector p'(0), ie dx(p'(0)) with this definition

>>9744125
It's all good mane.

[math]Area = Base * Height[/math]
Formula for area of rectangle.

[math]6x^3-30x^2 = 2x(x-5) * (Height)[/math]
We can factor the left side to make it easier to manage.

[math]6x^2(x-5) = 2x(x-5) * (Height)[/math]
Now it's easier to solve for height.

[math]\frac{6x^2(x-5)}{2x(x-5)} = (Height)[/math]
Simplify.

[math]3x = (Height)[/math]

>>9753941
I hate graphs.

>>9753983
>Wrong thread
May be, but I'm looking for some theoretic foundations no bizfag has; most of what I've found are explanations like "take into account market, promotion, objectives and other subjective stuff with no mathematical foundations and then use this shitty strategy based on what you think could be a good choice". I want to read something based on statistics, calculus, mathematical finance and serious observations.

>>9755185
>I want to read something based on statistics, calculus, mathematical finance and serious observations.
Wrong thread. Try economics threads and >>>/sci/sqt/ next time.

>>9752508
who fucking cares, you know they work intuitively

>>9755257
>they
What exactly are "they"?

>>9755318
Jews

>>9749005
Speaking of publishing, what's everybody working on (research, self study, prep for quals)?
At the moment I'm writing up some notes that'll be the basis for a paper (after my advisor picks it apart to try and find any critical errors).

>>9754742
Yeah, >>9754782 put it pretty well. I should've been more precise in that they're the basis of [math] T_p^*\mathbb{R}^3 [/math], which admittedly gives a better connection to your natural intuition of "differentials". But I was lazy since of course [math] T_p^*\mathbb{R}^3 \cong (\mathbb{R}^3)^* [/math] and the technical calculations turn out pretty much the same. It's pretty important to make explicit reference to the tangent space when you're working on non-Euclidean manifolds though.

Will I be homeless if I major in math and I don't go to a t20 school?

https://arxiv.org/pdf/1805.07329
>New explicit solution to the N-Queens Problem and its relation to the Millennium Problem
>Dmitrii Mikhailovskii
>(Submitted on 18 May 2018)
>Using modular arithmetic of the ring [math] \mathbb{Z}_{n+1} [/math] we obtain a new short solution to the problem of existence of at least one solution to the N-Queens problem on an N×N chessboard. It was proved, that these solutions can be represented as the Queen function with the width fewer or equal to 3. It is shown, that this estimate could not be reduced. A necessary and sufficient condition of being a composition of solutions a solution is found. Based on the obtained results we formulate a conjecture about the width of the representation of arbitrary solution. If this conjecture is valid, it entails solvability of the N-Queens completion in polynomial time. The connection between the N-Queens completion and the Millennium P vs NP Problem is found by the group of mathematicians from Scotland in August 2017.

So, suppose you have a piecewise function, is there any way to get a taylor series that represent both functions ?

Anybody who lives in the nyc/long island area:
Does any of you know a good place for math internships? I'm a going to be a junior in the fall, and want to get a head start on getting something next summer

>find stackexchange question about the exact detail I need to solve a problem
>poster provides no justification for what he did and I can't prove it works

>>9756505
Post link, maybe someone here can take a stab at it

>>9756535
Can't hurt, I suppose. It's about Lie algebras.
https://math.stackexchange.com/questions/311687/lie-algebras-and-roots-systems/311852#311852

In the second paragraph of this answer he claims a "we have a result" about a linear order on a sequence of roots which I can't think up any good explanation for.
The fact that it's such an off-hand remark makes me suspect the answer is extremely simple and I'm just a brainlet.

>>9756284
What schools are you thinking off?

>>9756555
Okay, I think I know what the answer is, take a look at these notes
https://en.wikipedia.org/wiki/Root_system
http://math.mit.edu/classes/18.745/Notes/Lecture_21_Notes.pdf
from the definitions given in pages one and two about simple reflections if [math](\alpha,\beta_{i_\ell})<0[/math] then by definition of what a reflection is [math]\alpha_{\ell-1}=s_{i_\ell}(\alpha) = \alpha - 2\frac{(\alpha,\beta_{i_\ell})}{(\alpha,\alpha)}\beta_{i_\ell}>\alpha[/math]

Does this help?

>>9756642
Fuck me, hopefully it works this time
If
[eqn](\alpha,\beta_{i_\ell})<0[/eqn]
then by definition of what a reflection is
[eqn]\alpha_{\ell-1}=s_{i_\ell}(\alpha) = \alpha - 2\frac{(\alpha,\beta_{i_\ell})}{(\alpha,\alpha)}\beta_{i_\ell}>\alpha[/eqn]

>>9756566
>>9756652
I guess I'm just a retard that doesn't know how to use latex on this board, just copy my first post into the TeX pop up at the top left hand corner of the reply box and you'll see the equations, apologies for the inconvenience

>>9756658
Pretty sure this just replaces one issue with another. There's no justification to assume any of the bilinear forms are negative to start. He derives this from the ordering.

How do I do multivariate calculus in my head

>>9756703
You don't, it's tedious and annoying.

>>9756658
Put math and /math inside [] instead of [eqn].

