/sci/ - Science & Math

Any1 here into integrable systems? Its an interesting and diverse subject

>want to get into math
>read textbooks
>try exercises
>brain fried
>give up

Holly fuck i'm too autistic for maths, why can't mathematicians build a standard for notations and definitions. I fucking hate reading graph theory, every single book use a different notation and slightly different definitions, triggering my autism. How I'm I supposed to work and gather results if there is no standards? At least computer scientists figured out this shit. Also fuck LaTeX, every LaTeX code longer than 4 lines is unreadable because of the shitty syntax. I will switch to cs whenever I can.

>>11235821

Shit's hard sometimes

>>11235839
>cant understand differences in notation and definitions
>graph theory
>I will switch to CS whenever I can
good, no one in meth will miss u

>>11235839
cs is fag shit, believe me, and its full of normies

How do I obtain the magistral patience to learn shit like control theory, C++ and get a job in industry?

>>11235816
the biggest question in that field is the proper definition of "integrable".

Why is math so cool bros?

>>11236179
It's only cool in comparison to your miserable life.

>>11236179
>>11236205

when did math become cool

In Macaulay2, how do you interpret a bettiTally of a simplicial complex?

https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.12/share/doc/Macaulay2/Macaulay2Doc/html/___Betti__Tally.html

Specifically, what is the information that this returns? I expected it to just give you a sequence of betti numbers.

I wasn't going on 4chan for almost 2 days. What did I miss in the world of math?

>>11235839
I actually have a dream of writing a huge treatise on standardizing all of mathematics, from the initial definition and symbols for effective computability (all mathematics is a computation) and then building up the standard axioms and finally notation to be used for all fields from the foundation in computability theory.

are there any set theories that make meaningful progress on the continuum hypothesis/GCH?

Also, are there any set theories that characterize uncountable sets further then ZFC?

It just seems like most uncountable sets are unremarkable, really interested to see if someone tried to look into higher cardinals

>>11236433
Sounds nice.
What's your take (apart from you liking computability)?

>>11236440
In a way, every new axiom restricts the realm of possibilities.
Maybe this guy gives you some ideas
https://en.m.wikipedia.org/wiki/Robert_M._Solovay

what are wheels?

>>11236497
fields that allow you to divide by zero

>>11236433
fuck off you computer science faggot. go make your own thread you plebian.

>why yes, I did years of research on wheels specifically to find the most relevant, universal and interesting wheels, just so I could name them hot wheels, how did you figure it out?

>>11236433
Isn't this what Bourbaki tried to do?

>>11236525
Bourbaki had
https://en.wikipedia.org/wiki/Epsilon_calculus
which is I think a choice principle of steroids.
Foundations such as Grothendieck conceived also have, roughly, a ZFC universe for every set (so that you can talk about big categories in your set theory)
Related is
https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory

I.e. this is far from questions of computability

The biggest math library is
https://en.wikipedia.org/wiki/Mizar_system
but the framework isn't nice.
There's similar such projects and websites.

Dependently typed languages like Coq and Agda have some math libraries, but those, then, are too far on the constructive sight for what that guy wanted.

>>11236440
Large cardinals cannot decide CH. Suppose large cardinal P implies CH, take any model of ZFC+P and force over that with Cohen forcing that breaks CH. This is a contradiction. This is true since Cohen forcing perseveres cardinals. I wouldn't say this says that uncountable sets are unremarkable, it just shows they are very sensitive to the model theory one is assuming. This makes them very interesting in my opinion. There are some axioms that are proposed that would settle CH, like Woodin's unlitmate L for example, but I really don't know much about it.

>>11236570
Forgot words, the large cardinal property P will hold in the forcing extension since Cohen forcing preserves cardinals, thus CH and not CH. This is the contradiction.

What's the best intro topology book? Is Munkres a meme?

>>11236736
i was slowly going through it now and it's decent, there is the topology without tears book but that doesn't seem to have any stuff about fundamental groups and such

Whenever I can't solve a problem I get angry like a sperg which makes everything worse. How can you stop that? I want to enjoy the struggle

>>11236523
This didn't actually happen right?

Last final was tonight. I think I did well.

I need a 93 or better to keep my 4.0 GPA ;_;

>>11237201
Most people lose their 4.0 eventually, focus on doing the best you can and LEARNING all you can not maintaining a perfect gpa

>>11236736
Unironically most of munkres shouldve already been part of your analysis classes. And then you should be able to pick up a book on algebraic topology, e.g. Hatcher

>>11237201
trying to get a 4.0 is one of the least productive things you could do in college

>>11237616
>>11237232
I just want to stay above 3.5. If I do that, I get a 40% discount on my tuition.

>>11237657
Besides when it comes to money, good grades aren't essential, see this thread right now for example >>11237481
Many people make their livings just out of blowing their skills out of proportions and making promises that they can't currently fulfill (kinda Ponzi scheme) — actual skills and self marketing are more important than silly pedigrees in the long run.

>>11237201
>>11237232
>>11237616
>>11237657
>>11237683
This is a math general, /mg/.
It's questionable whether such discussions have their place on /sci/ at all, but certainly not here.

>>11238078
>munkres, rudin
>taste of "real math"
jesus fucking christ burgers are retarded

>>11235821
>Want to run marathons
>Try running once
>Legs hurt
>Give up

This is what you sound like

>>11238102
>another insecure yuropoor compulsively bringing up that he was doing measure theory in 3rd grade at every possible opportunity

>>11238078
Why are there two proof books and why is it full of discord tranny anime shit?

>>11238132
how would I know?

>>11238132
you a transphobe?

Are there any books that are only available in Russian/Ukrainian and of a didactic quality that doesn't have anything like it in anglophone texts?

>>11238182
no
same answer for any other language

>>11238054
3.5 and 40% are numbers (maths)

>>11238204
elon lages books are great and they are in Portuguese.

>>11238145
>>11238142
>too stupid to write it in a readable colour
cringe

>>11238078
Need haskell or Scala book and wrote monads tutorial!

based
https://www.fuw.edu.pl/~kostecki/sdg.pdf

>practice question
>awards a mark for correct answer
>awards a mark for using a method not included in the textbook

why do they do this

>>11238102
the chart is supposed to be a joke

>>11238220
they might be great but I doubt you can't replace them with books written in English.

