/sci/ - Science & Math

What's the point of using numerical solutions to nonlinear systems like Newton's method as opposed to solving the system of equations algebraically? When should you use one over the other?

Can you somehow justify why it is ok to spend some of your university's money on abstract mathematics with no clear real world applications. I used to be proud of my uselessness, but now it just feels wrong, almost like commiting a crime. How do you convince yourselves it's ok to do your PhDs and research? Please help a faltering brother.

>>9380858
https://mathoverflow.net/questions/175847/how-does-one-justify-funding-for-mathematics-research

>>9380768
1. Not every equation can be solved algebraically.
2. Even if you can solve it algebraically, how do you think things like [math]\sqrt{x}[/math] are actually calculated?

What is the best path towards proving the negation of AC?

>>9381307
>What is the best path towards proving the negation of AC?
Just assume it's false

>>9381307
>proving something that is independant from your axioms

>>9381346
>>proving something that is independant from your axioms
What are his/her axioms?

>>9381346
And what would these axioms be? Are you mentally weak?

How should I respond if someone asks me for my preferred axioms? I know that respecting a person's axioms is vitally important to creating inclusive environments for all mathematicians

>>9380865
I was hoping for a few ideas, expecting a flame war and what I got was this one reply with nothing but a link. But the discussion you linked was able to ease my mind. Thank you, based anon.

>>9380858
google deloitte report on benefits of math research

how do i review linear algebra in preparation for projective geometry ? what should I focus on?

>>9381925
Skim through the lecture notes you took back then

>>9381698
Thanks to you, too. I think I no longer feel like a waste by doing math, or if I ever decide to apply for a PhD program.

>>9382272
>if I ever decide to apply for a PhD program.
Don't. It will ruin your life.

>>9382622
For a life to be ruined by doing something, it is required that it is not ruined before doing it. My life can not be ruined.

>>9380557
How can I get better at topology? I'm in the beggining of Munkres and it is being a terrible experience.

>>9382647
>it is being a terrible experience.
What do you mean exactly?

What matters more in Math? High Abstract Pattern Reasoning (Raven Matrices/IQ testing) or creativity(unorthodox insights into problems).

>>9382666
Both.

>>9382668
What field of math do you think requires more unorthodox insight? Would it be Combinatorics? Or?

>>9382671
>Combinatorics
You just lost your reproduction license.

>>9382671
Topology for example. Pretty much anything where you benefit from being able to visualize the objects, and then twist and turn them around in your mind.

>>9382662
Probably the worst part is the exercises. In most textbooks I've used in other subjects, they were no real problem and normally I could solve them. In Munkres' Topology, however, the same didn't apply. This probably reflects something deeper about my understandment of the text.

>>9382681
Point set topology exercises are usually garbage though. Much like the parts of it which aren't used in algebraic topology or any other superior fields.

>>9382681
Exercises in point-set topology are mostly definition-pushing. In most cases, it's just "write out all the definitions and do the only thing you can do". Are you struggling with remembering the definitions? Have you studied analysis before to give some motivation/intuition for the definitions?

When I started doing topology (also using Munkres) I thought each exercise was some super involved thing. But once you're familiar with it it all seems super trivial.

>>9382694
The same applies to Dugundji. It's good to have problems reinforce the definition in your mind, but it gets boring quite fast.

>>9382676
Explain? I'm really into Combinatorics right now because I like coming up with cool algorithms that can count things in polynomial time.

>>9382677
I assume low dimensional topology, right?

>>9382686
Are they? Can I still understand the text without being able to solve a part of them?

You used the example of algebraic topology. I'm really interested in this subject, and I've already a good knowledge of algebra- and even the most fundamental concepts of algebraic topology, like homotopy and homology. Isn't point set topology that necessary?

>>9382694
Maybe I'm struggling with some definitions. I can remember most of them easily. My real problem- I suppose- is with intuition. In fact, they sometimes don't look quite familiar- and that turns into a big problem when trying to solve the exercises.

>>9381307
It is fairly easy to do in ZFA (can't well order atoms), but showing that ZFA has the same consistency strength as ZF is nontrivial.

>>9382702
>Isn't point set topology that necessary?
In the beginning you should focus on learning about limits and colimits in the category of topological spaces. Also learn about compactness, connectedness, the compact-open topology and everything needed to define it. This should be enough to get started with algebraic topology.

What does it mean if I do better in real Analysis than abstract algebra? I can just read the proofs in real analysis and get a nice grasp of the material but abstract algebra just seems like a new bum fuck definition comes out of no where with no sense of meaning behind it. With real analysis I can see why someone would want to define the Bolzano Weierstrass Theorem and the cauchy sequence but why the heck would anybody care about cosets and all that stuff(yes I know it has something to do with quotient groups and the decomposition theorem).

What did I do wrong guys? Please don't tell me that I have an analyst brain or anything, I think that's bull shit and I like to learn both subjects. Thanks.

>>9382738
>define the Bolzano Weierstrass Theorem

>>9382738
>What does it mean if I do better in real Analysis than abstract algebra?
It means you're better at engineer garbage than mathematics, it's a pretty common condition around here.

>>9382741
Sorry my English isn't that good. What I mean, is that I can see how the theorem fits in with the rest of the material.

>>9382727
By what I've seen, connectedness sounds really more intuitive. Maybe it's the subject

>>9382745
Please no meme replies. Everyone knows that analysis is a respectable branch of Mathematics, anyone who thinks other wise is a retard or someone who did horrible in real analysis and thinks being good in Algebra is the only requirement one needs in order to be a Mathematician. The inverse is also true.

>>9382745
>implying analysis is closer to engineer garbage than to math

>>9382738
You have an analyst brain, use it to your advantage

>>9382738
Study harder in algebra, then you will feel the same intuition.

>>9382762
I find abstract algebra so sexy though, I don't want to pigeon hole myself into analysis. How true is this meme, I know your not being an ass hole but I don't like being told that I can only succeed at x being apparently I have x brain.
Let me ask this, how do I get good at Abstract Algebra with an analyst brain?