>>9756685
Basically the question asks that given the bilinear form [math](\alpha,\beta)[/math] is strictly non-negative for all [math]\beta[/math] then [math]\alpha[/math] must be the highest root of the same length. He then claims that there exists root [math]\alpha_{0}[/math] s.t. it is the highest among roots of the same length and it has the same length as [math]\alpha[/math], therefore they are related by simple reflections since simple reflection preserve the length . Essentially he is setting up a proof by contradiction. He then uses the fact that [eqn](\alpha_{0},\beta_{i_\ell})>0[/eqn] to prove the recurrence relation, however this then implies that [eqn](\alpha,\beta_{i_\ell})<0[/eqn] which is false by our assumption, so [eqn]\alpha = \alpha_{0}[/eqn]

>>9756711
Thanks
>>9756724
>>9756685
Basically the question asks that given the bilinear form [math](\alpha,\beta)[/math] is strictly non-negative for all [math]\beta[/math] then [math]\alpha[/math] must be the highest root of the same length. He then claims that there exists root [math]\alpha_{0}[/math] s.t. it is the highest among roots of the same length and it has the same length as [math]\alpha[/math], therefore they are related by simple reflections since simple reflection preserve the length . Essentially he is setting up a proof by contradiction. He then uses the fact that [math](\alpha_{0},\beta_{i_\ell})>0[/math] to prove the recurrence relation, however this then implies that [math](\alpha,\beta_{i_\ell})<0[/math] which is false by our assumption, so [math]\alpha = \alpha_{0}[/math]

>>9756724
Thanks, this is helpful. The part I was not seeing was that the order sequence was developed recursively from [math]\alpha_0[/math] having positive forms. I see what he did now.

>>9756732
No prob anon

>>9756705

>>9756705
Much like calculus itself.

Can someone explain how this solution works please? Im doing elastic collisions in two dimensions.

2 = 4e becomes e=2, what? e can only be between 0 and 1. Id say its just a mistake but it uses it again and if e =.5 the second calculation doesnt make sense.

>>9756755
Wrong thread. Try >>>/sci/sqt/ or >>>/toy/physics/.

>>9756755
If the first vector is v and the third vector is w, then v and w are parallel, so v=-ew for some e. Now, 2e=-(-4)=4, so e=2, and so we can calculate p from -ep=-5 <-> ep=5 <-> p = 5/2. There is a typo, as e should be with 2 and not 4.

>>9756566
Like schools ranked in the 21-30 range on this website
https://www.usnews.com/best-colleges/rankings/national-universities

Picked up a few text books for the summer.

Should I delve into topology or the intro to differential geometry book first?

>>9757661
You do realize diff geo uses topology?

>>9757669
Well, not unless you work in Euclidean space, then you can get away with not really knowing much topology.
>>9757661
>Should I delve into topology or the intro to differential geometry book first?
Why not both? Books like Fomenko's modern geometry or Novikov's text cover both topology and differential geometry. In general though I'd say to understand more advanced techniques in geometry having a grasp in topology is necessary.

>>9757777
>let me just quickly embed this 4-manifold in [math]\mathbb R^{35}[/math]

>>9757777
>Well, not unless you work in Euclidean space, then you can get away with not really knowing much topology.
What did he mean by this?

>>9757836
You can do it, it just requires a more classical picture of differential geometry
http://geometry.karazin.ua/resources/documents/20140424104758_c6db3e3f949.pdf
hell, like >>9757789 joked about, according to Nash's embedding theorem we can embed Riemannian manifolds in Euclidean space in such a way that it preserves distances. While it may not be pretty you can study differential geometry in this way, at least up to a point

>>9757882
I'm not even going to mention how the preliminary chapter begins with topological stuff, I will only meta-mention it like this.

>>9757893
The amount of topology mentioned at is essentially limited to defining what a homeomorphism and diffeomorphism is, he doesn't even define a topological space, that's pretty much what a meant when a said "not really knowing much topology", not "knowing no topology at all"

>>9757908
I'm surprised if he doesn't invoke the properties of complete metric spaces at some point. Nevertheless, diff geo sucks ass, so my suggestion is still to do topology instead.

>>9757908
>defining what a homeomorphism and diffeomorphism is
>he doesn't even define a topological space
You lie.

Could someone explain what the Fourier transform is actually doing? I don't just mean
>it takes a function into the frequency domain anon!
I understand that. But where does the integral come from?

>>9758257
maybe this will help
https://www.youtube.com/watch?v=spUNpyF58BY

and if you know a bit more about math, you could think of it as an inner product of functions
from a linear algebraic point of view, the inner product measures "how close" two vectors are, usually interpreted as an angle
this means that if you take the inner product of a vector and a basis vector, you can see "how much" the basis vector is in the other vector
and remember that you can decompose a vector into a linear combination of basis vectors
in your function space, the basis functions are trigonometric functions, but there are uncountably many of such functions, so that's a hint that you should integrate rather than sum
and that's what the fourier transform does
it takes the inner product of functions, decomposing the input function into its component parts (which are the trigonometric sin/cos functions), and measures how much of each basis function is present in the function

at least i think so?
got me through a complex analysis course so ive been sticking to it