>>11238383
can you explain what's the idea of this paper to a 2-year undergrad?
he provides some alternative framework for differential geometry?

>>11238465
>he provides some alternative framework for differential geometry?
Provides is incorrect. SDG is an old field.
If you want to know more about it, just read this https://libgen.is/book/index.php?md5=B1334DAF88F3ADDE5EB5C0760B3F450C
A good bit of the text is done without reference to topos theory, so just stop when you can't understand things anymore.

>>11236502
sounds fucking rad

>>11236497
>>11236502
Isn't this what Bourbaki tried to do?

>>11236433
>>11236558
Good fucking luck friend.
You will need it.
The greater Math community is hardly known for agreement, which is why you still get confusion over order of operations even in 2019, and non-SI + SI confusions IN SCIENCE COMMUNITIES all the damn time.

A true, universal consistent representation of math from the simplest concepts all the way through each advanced, discrete field is the dream.
This is why I always laugh when cunts say "math is the universal language", is it fuck, far from it. Most of maths is human-based conventions. Literally the "most used wins" scenario.
The most basic maths isn't even standard from one country to the next in some cases!
Order of operations alone has like 10 different (ENGLISH!) names and some of them are inconsistent with each other.
Never mind representations of large numbers, which there are 2 hugely competing formats. (one that is good, and one that is shit but older)
There's likely considerably more "standard" representations of Maths as there are date-time conventions, and you know how retarded those can get.

You'll definitely need all the luck you can get.
It's the Windows OS problem - people hate it but it works and is popular. Also the x86 problem. Also the Chrome problem. Also the everything-popular-ever problem.

>>11235839
>i will switch to cs
You mean the code-monkey class intended for gay retards who talk about Star Wars?

>>11236314
It was always cool just like God was always cool, atheists and mathlets are too retarded to understand either of them

>Yes I do view mathematics as a manifestation of the divine and treat its study as spiritual practice, why?

>>11238078
thanks

>>11237575
My analysis class did basic open / closed set stuff but only in the context of the standard topology. I assume there's more I should know before going more into fields that build off topology, no?

Is there a good way to factor a dirac delta function? I've been playing with QM and I came up with an interesting energy potential:
U(x,t) = Lambda*Delta(kx-cos(omega*t)). Can I split U into a function of x and one of t?

>>11238078
approved

>>11238816
i mean you can go over general point set topology but it's basically all the same thing, most things from metric space topology regarding open/closed/compact/connected generalize effortlessly. I guess going through separation axioms is useful, and infinite product topologies and quotient topologies.

>>11238078
you omitted being a tranny.

>>11238806
why is it always cs faggots who try to study category and type theory. do they want to feel superior to mathematicians by studying the "foundations" but are too stupid to into set theory?

>>11238816
You should have good understanding of quotients/pushouts and manifolds before starting algebraic topology. And an intro to abstract algebra, of course.

>>11238816
>>11238965
Ok, I will skim through Munkres and try to learn those things. I already have a strong background in algebra so that shouldn't be a problem.

what are wheels?

>>11238963
Type theory doesn't really study the same questions as set theory, and is obviously more relevant to computer scientists.

>>11236433
many people have had this idea before and it never works

>>11238963
CS people usually mean "type systems" when they talk about "type theory" in my experience

You have 44 slots and 44 pieces. You have to place the pieces on the right slot. What is the average amount of trials (placements) you have to do to place all the pieces correctly? A perfect game would take 44 trials and the worst game 990 trials.

>>11238986
>literally the Haruhi problem

>>11239002
What makes you attention whore every thread with le epic anime reactions? Is it because you're a tranny or you are just starving for attention? Why don't you get a trip so I can filter you? Honest questions.

>>11239002
you seem to be struggling with basic combinatorics

>>11239006
Why don't you go to a website where everyone has pretty little nicknames and you can just filter whoever you don't like?

>>11239023
You should go, since you don't seem to understand the point of an anonymous website.

>>11239032
>getting mad at touhou poster

>>11239002
Permutation is involved but no supermutation.
f(1)=1
f(2)=2.5
f(3)=4.5
f(4)=8+(1/6)

>>11239040
Oh, yeah.
But what you ultimately need to do is consider all possible routes of swaps that start from some given arrangement and go through all arrangements without repetition before stumbling across the solution.
It sort of has the same "flavor", you know?

I'm looking for a good textbook in PDE theory. Any recommendations?

I've done all of the methods stuff (calc variations, Green's functions, maximum principle, transforms, eigenfunction expansions, etc.) and want to learn more about the pure math side of things.

>>11239063
Evans.

>>11239056
Yeah and it gets bloated fast. There are apparently n! possible games you have to consider.
1 [1]
2 [1,1] [2,1]
3 [1,1,1] [1,2,1] [2,1,1] [2,2,1] [3,1,1] [3,2,1]
etc.

>>11239056
Kill yourself, asuka.

>>11239070
n! ?
Wait, correct me if I'm wrong, but it's like this:
>you have n slots and n different pieces
>you need to settle them into a precise arrangement with all slots filled
>every turn, you can either put a piece in a slot, or swap the pieces in two different slots
I thought that was why the best case was 44, and if you have infinite amounts of each piece it's pretty simple.

>>11238078
Add follow this jew
https://twitter.com/johncarlosbaez

>>11238078
just

>>11239111
Wtf I hate category theory now!

>>11239114
https://twitter.com/Daniel_Nikpayuk/status/1205039841521483777
Me too!

>>11238078
gayest shit ever

>>11238078
memes

>>11239149

does she sell feet pics?

>>11239152
I don't know

>>11239161
This is to definitely the face of a building female. Not masculine at all

>>11238610
>God is cool

Even if he were real (and its perfectly possible) he’d be the biggest tightassed buzzkill to ever be.

>>11239161
>>11239152
>>11239149
>>11239120
>>11239114
>>11239111
>>11239107
Fucking go back there, cancer. Stop bringing shit some random human said on twitter.

Stupid question. Is an inner product a linear functional?

>>11239425
The inner product is defined on VxV while a linear functional is defined directly on V.