>>9382756
>the only requirement one needs in order to be a Mathematician.
Not the only one, but it's a pretty major requirement.
>"inverse" instead of "converse"
Definitely an engineer.
>>9382760
That's indeed the case.

>>9382766
>how do I get good at Abstract Algebra with an analyst brain?
You have to start by actually learning mathematics instead of playing around with analysis. You're pretty far from your goal if you only "think" cosets have "something to do" with quotient groups.

>>9382774
Are you saying I should stress definitioms and thorough reading more than problem solving? Im really trying to learn, anon.

>>9382774
Unless you are saying that real analysis isnt math, then I dont really want advice from an idiot who most likely failed analysis but isnt mature enough to realize its importance.

>>9382778
>stress definitioms
What do you mean?
>problem solving
The only kind of acceptable "problem solving" at the beginner level is proving theorems. If that's what you mean, then go ahead.

>>9382785
>most likely failed analysis
How does one fail engineering courses without taking them?

What is a good introductory text for category theory?
No memes please.

>>9382738
To be fair, you have to have a very high IQ to understand modern algebra. The proofs are extremely subtle, and without a solid grasp of real mathematical reasoning most of the theorems will go over a typical viewer’s head. There’s also quotient groups, which are deftly woven into cosets- its deep meaning draws heavily from Lagrange's Theorem, for instance. The algebraists understand this stuff; they have the intellectual capacity to truly appreciate the depths of these groups, to realise that they’re not just interesting- they say something deep about POLYNOMIALS. As a consequence people who don't understand algebra ARE brainlets- of course they wouldn’t appreciate, for instance, the Isomorphism Theorems, which itself are a cryptic reference to category theory. I’m smirking right now just imagining one of those addlepated simpletons scratching their heads in confusion as Galois' genius wit unfolds itself on their textbooks pages. What fools.. how I pity them.

And yes, by the way, i DO have a Cayley Diagram tattoo. And no, you cannot see it. It’s for the ladies’ eyes only- and even then they have to demonstrate that they’re within 5 IQ points of my own (preferably lower) beforehand. Nothin personnel kid

>>9382803
>>9382795

Come here for help. All I get is meme replies = /. Can someone with a more mature view on Mathematics please help a nigga out? I may be stupid but I'm not enough enough to think an entire branch of Mathematics is engineering.

Thanks.

>>9382809
>All I get is meme replies
How exactly is >>9382795 a "meme reply"?

>>9382797
Leinster - Basic Category Theory

>>9382817

>>9382809
>not enough enough to think an entire branch of Mathematics is engineering
A branch of mathematics can't be engineering, this is correct. But real analysis isn't a branch of mathematics.

>>9382797
this >>9382820 and Steve Awodey - Category Theory. You can read both while skipping certain sections of Awodey depending on your needs.

>>9382797
mac lane - categories for the working mathematician

>>9382809
https://math.stackexchange.com/questions/355418/should-i-be-worried-that-i-am-doing-well-in-analysis-and-not-well-in-algebra

>>9382833
>>9382832
>>9382820
Thanks senpaitachi

>>9382803

>>9382797
>What is a good introductory text for category theory?
why bother? it's irrelevant to most of mathematics anyway

>>9382845
>most of engineering
ftfy

>>9382847
That's true too, but I'm no sure what it has to do with my post.

>>9382847
> I'm an elitist ass hole who thinks studying pure math at a third rate university makes me special.

>>9382852
How does it feel being so brown?

>>9382852
>> I'm an elitist ass hole who thinks studying pure math at a third rate university makes me special.
Who are you quoting?

>>9382854
Did that hit too close to home?

>>9382858
You are daydreaming.

>>9382918
The only one day dreaming are people who major in pure math and act overtly elitist in order to make up for intellectual short comings.

>>9382845
This. Leave the category theory to physicists, kids.

Arithmetic Gauge Theory: A Brief Introduction
Minhyong Kim
https://arxiv.org/pdf/1712.07602.pdf

>Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles appears to be closely related to an effective version of Faltings's theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of {\em Galois representations}, the structures linking motives to automorphic forms according to the Langlands programme. In this article, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory. For the most part, it can be read as an attempt to explain standard constructions of arithmetic geometry using the language of physics, albeit employed in an amateurish and ad hoc manner.

>>9382935
They told me I could learn physics with just calc diff equations and linear algebra :(

>>9383028
>They told me I could learn physics with just calc diff equations and linear algebra :(
Oh sweetie...
https://arxiv.org/abs/1111.5056

>>9382922
>pure math
There is no other kind.

>>9382922
This is me (except the last part).
Graduated with pure math three years ago but went straight into industry. Hate everyone here.
Would I do it again? Yeah I loved the major. People are autist so if you weren’t one you were already better. Def lots of lost potential tho.

>>9383133
>There is no other kind.
What do you mean?

>>9382933
>>9383034
>>9383140
>>>/r/taiwan/

>>9383133
Applied Math? Combinatorics?

Being overtly elitist is retarded and only serves to hamper your world view.

>>9383144
>>>>/r/taiwan/
uh?

>>9383145
>Applied Math? Combinatorics?
Neither of those are math.
>elitist
I didn't say anything negative about them. They simply aren't math in the same way history isn't.

>>9383138
I guarantee you would hate a life of being a research professor. The math purism view is just some kind of shitty Stockholm syndrome pushed by retards, people like perelmeme already BTFO academia faggots, so if you want to do pure math why not just do it without the illusions of grandiose that entails being a research professor?

>>9383153
> I didn't say anything negative about them. They simply aren't math in the same way history isn't.

I don't get how someone could have their head so far up their ass. It's quite funny to be honest, and sad at the same time.

>>9382809
>enough enough
the worst kind of enough

Brainlet undergrad sophomore here. What exactly is the difference between geometry and topology?