>>11239455
Thanks, the book I was using didn't specifically define it as one, I'm retarded

>>11239425
Not quite

Recommend me some cozy christmas reading like I'm an undergrad algebraist.

>>11239469
I'm reading Category Theory for Programmers rn. It's undergrad tier in terms of readability, but covers a large breadth of topics. Lots of comfy examples too.

[eq]1 + 0 = 0[/eq]
True/False? And why?

[eqn]1 + 0 = 0[/eqn]
True/False and why?

>>11239497
faggot

>>11239653
I don't have any recommendations for the mentally disabled though

>>11239617
>True/False
False

>and why?
It needs two to tango

>>11239306
but this random human talked about math, and the name of this thread is (literally) '/mg/ - math general'

>>11239469
aluffi algebra chapter 0

if youre comfortable with that stuff then either of

>morandi field and galois theory
>the first half of bosch commutative algebra and algebraic geometry, complementing it with the exercises from atiyah macdonald (the covered content is practically the same other than some minor exceptions, but a/m exercises are superior)
>a course on noncommutative rings by t.y. lam

>>11235839
>>11236433
Literally the only people who complain about notation are people who don't actually work in math at all. You fucks don't understand the first thing about what math is, you're just autistic little shits who want every excuse to not actually do math.
The most valuable thing I learned from grad classes was to be flexible in your notation, because everyone sees things slightly differently and math is all about communicating ideas and collaborating with others.

>>11239497
I went through a bit of that book a while back but didn't think it was too useful.

>>11240458
Thanks!

>>11240968
just so you know, morandi and aluffi have an internet errata that you should check out concurrently (and preferably in advance), and a/m has a community errata in math overflow that you should check out of you read it. The other two im not sure of, maybe they do too.

Are the betti numbers of:

A) the simplicial complex whose simplices are the edges of a graph (I.e., it will only have 1 dimensional simplices corresponding to edges)

B) the simplicial complex whose simplices are the cliques of a graph (which can have arbitrary dimensional simplices, the complete graph is a single simplex)

supposed to be equal?

>>11238204

>>11238204
I'm asking people who actually do have experience with those textbooks; either way, it isn't true: while certainly textbooks in English cover everything in material terms, the organisation and the presentation can differ wildly and make a textbook preferable to others.
I have a great example that is Prodi's "Analisi Matematica" which isn't akin to anything I've seen in English textbooks, the closest equivalent being Rudin (which seems to have more exercises, but doesn't have the same clarity and accuracy in presentation).

>>11241023
Did you try constructing a homotopy between those two simplicial complexes by contracting the cliques?

>>11241035

I know what a homotopy group of a topological space is but I don't see how you could "form a homotopy" or how any of those concepts apply here

>>11241023
No.

>>11241076

That is what I thought

I asked because I have to use Macaulay2 (which is quite shitty) to compute some betti numbers and it's giving me an output that doesn't make sense

(i.e. there is no betti number for number of connected components or cyclotomic number when fed a 1 dimensional simplicial complex )

Anyone has a good reference for oscilatory integrals that uses distribution theory?

>>11238963
Because type theory has little to do with set theory, but a lot to do with computer science

Can anyone recommend me a good Combinatorics book? I've only been reading my prof's book in my native tongue this far.

>>11241440
>Can anyone recommend me a good Combinatorics book?

>>11241440
Lovasz.

>>11241446
>>11241447
Thanks, it seems that my faculty's library has both of those covered, only one available at the time though.
Good thing I'm in no rush to read them.

>>11241446
>>11241447
Neither of these are particularly good suggestions. Stanley is the Bible of the field (in America, anyway), but suggesting somebody halfway through his first course on the subject try to sit down and read it is about as sensible as suggesting that a new church convert plow straight through the old testament.
Lovasz is a problem book. You're expected to already know combinatorics if you're reading it, and it's also almost entirely skewed towards Hungarian-style graph theory and goofy extremal problems.

>>11241440
Depends how serious you are. If you just want to look at some pretty asses Cameron and van Lint/Wilson are both nice books that are basically just a grab bag of cool results.
If you want something more systematic maybe take a look at Miklos Bona's book. It's a reasonably accessible undergrad intro and Bona is one of the better mathematical writers I've ever ran into.

>>11241534
>Lovasz is a problem book.
I was actually talking about the handbook.

>>11241549
Lovasz only ever means one thing to anyone I've ever spoken to, but whatever floats your boat. Reading a 2200 page encyclopedia from 25 years ago seems like a very reasonable course of action.

>>11241440
https://doc.lagout.org/science/0_Computer%20Science/3_Theory/Mathematics/Discrete%20Mathematics/A%20Walk%20Through%20Combinatorics%20-%20An%20Introduction%20to%20Enumeration%20and%20Graph%20Theory%2C%202nd%20Ed.pdf

This is basic

>>11238975

> category theory deals with morphisms, categories, functors, etc. and it's important for algebraic geometry or something
> Functional programming languages like Haskell often use words which sound like they're from category theory, like functor or monad
> Type theory refers to stuff like calculus or inductive constructions and theoretical frameworks for proof assistants


Why is there stuff like "Category theory for programmers"?

Is there actually any connection between category theory and computer science?

>>11241579
>why is there stuff like category theory for programmers
take a wild guess

>>11241579
Haskell doesn't just use words from category theory, the universe of haskell types IS a category (or at least a slight restriction of it is).

>>11241596
faggot

I still don't get the monty hall problem

>>11241598
>stating facts is bad
I neither use nor particularly enjoy haskell but keep seething, autist

>>11241596
>or at least a slight restriction of it is
translated: ok it's not REALLY category theory but if you really push hard you can still squeeze it into the hole

haskell is a cult

>>11241605
I believe it's a category if you removed shit like "undefined" (why that exists in the first place idk, my haskell knowledge is not extensive).

>>11241601
Please explain it to me asap.

>>11241610
undefined exists so that we can encapsulate the nondeterministic nature of computation - the program may return a value or keep running indefinitely - it's fixed point is undefined aka 'Bottom'.

>>11241611

>>11241565
>>11241534
It's a shame that it's not available in the lib - looks great and in line with what I've learned so far

>the undergrad anti-category zealot

If a number disproves the Collatz Conjecture, then let’s assume we’re dealing with the smallest such number, C. Therefore it must either grow to infinity without ever going below itself (since the numbers below itself all reach 1), or cycle. But if it cycles, then each number in the cycle has to be unreachable by elements outside of the cycle. So for each element in the cycle, its multiples of 2 are also in the cycle. So ALL of these elements must somehow never interact with non-cycle, collapsible numbers, which seems unlikely. Discuss

>>11241623
just get it off libgen

>>11241642
What if it falls into a cycle but isn't itself in the cycle?

>>11241618
what

>>11241648
Well that or just use the copy anon provided in this very thread.
I simply prefer them physical

>>11241611
The one you've picked has 1/3 chance of being correct, so naturally the alternative has 2/3 chance of being correct, since the probabilities sum one.

>>11241649
yes that’s right but it still means that the number doesn’t escape the cycle and can’t collapse to 1. So it’s not technically in the cycle but it ends up in the cycle.

>>11241642
we know that the cycle begins with an odd number C and ends with an even number, which must be divisible by 4, else it would grow beyond the cycle after 2 steps. We also know that 2C is in the cycle, and that 2C can not be represented as 3n + 1, else the cycle would be reached by some number below itself. I don’t know if this leads anywhere

>>11241642
This is nonsense. You're literally just saying
>if the Collatz conjecture were true, then the Collatz conjecture would be true, which feels unlikely to me

>>11241673
obviously I’m not saying it’s a proof or anything, but if it seems unlikely then maybe we can prove that it’s impossible. Either way a collatz number will generate many other collatz numbers, and all these numbers must not be reachable by non-collatz numbers, which seems unlikely, that’s all I’m saying.

>>11241642
Since C cannot be even, it cannot be of the form 2n. So now we test 2n + 1 (or 2n-1 , not sure which is best).
2n + 1 -> 6n + 4 -> 3n + 2
If n is even, then 3n + 2 is even, and dips below 2n + 1, so it cannot be a collatz number. So n is odd and we replace 2n + 1: 2(2n+1) + 1 = 4n + 3
Repeating the same process as above, we find that 4n + 3 is not always possible, so now we must test (iirc) the forms
16n + 7
16n + 11
16n + 15
and each of these forms will generate newer candidate forms and so on.

I did this a while back and I wondered if you can show that the process goes on forever, and the constant (ex: 3 in 4n + 3) goes to infinity, proving that no representation can be made for collatz numbers. But if n = 0, then the constant may be a collatz number itself, and if we try this process then all we will do is keep changing the power of 2 coefficient forever

>>11241678
>which seems unlikely
Why? Because if there's one counterexample, then there would be a lot of them? Why does that make any difference?

Thousands of professional mathematicians have dedicated thousands of hours to trying to figure this shit out. It really baffles me where so many totally untrained people get the ego to think whatever vague waffley nonsense they come up with after thinking for 2 hours is going to make progress.

>>11241698
Because from all the numbers below the collatz number, we know that there are many numbers beyond the collatz number that collapse to 1. So for every k below the collatz number, 3k + 1 and 2k are collapsible. So the collatz number would have to avoid every single one of these numbers, and the numbers generated from those, and so on.

This posting is not e-stat. Please give serious replies.

Should I worry about GPA that much? All my friends that have GPAs around mine (4) seem to super K/D whore over it.
This gives me mixed feelings:
Are they autistic?
Am I autistic?
Should I put more effort? I think I could do better. I don’t really study that much.
Is the diminishing returns close to 5 worth it or should I just focus on getting a firm and broad foundation instead of K/D whoring?
Is 4 not that good and I’m just using high school standards to measure it?

Thanks for reading my blog post.

>>11241858
> Should I worry about GPA that much?
No

I've been enjoying learning inequality tricks with The Cauchy-Schwarz Master Class book by Steele.

Does being proficient with certain real analysis exercises mostly involve spamming these counter intuitive inequalities, identities, and other tricks? Are we meant to somehow commit them all to memory? I understand that I am oversimplifying here. I also understand that some sections of an exercise may require arguments that are fundamental to the subject. Limiting arguments, working with some structure, or the tricks provided here https://terrytao.wordpress.com/2010/10/21/245a-problem-solving-strategies/..

It seems cheap that the crux of some exercises require prior knowledge of an obscure identity.

Im trying to find the dual lattice of [math]Z_m\times Z_m[/math] because it is generated by [math]\{(0,1), (1,0)\}[/math] with addition modulo m over the integers. If I take the definition I found, as "linear" functions to the integers, then I think the dual lattice would actually be all of [math]\mathbb{Z}[/math]. Now I have no clue if this is actually the definition or what sort of lattice I have, but the paper I'm reading says you can define the dual lattice as the "intermediate" lines between points. I just want to see what sort of lattice the paper is talking about and what is the abstract definition of its dual space. Any thoughts?

Given a linear system of equations Ax=b, what is the derivative of A wrt b?

>>11241970
>it seems cheap
Life isn’t fair anon-kun desu

What the fuck is an operad? I ran in to this term a few times on a research rabbit hole into programming language theory and its categorical semantics.
Can someone give me a quick rundown?

>>11242088
It's autism.

>>11242106
What I've read on them seems intriguing, even if I don't follow all the examples completely. Mostly, because I am intrigued at its potential to generalize seemingly any operation that a computer can do. Seems useful to me from a program analysis standpoint.

>>11241992
>addition modulo m over the integers
Anon, that's not a lattice.

>>11242141
wtf is a lattice

>>11238078
Faggot design and faggot book choices

>>11242148
Discrete subgroup of [math]\mathbb{R}^n[/math] or [math]\mathbb{C}^n[/math]. Arbitrary vector space if you're willing to force it.
https://en.wikipedia.org/wiki/Lattice_(group)

>>11242154
https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup) What about this for the torus?

>>11242160
Oh shit, completely forgot about that definition.
But how do you even talk about the dual lattice of a lattice on a torus? Wouldn't you need some specific choice of inner product?

>>11242172
Never mind, the dual lattice is in the dual group.