>>9383180
>What exactly is the difference between geometry and topology?
Nothing

>>9383170
You might be retarded. As in medically.

>>9383180
topology is a subfield of geometry

>>9383195
Get mad fag. Take your pseudointellectual elitism some where else.

>>9383153
define "math"

>>9383204
>fag
Why the homophobia?

>>9383204
Do you deny your nature?

>>9383150
>>9383210
see >>9383144

>>9383210
>>9383215
No. I just think you are a faggot; as in, you are being an elitist cuck just for the sake of showing off your epeen.

>>9383180
Geometry is local while topology is global. Though this doesn't mean that they're mutually exclusive.
https://arxiv.org/list/math.GT/recent
Pretty based field actually.
https://arxiv.org/abs/1709.04306

>>9383225
see >>9383144

>>9383222
>No.
Props for your honesty. At least you realize your own retardation, too bad that doesn't help you understand that combinatorics is not mathematics.
>you are being an elitist cuck just for the sake of showing off your epeen.
Everyone with a brain begs to differ.

>>9383215
>Do you deny your nature?
https://en.wikipedia.org/wiki/Homosexual_behavior_in_animals

>>9383238
>>>/r/taiwan/

>>9383180
Geometry = Topology + additional structure

"additional structure" varies, generally a sheaf encoding some algebraic or analytic structure

>>9383237

Combinatorics is real mathematics, sorry if that statement triggers you and is deeply problematic.

>>9383252
I can say with confidence that you are wrong.

>>9383254
How do you explain this?

>>9383254
Show me on this toy where Combinatorics touched you?

>>9383260
How do you explain this?
*inserts a random screenshot from a psychology book*

>>9383270
You had a good gig but now you ousted yourself as a psued.

>>9383279
Do your own homework, scrub.

Based role-player.

>>9383293
>scrub
cringe

>>9380858
stop falling for the jewish propaganga that you're an evil person unless what you do is profitable for the system

If i have a von Neumann regular ring, can i realize its refinement conical monoid, provided it is countable?

What does a typical four year applied math undergrad curriculum look like? Emphasis on typical, not "ideal". I'm curious what I missed and want to self study.

>>9383368
a typical one looks like an ideal one with some shit left out and done badly

Barebones, core:
- Linear Algebra, e.g. H&K
- Real Analysis, e.g. Tao I & II (with an introduction to set theory and the axiom of choice)
- Complex Analysis, e.g. Ahlfors
- Abstract Algebra, e.g. D&F
- Topology, e.g. Munkres
- Differential Geometry, e.g. Pressley

Other important stuff for undergrad level:
- Differential Equations, e.g. Hirsch & Smale
- Operations Research, (simplex, lagrange mults, convex opt)
- Measure Theory and Probability, e.g. Athreya&Lahiri
- Functional Analysis, e.g. Analysis Now

yeah

>>9383368
>>9383423
oh you said applied math
the same I guess, depending on how serious you are. maybe more stats

Why is [math]\int\int\int[/math] a meme?

>>9383180
Geometry includes measuring distance (and other quantities like area and volume) in general; topology only recognizes infinitesimal closeness aka indistinguishability.

>>9383423
> - Differential Equations, e.g. Hirsch & Smale
Didn't know Smale wrote a textbook on DFQ.

Why is the change in f(x) of the antiderivative equal to the area under the line in that region in the first function? I'm not sure if there's something intuitive about this relation or if the explanation is an unintuitive proof.

>>9383423
Doesn't applied math usually include more numerical computing?

>>9383442
yeah it's a good one

>>9383445
I fucked up and wrote a regular math one
add Numerical Analysis from e.g. Burden and some stuff on algorithms and finance maybe??? i dunno

>>9383444
integrating is like adding up a bunch of trapezoids, so you need to know the function which has that slope (of the top part of the trapezoid)

>>9382666
Depends onthe type of math. Raven's is a pretty shit test by the way. Can't measure jack shit beyond 140 IQ (st. dev. 15) and is really unreliable above 130.
Abstract pattern reasoning is loaded on the Verbal factor of IQ.
Geometry -> Spatial IQ
Algebra -> Verbal IQ
Number Theory -> Performance IQ

>>9382809
No one can help you out with anything. It's pretty typical of mathematicians to be adept in one branch/domain of mathematics but lacklustre in others.

>>9382922
Math is pure math. Applied mathematics is an ill-defined term.

>>9383444
well basically when you integrate you multiply the function value by a length of a small interval. when you take a derivative, you divide by a length of a small interval. that's the intuition and the proof actually goes like that.

>>9383153
Combinatorics is math though.

>>9382738
early specialization is a mistake, work harder on your algebra. That aside it is normal to better at some things than other.

>>9383444
Because [math] F(x)= \int_{a}^{x} f(t) dt [/math] is an antiderivative of F(x), and if G is another andiderivative then G=F+c, where c is a constant number.

Why is F(X) an antiderivative?
Let x be a certain point.
For points x+h with h being small (i.e. x+h is close to x) you can approximate (due to the continuity of f) the area under F from a to x+h as:

The area under F from a to x
plus
The rectangle with height f(x) and length h.

Symbolically
[math] F(x+h) \approx F(x)+f(x) \cdot h \\
\frac{F(x+h)-F(x)}{h} \approx f(x) \\
\displaystyle{\lim_{h \to 0}} \frac{F(x+h)-F(x)}{h} = f(x) [/math]

>>9382738
It takes a while, but you see the need for such ideas in time as they come up. With group theory in particular, it helps to read about how groups are used in other fields, e.g. groups acting on spaces.

>>9382738
Yes, Abstract Algebra is not as "useful" as Analysis.
Your motivation should be "fun".
Maybe your textbook sucks. Personally, at first I used a shitty book where stuff seemed very unmotivated and dry, which made me hate Algebra. Then I picked Fraleigh's book and everything flowed nicely.

>>9382933
fuck off hankel

>>9382741
An infinite bounded set has at least one accumulation point.