>>11242031
This doesn't really make sense because A is not a function of b; changing b does not force a unique change of a.

What are the prereqs for learning Operads? I made a thread but I realized it should've belonged in this thread. Anyone?

>>11242824
can't you find out by lurking the nLab articles on it?

>>11241579
Yes, in the same way category theory ties to logic. Many type theories are also logics.

>>11242152
thatsthepoint.png

>>11238795
"why yes" is a phrase and doesn't mean there's "why? question

>>11240875
>math is all about communicating ideas and collaborating with others
Yeah and that's why we should agree with a common language retard.

>>11238963
yes

>>11242924
or you know...just not be retarded?

>>11241970
Really mastering analysis means reaching the point where "counter-intuitive identities and tricks" stop being counter-intuitive and start looking like sensible methods instead of asspull tricks.

>>11241970
you get used to it.

>>11238981
Why not?

I am quoie sure that the border of a convex set is not, in general, a zero set.
How about the border of an open convex set though? If I close an open convex set, can I gain measure?

>>11243212
in R^2 you can prove that boundary of convex set consists of at most countably many C^1 curves
in higher dimensions there probably are similar statements

>>11243218
What about this: take all the rational points in [math] B_1 \subset \mathbb{R}^2[/math] and form the convex hull.
Then the convex hull is a countable union of line segments (connections between rational pts) and therefore a zero set, in particular the convex hull is a weird, dense subset of the ball and its border has the measure of the ball.

>>11243212
Take the closed convex hull, and its set of extreme points.
Conjecture one: if a point is in the boundary of a closed convex set in R^n, it can be written as a convex combination of at most n points.
Prove it by contradiction. Namely, prove that a convex set with n+1 extreme points satisfies that, and do a pretty little inclusion.

This gives that, if the set of extreme points is countable, the boundary is contained in a countable union of hyperplanes.
Now consider a convex set with an uncountable amount of points, and show that you can approximate it to arbitrary measure with a convex set with n extreme points.

>>11243237
>convex combination of n points
*n extreme points.
In retrospective, conjecture one really just says that the boundary of a simplex is the union of the faces.
The measure thing is either simple or an absolute pain to prove, but I don't really remember if there was any theorem about measurability of convex sets.

>>11243224
convex hull of rational points inside unit ball is the open unit ball
do you know what is a convex hull

>>11243270
ah yea of course nevermind

When will I get gud at proofs?

>>11241970
Fuck Carleman inequality desu, tried proving it for 2h until the professor came in and said it's trivial

>>11243291
be smart
read proofs
write proofs
rinse and repeat

>>11243291
trust ur gut, read the theorems and definitions in the fucking book, experiment, trust ur gut

>>11243291
wave them hands boy, wave them harder

>>11243350
this, I never realized how good this is. when you're confident you just wave your hands and people trust you

>>11243351
he means to invoke autism in oneself so as to channel great unnatural powers

>>11241605
Haskell is modeled entirely upon the category Hask.

>>11242824
Really not many, categories for the working mathematician. Then I would suggest General Operads and Multicategories by Leinster. A little domain specific motivation may be of help, which of course would depend on why you are interested in them in the first place.

>>11241970
>counter intuitive inequalities
look at this:
[math]x^2 + y^2 \geq 2xy, [/math]
[math]x^3 + y^3 + z^3 \geq x^2y + y^2z + z^2x + xy^2 + yz^2 + zx^2 \geq 6xyz[/math] etc,
AM-GM,
Cauchy-Schwarz inequality,
all of this can be summed up by "if you multiply variables with each other, it's smaller than having them separated"
I haven't seen carleman inequality before but I'm 99% certain you can prove it by summing weighted AM-GM's

>>11243303
kek, based prof

>>11241596
>>11243355
I don't think people reasoned much about Hask before they designed Haskell. In fact there's a super nice paper/essay documenting the creation of Haskell, I highly recommend it:
https://www.researchgate.net/publication/221501761_A_history_of_Haskell_Being_lazy_with_class

I've summarized some issues of the would be category here
https://axiomsofchoice.org/hask
and here
https://ncatlab.org/nlab/show/Haskell
and there's quite an angry blog about the topic by Andrej Bauer here
http://math.andrej.com/2016/08/06/hask-is-not-a-category/

As it's higher order in the functional programming sense, there's some nice questions one is naturally led to asked w.r.t to logical design,
e.g., waging extensionality
> ∀x. f(x) = g(x) ⟹ f = g
against indiscernibility
> ∀h. h(f) = h(g) ⟹ f = g
for function terms

For a relaxing Saturday evening, here's some 20's German University life reflections


https://youtu.be/PrF5eMXr0Bc

Guys I'm worried, I went to a shit High School and never really got a solid foundation in math. I failed precalculus 3 times in college because I didn't want to do the exercises or study. Now I'm trying to learn precalculus on my own and I just can't grasp it. Functions and polynomials are okay but I can't factor and trig wrecks my brain. I want to learn Calc and move on to higher math but I'm getting kind of hopeless. Is there anything I can do to improve mathematical thinking?

>>11243945
lots of practice, and try to get insight from several different sources. one of the biggest changes I made in the way I study mathematics at a higher level is to always have at least two textbooks on the subject in addition to lectures/lecture notes. Sometimes seeing something from a different point of view, or even just with a bit different exposition can do wonders for understanding.

Importantly, figure out what it is that you're struggling with in your studies. This can be difficult to self evaluate, but try to find out if you are struggling with the concepts you are currently studying, or perhaps you have not mastered the underlying techniques as well as you think you have. e.g. don't move on to study calculus if you struggle with algebra like factoring, solving polynomials, etc, which sounds to be part of your issue. Take a step back and really nail trig and algebra concepts before trying to move on to the "fun stuff."

Also, luckily for you, since you're studying math at a lower/undergraduate level, there are bountiful resources to be had for free on the internet. YouTube is your friend.

Don't lose hope, math can be challenging, especially without a firm understanding of what you're doing which many primary schools fail to provide. Just keep at it!

>>11243982
>always have at least two textbooks on the subject in addition to lectures/lecture notes. Sometimes seeing something from a different point of view, or even just with a bit different exposition can do wonders for understanding
same desu, greatly recommended

Any books to learn statistics along with prob. theory, with examples and exercises? We had a lot of statistics in university, but I always insufferably failed in understanding (we didn't have measure theory though). Last thing I remember we did is uniformly most powerful tests, minimax statistical decision, Neyman-Pearson, some probit/logit, ...
But I need to pick it up from basics, starting from probability distributions (I don't even know what's the difference between PDF and CDF), things like law of total expectation, etc.

>>11238182
Шиpяeв.

>>11243456
Interesting. Would you say that Agda has done it better in terms of a sound categorical foundation?

WHAT THE FUCK IS A MANIFOLD

>>11244227
I don't know it enough to make a "better" judgement. While type-families are implemented as the object-map of functors (say the List functor giving lists of Ints and String and so on), the categorical semantics for dependent languages like Adga are more complicated. E.g., I tried to explain dependent products (universal quantification in categorical logic), as they are defined in terms of adjoints, here

https://youtu.be/fTBL4TmZNRo

>>11244256
the easiest way to think about it initially is that it's a subset X of R^N for some N where, if you look really closely at any point of X, that small neighborhood "looks like" a small neighborhood in R^n for some n <= N. By "looks like" i mean "is homeomorphic to" or "is topologically deformable to." so curves and surfaces are examples of manifolds of dimensions 1 and 2 (the dimension is n, not N) and there are higher dimensional manifolds sitting in higher dimensions of R^N.
in full generality we can just have X be a topological space, not a subset of R^N, and we require X to be second countable and hausdorff. but then it turns out that X is able to be viewed as a subset of R^N (it embeds into R^N).

>>11244308
interesting. thanks anon

If two subsequences converge to two different values, then the sequence does not converge.

Can someone please write a proof for that using just the definitions of a sequence and subsequence? I'm guessing proof by contradiction would be the easiest way?

>>11244330
>>>/sci/sqt

>>11244312
here is a picture of what i mean. notice that the whole shape is not going to be deformable to the whole of R^2, since the shape is kind of "closed up" and R^2 is kind of "flattened out." you'd need to tear the manifold to get R^2 (we formalize this difference by "compactness")
however, for any point you give me on the manifold, i can give you a neighborhood of that point which might need to be really small, and a way to deform just the neighborhood to be part of R^2.
ignore the bit at the bottom, it's irrelevant to the point.

you would learn about manifolds in a class like "differential topology," "differential geometry," or "algebraic geometry." (classical) differential geometry is probably the most accessible, you can learn all the basics after you've taken a multivariable calculus class.

love me

I just finished my undergraduate degree (like last week), studied CS and Math.

I want to study algebraic geometry. Can anyone give me a book recommendation for someone around my level?

>>11244625
I feel nothing for anyone anon you aren’t an exception sorry.

>>11244795
I used an invitation to algebraic geometry. Good intro

>>11244983
Thanks!

>>11244795
algebraic geometry is a meme
learn french and read EGA obviously
if you're too much of a pussy for that i guess just read hartshorne.

>>11244256
it's an n-dimensional space. n-dimensional means you can use n numbers (coordinates) to describe a point. usually you can't do this in a consistent manner in the whole space, this is the difference between a manifold and whole R^n.

>>11244795
>CS and math
Conics and Cubics: A Concrete Introduction to Algebraic Curves

>>11244795
as the other anon said, invitation by smith is pretty good, albeit lacks content. Another very introductory and short treatment is Reid's undegraduate AG, but covers slightly different things. So read that first (if you go hard on it, should take less than a week I bet), then follow it by a more complete treatment like shafarevich basic algebraic geometry 1.

Learning schemes without knowing the classical counterparts is a meme.

>>11236736
Munkres is quite clear and contains more than enough material on point-set topology.
>>11236850
There's no much you can gain in studying concepts of algebraic topology in point-set oriented books anyway.

>>11236736
it depends on what you mean by topology. if it's "klein bottles and shit", you're gonna be surprised.

>>11239063
Arnold.

>>11244125
bump

Anons I’m very happy. I’m publishing my first paper soon. I do it for you

>>11245730
good for you anon. what's it about ?

>>11245739
It’s going to give visual representations of effective cones for a few smooth varieties and proofs of how they’re generated

What's the best exposition of Fourier Analysis you guys have seen?

>>11238182
MIR publishers, soviet books.
Every serious physicist/mathematician know them like.

>>11239063

Pure math is for prodigies, you just like to suffer kike

>>11244795
http://www.math.ens.fr/~benoist/refs/Perrin.pdf
This webpage is from autor book, 100% legal!

Is taking notes pointless? If you're forgetting something then you probably don't have a use for it in the first place. But if you have a use for it, you'll see it so often that you never forget.

>>11246273
No.
Wrong.
Wrong.

>>11246304
Elaborate.

Is rational trigonometry any good? https://en.wikipedia.org/wiki/Rational_trigonometry

>>11246421
Rational trigonometry is the best.

>>11246421
Rational trig is a meme

whats the point of type theory? Im really well studied in set theory and category theory and im also a programmer, It just seems like type theory is constructionist set theory. Am i wrong in thinking this?

>>11246436
The endgame of type theory is the classification of races into types and the systematization of aryan superiority.

>>11246436
It's non-classical logic isn't it? Do you know enough set theory to formulate ideas that include formal logic?

>>11246436
Types = proof = programs = categories
Build formal proof for mathematics

I understand how a simplicial complex can have betti numbers, but how are the betti numbers of a monomial ideal defined?

Perhaps what I'm seeing is actually a special case of some notion of betti numbers of a commutative ring?

>>11246436
Type theory forms the theoretical backend of type systems, which are obviously useful.

>>11246621
This is incoherent.

A type theory is a formal system of term construction rules, type construction rules, a pile of reduction rules for the former, and typing judgements that relate terms and types.

You can build type theories that model different constructive logical theories (see: BHK interpretation of constructive logic). This is obvious and not very interesting: in constructive logic a proof is a witness, so you use type construction rules/typing judgements to restrict the kind of witness you want (i.e. encode the proposition) and use term construction rules to actually build the witness. You can obviously build type theories that just correspond to some absurd logical theory, Curry-Howard is not a magic buzzword.