>>9384186
Uh, groups are ubiquitous in mathematics. I don't think analysis is any more "useful" than any other basic branch of mathematics.

>>9382753
As a rule of thumb, Compactness is a property that makes local properties global, since in some sense, it implies your space doesn't need too many open sets to describe it fully

>>9383777
Why because some elitist shit poster on a Mongolian basket weaving forum said so?
>>9383757
Let me guess, you pulled that out of your ass?


All of you guys just talk out of your ass, no substance.

>>9384625
math master race, brainlets git out

>>9384625
I don't have to hold your hand faggot. I've done that plenty of times on this shit ass board to realise that it's a total waste of time.

Fuck off.

>>9384651
>faggot
Why the homophobia?

>>9384861
Guess.

>>9384629
See that's what you fags do, after putting you fags in a corner all you can do is scream meme phrases with no substance. You faggots disgust me.

>>9383180
Topology is the study of the fundamental nature of a space while geometry studies the more "metric" notions of it.

Topology only cares about spaces that aren't the same up to a continuous deformation of one onto another - all properties of the nature of the space are the same. The ultimate goal is to classify all the different spaces in this way. We consider fundamental in a space things like connectedness (are there different disconnected components in a space? Can we find a path between any two points?), compactness (how many pieces of a space do we need to know its full nature?), separability (how much "space" is there between any two points/subsets of our space?). Topology finds the most basic notion of what continuity means, and important theorems from elementary real analysis turn out to be purely topological, such as the extreme value theorem (compactness), or the intermediate value theorem (connectedness).

Once you introduce a metric to a space, you can start defining lengths and curves, and define interesting ideas, such as minimizing curves, integration over abstract spaces, curvature, etc.

Then you find out that there are surprising connections between these geometric and topological ideas in theorems like Gauss-Bonnet or Hopf-Rihow.

should i into pure or financial math? also why did some faggot say that pure math is useful for financial math

>>9383757
Actually, algebra and number theory aren't about these... Even when doing these you must have some picture in your mind. If you're not in some sense "seeing" what you're working with you're lost. Even if the picture is pretty vague. It's often about coming up with the right "pictures".
Source: doing maths.

https://www.youtube.com/watch?v=tdOaMOcxY7U

I watched this and now I'm a genius???

>>9380557
/sqt/ <=> /mg>

how the fuck do i intuitively learn integrals

all I learned is some rules and u-substition -- 'methods'
my problem though is that
IT'S TRIVIAL
I wan't to have a deeper understanding of the integral more meaningful than
>it's just the area under the function bro
>it's the deriviative's opposite bro
>you can represent it as a reinmann sum
Because that sums up my understanding, and it looks so painfully plain

>>9385749
Learn measure theory then.

>>9382803

> personnel

>>9386166
What is confusing you?

>>9383432
Because there are three integrals.

For real though, they are pretty helpful in calculus III.

>cantor-dedekind AXIOM
worry.jpg

What are my fellow applied combinatorists planning to read during the winter break? I'm gonna start these.

>>9385749
That is what the integral is, though. Just because you type it like a retard doesn't make it any less meaningful.

>>9386250
>Fomenko Homotopic topology
>Baues algebraic topology
>Some cute stuffed animals
Great taste anon, you should check out Fomenko's mathematical impressions at some point if you're a fan of topology. The fact that your socks are different colors is triggering my autism though, so change that shit

>>9386250
Nice thighhighs.

>Scholze confirms Mochizuki is a hack

https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/?psincomments#comment-4619

>>9386754
>claim he confirms he's a hack
>all he says is he can't follow his logic

>>9386250

>wears homosocks
>reads homobooks

/sci/ is a Christian board. Go be a sodomite somewhere else

>>9386759
His claim is that there is no logic.

>>9386759
if the top people in the world can't follow the logic it's safe to say the logic is unfound. See Brian Conrad's reply a bit further down.

>>9386765
>His claim is that there is no logic.
No it isn't: "I am pointing out that I am entirely unable to follow the logic"

>>9386776
It's difficult to follow what is not there.

>>9386768
>if the top people in the world can't follow the logic it's safe to say the logic is unfound. See Brian Conrad's reply a bit further down.
Yamashita seems to be able to follow it

>>9384861
>>9386213
>>9386759
>>>/r/taiwan/

>>9386250
What are the prerequisites for the right one topology-wise? It seems to go faster than Spanier.

>>9387139
Not him but I've read the book, it really only assumes basic topology and algebra

>>9383237
>At least you realize your own retardation

>>9387161
lay off the carrots bugs...

How do distance metrics work in barycentric coordinates?

>>9386801
What does all this shit like "nubmer fields" mean?

>>9387462
Take an algebra course.

>>9386250
Tell me about applied combinatorics.

>>9387462
>What does all this shit like "nubmer fields" mean?
https://en.wikipedia.org/wiki/Algebraic_number_field

>>9382756
>>9382760
This is delusional thinking. Once you start laying down unquestionable axioms and start logically deducing results you've quit thinking and you've begun turning the crank of determining what they imply. At best, really good at following rules. You're useful though to the creative, intuitive type mathematicians who don't have the patience to be bothered by this menial work. So, at least some day you might have the privilege of working with some really brilliant guy and picking up his sloppy seconds and making them "rigorous".

>>9387663
>working with some really brilliant guy
or really brilliant girl

>>9386376
My animal friends keep me company while I read. I often have them on my lap, and I ofter stroke the polar bear like Blofeld strokes his cat while thinking. If you mean some book by him, I can try to find it, but if you mean his drawings, they are really cool. I remember seeing something similar in some gallery once, but sadly I can't remember when or by whom, as it was ages ago. Sorry for the symmetry violation.

>>9386740
Thanks. They are good for keeping your thighs warm in the coldness of winter.

>>9386763
My future accomplishments will be dedicated to Tengri.

>>9387139
I haven't had much time to read it yet, as I had to wash my dishes and do some general tidying up, but the beginning seems like a quick review of homotopy theory's basics like homotopy equivalence, loop spaces and suspensions, homotopy groups, and stuff like that. Therefore, the prerequisites so far have been very minimal, just a little bit of topology and algebra, but it helps to know what those homotopy thingies are, so you won't get scared while on board the definition express. The illustrations are G O R G E O U S! Reminds me of the Topological Picturebook, a nice read.

>>9387589
The one true form of mathematics.

>>9387670
Girl (male)*

>>9387676
Basically Fomenko made a book called mathematical impressions that's full of his drawings (about two hundred pages, took a pic), much like the ones in homotopical topology. If you're interested in the intersection of combinatorics and more abstract mathematics you also may want to check out
https://www.youtube.com/user/federicoelmatematico/playlists
https://arxiv.org/pdf/hep-th/0408145.pdf

>>9387676
Can you please tell me more?
I give this as tribute.

Imagine doing topology without drawing any pictures.

>>9387768
Possible, just not nearly as fun. Just Monika, best girl, only girl

>>9387807

>>9387768
>>9387807
>>9387860
>reading reddit garbage

>>9387860
Strange Yuri doesn't like Zorn's lemma considering I'm sure she loves Dedekind Cuts

>>9387872
Yuri doesn't want to stop cutting.

>>9380858
Question: How did your shitpost get to us error free?

>>9387919
My brain uses error correcting when it transmits posts.

>>9387768
this is unironically possible. we have a mandatory course called simply "topology", but only point-set topology is covered and then shit like uniform spaces. it's absolutely possible to ace that course without realizing that you're actually doing something related to geometry.

>>9387768
Do diagrams count as pictures? Because if so it would be impossible.

>>9387422
pls respond

>>9388194
Those are the only kind of pictures which actually matter.

>>9387754
I see. I could try to find the book somewhere! I'm not really into combinatorics per se, but it pops up everywhere, so I could check those links out later. Thanks!

>>9387759
Assuming you are referring to applied combinatorics, its clandestine nature is only revealed to the chosen few. If you don't already know it, then pic related.

>>9387919
That was actually a real question I was wondering, and it was enough to crumble my confidence completely. Now I've recovered.

Let M be an even dimensional smooth manifold.
I want to find an example M such that "Kahler cone ≠ symplectic cone" with non-empty Kahler cone, satisfying the following conditions:
M admits a Kahler structure.
ω is a symplectic form on M.
There is no Kahler structure (M,ω,J) such that [ω]=[ω]∈H^2 (M;R).

>>9387422
>>9388195
You're gonna need to clarify things then, when using barycentric coordinates you're already using the euclidean metric, what else are you asking about
>>9388305
Oh, I assumed you were into combinatorics since you joined the "discussion" about it. What math are you interested in? Algebraic topology?

>>9388622
Algebraic topology and cat theory meow. They support one another nicely and make each other's stuff more intuitive. And I was just teasing the people who get triggered by combinatorics and applied math, but, combinatorial arguments do indeed pop up in a lot of places, so it is good to see how it can be applied to other stuff.

>maths general
Fucking losers, math is literally the worst subject to major in
t.physics major

>>9388650
>Algebraic topology and cat theory
Then you chose one of the best books, based fomenko makes some stellar topology and geometry books. Tom Dieck also has a pretty good book that includes some homological algebra in the mix

>>9388694
I liked Rotman's books on AT and homo stuff, too. I think his way of writing is easy to follow.

>>9387589
Just take a look at statistical physics (e.g. the last section in Aigner's "Course In Enumeration").
You may also like experimental design (finite geometries are indeed of a combinatorial nature). Combinatorial Optimization (look at Korte and Vygen's book) is nice, too - shortest path algorithms are obviously applicable in "real" world.
Enjoy learning combinatorics!

Is there an operator for multiplying infinitesimals, kind of like how the integral is made to add up infinitesimals.

>>9388721
Thought about that once, too... the infinitesimals should then be something like 1+dx (instead of dx).
But I ended up with another way of calculating
[math] \exp (\int_a^b \log f(x) dx) [/math]...

>>9388622
How would I calculate a vector norm in barycentric coordinates?

>>9388740
You're correct in a sense though. You can think of a Lie group whose "infinitesimal" about the identity lies within its Lie algebra. If the Lie group [math]G[/math] acts on a manifold [math]M[/math], then there is a one-to-one correspondence between the Lie algebra [math]\mathfrak{g}[/math] of [math]G[/math] with the vector fields [math]\xi \in TM[/math] of [math]M[/math] by the action [math]a \mapsto a_\xi[/math], where
[eqn]
a_\xi(x) = \left[\frac{d}{dt}\exp(t a)\cdot x\right]_{t=0},[/eqn]
and under some technical conditions this actually generates all actions of [math]G[/math] on [math]M[/math], so you can WLOG look at Lie flows by the vector fields [math]a_\xi[/math] instead of the group elements themselves, which are usually harder to manipulate (the former is linear while the latter is multiplicative).

>>9388795
Can you explain point symmetries of ODEs and symmetry generators.

>>9388795
It took me a little bit to get a feeling for what you mean. I'm not really familiar with Lie group theory (although when making my post I had a vague feeling that there might be something going into this direction)
Nice generalization, indeed. Thanks!

>>9388740
Yeah that's why this generates a rotation 90 degrees and spins you in a circle around the origin in the complex plane
[eqn]\lim_{n\to \infty} \left(1+\frac{ix}{n}\right)^n=e^{ix}[/eqn]

>>9388755
For a triangle ABC and point P in ABC that defines the point you're looking at using barycentric coordinates there are a few things you can do. For one if your triangle is embedded in some vector space then if you know the coordinates of your vertices you can get the coordinates for your point using the formulae on slide 21
https://andreask.cs.illinois.edu/cs357-s15/public/notes/section_m_notes/CS357Lecture3_Norms.pdf
After which it's just your standard norm calculation
If not and you're just looking at a triangle as some affine space then you can assign a vector to the point P with whatever length you want.