NLab likes to talk about type systems as the internal language for categories, so that types (or "contexts" of types, variable bindings basically) in the theory correspond to objects in the category and terms correspond to morphisms. This lets you treat an appropriate category like the "semantics" of the type theory's "syntax".

>>11244436
neat. where is that picture from.
Also, you cant map that whole shape in R2 cause the wedge in the bottom left would be squished onto the plane?

How do I solve Ax=b where part of x is unknown and the complementary part of b is unknown, but neither all of x or b are known? I remember learning how to do this, but I'm rusty.

>>11247117
I just found it online. You can search "manifold math" on google images.
Uhhh, actually maybe you can flatten that one out to R^2, I was thinking it was like a partially deflated ball but maybe it has no inside.
Either way, it's a manifold. All you need to be a manifold is to be "locally flat," and the global (large scale) behavior can be a whole ton of things.

so, I didn't want to make a thread
am pretty damn smart myself but never really applied to mathematics, never studied it nor practiced it, now I want to get into electronics engineering and I feel like I'll get fucked on the ass real hard by math
is there some brainlet-to-wizard sort of guide about shit I can learn in order of complexity? from, like, arithmetics to big brain math?

>>11246436
Types "hold" terms, but they are not sets: Every terms necessarily has a type. You can't form unions of types (in the set theoretic meaning of the word), etc.

>>11247358
Check /sci/ wiki mathematics section. Make sure to check the links below it too.

>>11247312
If someone could even provide me with the right search terms for what it's called, that would be really helpful too. All I get is a bunch of unrelated BS when I search the internet,

is complexity theory welcome here? Has little to do with cs imo

>>11248066
I genuinely can't even read your question.

In this picture, math]f_i[/math] is a sequence of generators of R-module M with [math]a_i= deg (f_i)[/math].
[math]M(a)_d[/math] denotes [math]M_{a+d}[/math] (i.e. it's shifting the degrees of a Z-graded module.)

Wouldn't [math]F_0[math] depend on the order of the sequence [math]f_i[/math]? In other words, the order of [math]f_i[/math] determines which [math]a_i[/math] goes to what [math]R_i[/math].

Wouldn't you normally treat the generators of a module as a set, because this is commutative algebra after all?

>>11248247

[this picture]

I have a linear system (mod 2). I have about twice as many variables as equations, but I also know exactly how many are odd and how many are even. Can anyone tell me a way to encode the last bit of information into more congruences to solve this efficiently?

>>11248303

this is some basic coding theory. If x_i are your variables,

[math]\sum\limits_i{x_i}=1[/math] tells you "an odd amount of variables are odd."

Here is another approach which would give you the most precise congruence that you could possibly come up with given that information. You could write:

[math]f(\vec{x})=1[/math] where f is the boolean function "[math]\vec{x}[/math] has population k".

f is nonlinear, but it's permutation symmetric. Permutation symmetric boolean functions form a vector space. A basis for this vector space would be [math]M_{1\leq i \leq n}[/math] where [math]M_i[/math] is the sum of every degree i monomial in the [math]x_i[/math].

I'll leave it to you to find out what exactly f is in terms of M_i.

>>11248390
Thank you

>>11248504
I thought the coomutative algebra one was the least funny coomer edit it was possible to make. You've proven me wrong.

>>11248504
I'll admit, this one is excellent.

>>11248247
>>11248251
It looks like F_0 does depend on the order of generators but any choice is obviously isomorphic to any other choice. From memory the thing we care about with free resolutions is the (co?)homology and you can show that any free resolution gives you the same (co?)homology.

>>11248518
Can you post it?

>>11248529
explain, is there a base to the punch lines

>>11248994
The base of my dick lmao.

what on earth is a hom functor????

>>11249021
But anon, any log of base 0 is 0

>>11249034
It's a representable functor.

>>11249034
The mirror image of any object, in the sheaf category into which the original category fully faithfully embeds into

>>11249115
In English, Doc!

>>11249034
Hom(A,B) = the set of all maps from A -> B.
if A and B have some extra structure, the maps are required to preserve this structure (e.g. if A and B are groups, we're only talking about group homomorphisms)

>>11249181
I was trying to give a cheeky meme answer.

Although it's true:
If X is any object in your category and Y is another, then the Yoneda lemma says that
>Hom(X, Y) = Hom(Hom(-,X), Hom(-,Y))
so the objects of the form Hom(-,X) behave exactly like X themselves.
The Yoneda embedding takes any category and embeds it in a bigger category of functors, i.e. it embeds every category C fully faithfully in the topos Set^C.

The image of the hom-functor Hom(-,X) is defined as the maps into X (see >>11249230), but it ends up being X itself.

>>11240875
Computers complain about notation.

how many pictures do /math/ draw daily in average?

>>11248987

Wrong.

You can see in the picture that R is constant in the direct sum.

I.e., it's not R_i, that appears in the summand, it's just R.

Man, Probability Theory is fucking huge.

>>11248098
Solve the system of equations Ax=b where half the unknown variables are found in the vector x and the other half of the unknown variables are found in b.

>>11249782
Something like this? More context may be needed here.

https://math.stackexchange.com/a/1727212

>>11248504
absolutely awful. After the coomutative algebra "meme" you should have learned that you're too low-IQ to meme.

>>11249681
and to think its all fake and gay

>>11247312
mult Ax out using the usual rules of matrix mult and then compare the result and b piecewise

I want to do nothing but mathematics research but I got a degree in physics (took analysis/abstract, other math courses as well) and screwed up my GPA (2.88) from mental health problems prior to my junior year. Any tips for someone with a low GPA to get into a math PhD program?

>>11248504
Just fucking stop, it is plain unfunny

Good books that cover set theory and logic?

>>11250181
Jech

>>11250036
You are going to struggle a lot to get into any real university with a GPA that low. Anything below 3.0 is essentially auto-rejected from most respectable programs, especially since your math background is weaker than most students to begin with.
Presuming you're American you can try looking into Master's programs. You won't get paid and likely won't get a tuition waiver like a PhD would, but (because you pay for them) the admissions standards for Master's degrees are much lower, and doing well in those classes/writing a good Master's thesis/getting references who actually think you're a good student will give you another chance to put together a usable application.