>>9380557
Topology is the devil's math.

>>9389088
Wrong. It's the only math worth doing.

>>9388721
>>9388740
https://en.wikipedia.org/wiki/Product_integral#Type_II

>>9388721
>>9388740
>>9389476

My friend and I explored this once. We decided to call it a "mintegral" (multiplicative integral).
We basically used what >>9388740 came up with.


If we denote [math]\mathcal{M}[\math] as the mintegral operation you get:

[math]\mathcal{M}a=ca^x[\math]

[math]\mathcal{M}x=c{x^x}e^{-x}[\math]

[math]\mathcal{M}e^{f(x)}=ce^\int{f(x)dx}[\math]

But what most interesting is when [math]f(x)[\math] goes negative. Here, you're basically multiplying a negative number to another negative number over and over, so it should oscillate between positive and negative, and it seems like it does, but in a weird non-defined way. For example, if [math]f(x)=-1[\math] we get:

[math]\mathcal{M}(-1)=e^{\int{\log(-1)}dx}=e^{\int{\log(1)+in\pi}dx}=ce^{in\pi x}[\math]

Where [math] n [\math] is an integer. I didn't know what to make of it past this, though it looks neat that it does what you'd expect.

>>9388721
>>9388740
>>9389476

>Fuck me and my slashes going the wrong way

My friend and I explored this once. We decided to call it a "mintegral" (multiplicative integral).
We basically used what >>9388740 came up with.


If we denote [math]\mathcal{M}[/math] as the mintegral operation you get:

[math]\mathcal{M}a=ca^x[/math]

[math]\mathcal{M}x=c{x^x}e^{-x}[/math]

[math]\mathcal{M}e^{f(x)}=ce^\int{f(x)dx}[/math]

But what most interesting is when [math]f(x)[/math] goes negative. Here, you're basically multiplying a negative number to another negative number over and over, so it should oscillate between positive and negative, and it seems like it does, but in a weird non-defined way. For example, if [math]f(x)=-1[/math] we get:

[math]\mathcal{M}(-1)=e^{\int{\log(-1)}dx}=e^{\int{\log(1)+in\pi}dx}=ce^{in\pi x}[/math]

Where [math] n [/math] is an integer. I didn't know what to make of it past this, though it looks neat that it does what you'd expect.

>>9389614
W/e close enough

>>9388922
Suppose we have a linear operator [math]\mathcal{L}[/math] acting on a separable Hilbert space [math]\mathcal{H}[/math], and put [math]\mathcal{W} = \{f \in \mathcal{H} \mid \mathcal{L} f = 0\} \subset \mathcal{H}[/math]. Suppose there exists a group [math]G[/math] acting on [math]\mathcal{H}[/math] such that [math]\mathcal{G}\mathcal{W} \subset\mathcal{W}[/math] by left translation. When this happens the solution space [math]\mathcal{W}[/math] (or equivalently the equation [math]\mathcal{L} f = 0[/math]) is said to have symmetry [math]G[/math]. Given a representation [math]\rho[/math] of [math]G[/math] corresponding to a representation space [math]V[/math] of [math]\mathcal{H}[/math], the group [math]G[/math] acts by the adjoint representation on the linear operator [math]\mathcal{L}[/math], hence [math]\mathcal{L} \rightarrow g\mathcal{L}g^{-1}[/math] such that [math]g(\mathcal{L} f) = (g\mathcal{L}g^{-1})(g(f)) = \operatorname{Adj}_g (\mathcal{L})(g \cdot f)[/math].
From this we see that if [math]\mathcal{W}[/math] has symmetry [math]G[/math], then there exists a subgroup [math]H[/math] such that [math]h \mathcal{L} h^{-1} = \mathcal{L}[/math] for [math]h\in H[/math], namely [math][h,\mathcal{L}] = 0[/math] (in the adjoint representation). This group [math]H[/math] is called the stable subgroup of [math]\mathcal{L}[/math]. In some cases [math]H = G[/math].
Therefore an ODE [math]\mathcal{L}f = 0[/math] has symmetry [math]G[/math] if [math][g,\mathcal{L}] = 0[/math] for [math]h \in \operatorname{Adj}(G)\cong \operatorname{Lie}G[/math].

>>9389751
Catching the typo is left as an exercise for the reader.

>>9389751
>>9389766
Shouldn't the commutator in the last line have an h instead of a g?

>>9389772
The other way around.

>>9389772
Err actually you're technically right but it should be a g in the following sentence for consistency of notation.

>>9382738

I never understood the point of abstract algebra either until I read this book.

A First Course in Abstract Algebra: Rings, Groups and Fields - Marlow Anderson, Todd Feil

All the abstraction is motivated by very basic number theory and it is designed to be easy to understand.

>>9389902
>All the abstraction is motivated by very basic number theory and it is designed to be easy to understand.
In other words it's for brainlets?

>>9389614
Should be [math]n[/math] is an odd integer

https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/

>>9390092
Already read it, not terribly surprising, Tao announced blow up results for a version of Navier stokes a couple years back. Blow up solutions for nonlinear PDEs aren't uncommon. The results established in those papers aren't strong enough to actually disprove the millennium problem though

>>9389911
Compare Jacobson's and Humphreys' books on Lie Algebras - both cover exactly the same theory and do the same proofs. But Humphreys' book is structured in a much clearer way - you might immediately think Humphreys is "easier".
Hopefully you understand what I want to tell you.

>>9389614
I think the problem is that there is neither a globally defined [math]n[/math]-th root nor a globally defined logarithm.
If you extend your integration to complex-valued functions instead and choose a branch of the logarithm for your integration you can resolve this problem - at the cost of making your integral non-unique.
I think, there might be some nice topological (de Rham cohomology?) obstruction for that but I should think a little bit about this before posting bullshit.

>>9389911
Indeed, only morons are interested in learning. Intelligent beings prefer to posture in effortless reciting of definitions.

>>9390406
This, unrigorous "intuitions" (preferably without proofs) have been shown to be the best way of learning.

The physical intuitions meme seems to have pretty long legs. When will it end?

>>9390339

Yeah, this apparently gives you have an arbitrary choice for [math]n[/math] in [math]e^{in\pi x}[/math].

I mean, you're taking the limit as you multiply [math]-1[/math] to itself infinite times while taking the exponent of each [math]-1[/math] to zero, so this choice is probably an artifact of multiple roots between integers.

The integer values should give you all the same solutions though, and exactly what you'd expect. If most generally we have:

[math]ce^{i(1+2n)\pi x}[/math]

Where [math]n[/math] is an integer, than if x is an integer it just reduces to (since [math]e^{i2n\pi x}=1[/math] because xn is an integer):

[math]e^{i \pi x}=c(-1)^x[/math]

Which is exactly what you'd expect, and holds for any choice of [math]n[/math] when [math]x[/math] is an integer.

>>9390429
Spoken like a kid who just took his first into to proofs class. Mathematicians constantly talk in terms of intuition, knowing that they can fill it in. Nobody learns just by throwing themselves into rigor without trying to dig into the ideas beneath.

>>9390479
>took his first into to proofs class
I don't like or use "proofs" since I find them unnecessary in mathematics.
>Mathematicians constantly talk in terms of intuition
Indeed, and the best of them realize that intuition can easily make up for proofs.
>trying to dig into the ideas beneath.
Yeah, I like deep and beautiful math like [math]e^{i \pi}[/math] too. Sometimes I try to think deeply about these things using my intuitions to dig into them.

>>9390479
>Nobody learns just by throwing themselves into rigor
Speak for yourself. Not everyone is mentally challenged.

>>9390479
>Nobody learns just by throwing themselves into rigor without trying to dig into the ideas beneath.
Rigor is overrated anyway. It's not relevant to most of mathematics anyway.

>>9390524
>>9390548
I'm impressed by what's either your lack of reading comprehension or your ability to completely change someone else's words into something easier for you to refute. In any case, congrats for being such big strong mathematicians. I bet you even did Rudin all by yourselves!

>>9390553
>I bet you even did Rudin all by yourselves!
Rudin is a meme.

>>9390553
>In any case, congrats for being such big strong mathematicians.
I'm not a "mathematician".

Straying into role-playing territory...

>>9390553
>I bet you even did Rudin
I wouldn't even touch garbage such as analysis.

We're officially in role-playing territory.

>>9390583
analysis > algebra

>>9390583
autistic plebian

>>9390632
>>9390655
You literally have shit for brains, don't you?

>>9390632
analet

>>9390632
No.

What books should I read for about 2-3 years to develop my intuitions before actually touching real math? Preferably something which motivates all of the (nonexistent) abstraction using very basic number theory and is designed to be very easy to understand.

>>9390784
Lol at this smug, petty child

>>9390784
>What books should I read for about 2-3 years to develop my intuitions before actually touching real math?

Quick question: should the last equal sign and congruent sign be switched?

>>9390934
No, it's just awkward formatting. You have a^(phi(m)) congruent to 1 mod m, but the author just put the (mod m) at the end of the entire string of relations.

>>9390938
Could you explain what the author is doing from the 3rd to 4th step? Thank you
>t. brainlet studying number theory

>>9390955
Euler's theorem lol

How would I best go about learning about lattices and lattice basis reduction specifically?

I don't study Maths or have a strong background in the area.

>>9390998
Go to CS department, they use plenty of lattice theory and can explain in brainlet terms.

>>9391220
>they use plenty of lattice theory
Not the case for my univeristy.
I've already tried what you're proposing.

Could someone give me the mathematics book guide? Wanna get into logic and category theory.

Some showe a single instance were the lebegue integral is used in probability theory?

What are some important theorems in number theory? Elementary-level if possible

>infinitude of primes
>Chinese remainder theorem
>Fermat's little theorem
>the Euler one
>Bezout's identity (not exactly a theorem but)

>>9391670
if a = bq + r, gcd(a,b) = gcd(b,r).
Euler did this, I was impressed.

>>9391541
>Boolos, Burgess, Jeffery Computability and Logic first
>Awodey Category Theory next
>Goldblatt Topoi a Categorial Analysis of Logic

>>9391674
That's Euclid's method

>>9391541
http://www.logicmatters.net/resources/pdfs/Appendix.pdf
http://4chan-science.wikia.com/wiki/Mathematics

>>9391677
Lmao Euclid sorry
but isnt it weird that that actually works?

>>9391670
>What are some important theorems in number theory?
https://en.wikipedia.org/wiki/Quadratic_reciprocity

>>9391676
>>9391678
Thanks. What do you guys think of that copypasta image of the guide on learning maths?

>>9391688
Which shitty guide?

>>9391694
like the one that's like learn spivak then learn proofs (the guide has like 3 proof books)

>>9391704
>(the guide has like 3 proof books)
Then it's a waste of time.

>>9391704
Probably, any image with baby spivak is a meme.

I'm bored AF
Is there a transformation matrix that turns a hypercube in n-dimensional Euclidean plane into a hypersphere

>>9391670
>not exactly a theorem but
What did he mean by this?

>>9391946
Not a matrix (for a matrix would transform lines into lines)... but you can simply use the vector norm by putting
[math]f: \mathbb{R}^n \setminus \{0 \} \to \mathbb{R}^n \setminus \{0 \} [/math] ;
[math] v \mapsto \frac{v}{|v|} [/math].
Then the "surface" of a hypercube becomes a hypersphere.

>>9391946
do you mean linear transformation ? well linear transformations map lines to lines which means that polyhedra necessarily map to polyhedra, so what do you think ?

>>9391715
Why? I heard those books are pretty good since they try to motivate the abstraction using basic number theory.