If you spammed applications absolutely everywhere you might fluke your way into a couple mediocre universities, but doing a PhD at a crappy school isn't really worth it. Your chances of getting any sort of research position with a PhD from flyover McUni are functionally zero.

Can any of you anons help me decide on choosing good proof books?
>>11250199

>>11250227
fuck proof books, just start with real analysis and linear algebra like the rest of the world

>>11250227
no, proof books are complete meme. see >>11250241

>>11250256
>>11250241
O.K thanks for the advice maybe I'll reconsider this. I was told it's smart to do some proof book before jumping into this.

Merry Christmas, /mg/!

>>11250222
thanks for the advice. I took enough math courses to be able to get a math bachelor's with only 3 additional semesters, so I'm working on that now. I considered the master's route, but a lot of program's won't even let you apply to them and just award them to PhD students who are partway through the program, or they only allow 1 or 2 master's students and reserve research spots for PhD students. Should be able to beef my GPA up in my last 2 semesters though so that may increase my chances.

>>11250513
merry christmas, faggot

>>11248504
good for the soul purpose of making /sci/ autists seethe

>>11250582
>I took enough math courses to be able to get a math bachelor's with only 3 additional semesters, so I'm working on that now.
This info wasn't included in the original post so I didn't think to suggest it but it's a probably a better idea. If you can get to like a 3.2 or 3.3 you should be safe that everybody will at the bare minimum LOOK at your application.
You may want to consider retaking some of the courses you did poorly in if your schedule has room. Retaking a course you got a 57 in does way more for your GPA than adding on another new 95, and it's way less work.

Get to know your profs as well as you can in the time remaining too. You'll need to for good rec letters anyway, but there's way more value in it than just the letters.
I was in a situation somewhat similar to yours (I was on academic probation after sophomore year) and a huge part of why I still made it okay is one particular professor who took an interest in helping me out.

>>11250312
plenty of people will tell you all kinds of shit
the only way you can know what will work is to try something and see if it works for you
if it's not working, try something else
but keep in mind that you're supposed to feel uncomfortable and awkward when you're learning how to do proof based math for the first time, it gets better quickly.

>>11250582
how is your major GPA just for math? if it's better and your early shit was mostly physics, that could benefit you if you make it clear in your applications that your math major gpa is a lot higher.

>>11250634
this isn’t reddit, treat others like they’re adults or retarded animals but please stop posting gay shit like this

I am working on a project right. I'm an undergrad and my hope is that it will get accepted into a certain journal.

This project was not something I just decided to do on my own. I was motivated by a project I worked on with a professor at my uni and I have talked with a PhD student who has published in that journal before about whether or not it would be viable. The PhD student and the professor thinks it's an OK idea.

However, my concern is that neither of these people will want to be a coauthor or author on this. It will definitely be weird if I try to submit an article as an unknown undergrad.

I think that I shouldn't really be trying to do the math and write this paper myself because that doesn't leave room for anyone else to contribute. Right now, all I want to do is work on it since I don't have class for an entire month.

>>11250680
I’m in the same boat. I’m the only author on my paper, but the professor whose work the paper is based is encouraging me to attempt publishing alone.

>>11250632
thanks anon that gives me hope. I think the policy at my school is to average the GPA of retaken courses. I've never gotten below a C in a course though, but I got a lot of them my freshman and sophomore years. Will definitely keep this in mind

>>11236068
Self discipline

what's harder pure maths major or theoretical physics major

>>11250739
pure maths major

>>11250312
it is, dont listen to them. they probably forgot already, but one of the first things you do in your math UG is going over proof methods. There's at least one class that covers it. I heard in USA it's typically discrete probability that covers this, in the UK for me at least, it was real analysis, and in the university im doing my phd in right now (top 10 in world, mind you), they cover it in a class called something like "logic, sets and numbers". Although it is correct that you shouldnt spend too much time on this, so fast track through one of these books.

>>11250739
physics because of the unearned insufferable elitist attitude of your peers. at least the mathematicians earned the right

>>11241651
Let's say you pick door number one. The host will always pick a door that doesn't have the car. There are three outcomes:

The prize is behind door 1. Host picks door 2 and you switch to door 3. You lose.

The prize is behind door 2. The host picks door 3 and you switch to door 2. You win.

The prize is behind door 3. The host picks door 2 and you switch to door 3. You win.

So switching wins 2/3 times.

>>11250739
Theoretical physics its not even close.

>>11250635
Yeah that's kinda a problem. My 100 and 200 level courses are garbage (c's and b's) but my 300 and 400 levels are better (mostly a's and b's). I'm definitely going to try and get a good GRE score as well because that might cover up some of my weaker class grades.

>>11250739
math, by a mile and a half

>>11250636
trying to get the hopeless moron to move on his way and feel like he has a chance so it's all the more crushing when he realizes he will never make it
maybe if you lurked more you'd understand board culture?

>>11250739
each is harder than the other for different reasons
in math you have to know how to think creatively
in physics you need endurance

>>11250680
>>11250688
When a professor gives a task to new student, he usually chooses a task well below the level he would consider appriopriate for himself, and he doesn't really care about being coauthor even if you did great job
>It will definitely be weird if I try to submit an article as an unknown undergrad.
There's nothing wrong in publishing by yourself, the journal editors and editors won't think less of you as long as your work is decent. Most likely they won't even realise you're an undergrad, they'll just think "oh that's paper written by someone i didn't hear about".
You should definitely ask the prof or the PhD student to read your paper carefully and help you with getting style/format correct (i assume this is your first paper). Be sure to put them in acknowledgements. But you
t. published as a single author in undergrad

which one would you rather be?
>a good student (say, top1 of the year) at an average university
>an average student at a top university

>>11251373
Average at top.
Plus ace all of your coursework and don't care too much about exams. Then you'll have a record of your writing to show a professor if you want to do a PhD.

>Mathematicians laughed at the Dirac delta function, which was made up on the
spot by the great physicist P. A. M. Dirac to solve some differential equations. They stopped laughing when they discovered a completely new
branch

>>11250181
Kunen is another good book for set theory

thred is almost ded
new >>11252098