>>9391946
>projecting

>>9389155
well kinda ... its another way of coming at "it"

that just happens to have a bit more asthetic

>>9390094
of course NavStokes are a statistical approximation of a quantum continuity (see the problem) and so they don't work at low scale, probably at all.

sup piggots

>>9392085
Wouldn't [math]\mathbb{R}^n[/math] have to exist for arbitrary [math]n \in \mathbb{N}[/math] first though?

>>9392096
>I heard those books are pretty good since they try to motivate the abstraction using basic number theory.
Just read Hungerford's "Abstract Algebra: An Introduction" if that's what you're looking for

>have a math degree
>stuck in a boring ass programming
>want to do more math in grad school
>have a dogshit GPA

HOW THE FUCK DO I INTERPRET EQUIVALENCE CLASSES GEOMETRICALLY. IT'S BEEN AN ENTIRE YEAR AND I STILL DON'T UNDERSTAND

>>9393170
>HOW THE FUCK DO I INTERPRET EQUIVALENCE CLASSES GEOMETRICALLY. IT'S BEEN AN ENTIRE YEAR AND I STILL DON'T UNDERSTAND
Give an example of an equivalence class you want to interpret

>>9393184

>>9393170
fuck modulo

>>9393170
>INTERPRET GEOMETRICALLY

>>9393170
Lines and planes, or pretty much any level set, are equivalence classes.

>>9393170
Equivalence classes of what?

I'm looking at course schedule for next terms. I'm given the chance to choose between Applied Complex Variables or Complex Variables. Names should speak for themselves. I can't choose between the two. I looked at the syllabus for both and Applied allows for an index card to be brought into exams which turns me off of it since that suggests its just for engys who don't want anything more than the tools that math provides. But my course load for that semester is already pretty heavy. Should I just take Applied or stick to the foundations?

>>9380557
I find it extremely hard to abstract things to a geometric reasoning. Calculus II for example, was all flowers and baby penguins until the professor started with the geometric bullshittery.

Check my prime

>>9393502
missed by 55, fuck

Also, found out by talking to a third world math PhD that he didn't properly understood concepts like surjection. He failed me in his class.

>>9393489
How did you get understand any of calculus without geometric intuition? Sounds like you were just good at doing what you're told. Also
>abstract to geometric reasoning
What the hell are you talking about? Calculus has always been about geometric reasoning.

>>9393486
What's on the syllabus for your complex analysis course? When I took it we went along with parts of ahlfors book and looking back the material could've been compressed into a few weeks without making it difficult. Find out the book and what chapters and that'll give you a good indication for how hard the course is but from what I've seen most undergrad complex analysis courses are pretty easy.
>>9393502=2*433*10847

What am I missing in the definition of. Closed plane curve that prevents a triangle from being a contradiction to the four vertex theorem?

>>9393508
I find it easier to just interpret it all as purely algebraic concepts. I know the origins of it, but today calculus is well defined in algebraic terms.

>>9393557
But calculus is very much non algebraic. The idea of a limit is as non algebraic as it gets.

>>9382809
1) I guess you just spent more time on analysis than on algebra
2) No one is learning abstractly, you should study concrete examples first and then with enough experience abstract your knowledge, maybe that's what fucked you up with abstract algebra i dunno

>>9393517
Applied uses Churchill and Brown 8th edition. First 10 chapters. Focuses on complex, analytical, and elementary functions, residues, and conformal mappings.

Complex Variables uses S.D Fisher published 1999. First 3 chapters. Focuses on the complex plane, basic properties of analytical functions, and mapping of analytical functions.

>>9383368
Calculus 1&2
Vector Calculus
Ordinary Differential Equations
Matrix Algebra
watered down Analysis/"Advanced Calculus" (eg. Gaughan, Abbott, Bartle)
watered down Algebra (eg. Pinter, Gallian, Fraleigh, baby Hungerford)
>electives (pick ~6)
Numerical Analysis
Linear Programming
Applied Linear Algebra (Strang)
Partial Differential Equations
Game Theory
Probability 1
Probability 2 / Stochastic Processes
Mathematical Statistics
Mathematical Finance
*Biological Modeling
*Mathematics for future High School teachers
*watered down Combinatorics (eg. Tunker, Niven)
*watered down Graph Theory (eg, Trudeau, Chartrand)
Integral Transforms / Fourier Methods / Signal Processing
*Discrete Mathematics (for CS majors) 1 & 2

The easiest courses and the ones always done by CS double majors are marked by *

Should I take linear algebra or Differential equations for my 2nd year level courses
I only need one of them for my Bachelor's

>>9393614
Bachelors in what, Chemistry or CS?

Take both.

>>9393614
Linear Algebra for sure

>>9393616
Dual major in Maths and CS

Is there a way of numerically approximating roots of high order (>5) polynomials in your head?

>>9393672
binary search

>>9393576
Okay, I looked over both books, each of them look very easy, so I don't think you'll have a problem with either class, but just in case I've linked the two books, try giving them a skim and see which one you feel comfortable with
http://math.unice.fr/~nivoche/pdf/Brown-Churchill-Complex%20Variables%20and%20Application%208th%20edition.pdf
https://archive.org/stream/complex-variables-2ed-dover-1999-fisher/complex-variables-2ed-dover-1999-fisher-s-d#page/n3/mode/2up

>>9393717
Thanks!

>>9393663
Aren't math majors required to take both diffeq and linear at your clown college?

>>9393701
>not secant method
>not Brent's method
>not Newton's method
>not Halley's method

>>9393545
Halp

>>9394029
Triangles aren't smooth.

>>9394204
What if I smooth out the discontinuities at the vertices— how do I end up with 4?

>>9394216
The middle of the bottom edge and the vertices.

why is everybody working on higher category theory high as fuck

Like how can you spend so much time proving something is n-trivially trivial because its n-trivial

If a polynomial takes integer values for an infinite number of integer arguments, must it take integer values for all integer arguments?

If it must, how can I prove it?

>>9395025
How about [math] f(x) = \frac{x}{2}[/math] which takes integer values at any even integer?