/sci/ - Science & Math

First for tropical curves

>>10671823
doing a PhD thesis on any problems around Langlands program is a mistake

>>10671832
any good books which cover these? They seem to be an interesting microcosm of a lot of cool stuff.

>>10671857
It has sunk the careers of many math PhDs at elite schools. As a grad student you basically need to have some sort of edge to make real progress like you're an expert in X and it just so happens to be useful for something related to Langlands.

>>10671868
Yeah, it's a high risk direction for even exceptional students. It makes me uneasy when beginning grad students say they want to do something in Langlands program for their thesis.

Math is too hard but everything I want to learn is related to math somehow. Really sux bros

https://www.youtube.com/watch?v=qQXhNI7cyCQ

>>10671952
gay and stupid

>>10671959
based

So what happens after phd?

What are some good resources on cryptography and complexity theory? Im decently versed up to elliptic curves but want to learn more.

>>10672021
post doc treadmill until death

>>10672021
Endless shitfest, then the sweet release of death.

>>10672021
callcenter

Give two sets of positive multiples of integers [math]n\mathbb{Z}^+, m\mathbb{Z}^+ [/math] where m and n are positive integers, is there a way to determine, in general, some number [math]a[/math] such that for every integer greater than or equal to [math]a[/math], those numbers are the sums of elements from each of the sets of integer multiples? Does it help if n and m are mutually prime?
I'm trying to generalize the problem about what is the minimum bet that can't be made with chips of $8 and $5. In that case the answer is 28. For $5 and $6, I believe it is 19. For 7 and 9, I think it is 47. In both the latter two cases it happens that you need 3 of each minimum to make those bets, but with $5 and $6, you need at least 4x$5. Also, I counted 11 numbers you couldn't bet in the case of $5 and $6.
It seems like this is a cyclic group/generator problem, but I don't know what to do with it.

>>10672390
Oof. Correction. m and n can be 0.

I know literally nothing about algebraic geometry, should I read the Grothendieck meme text or will I understand literally none of it?

>>10672021
>he fell for the PhD meme
I hope your parents are rich

>>10672446
It's a little arcane. Hartshorne is the abridged version.
If you are a complete beginner, start with some commutative algebra and classic AG. Then go to schemes.

>>10672390
https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity

>>10672390
Sounds like this in the relatively prime case
https://artofproblemsolving.com/wiki/index.php/Chicken_McNugget_Theorem

>>10672505
Thanks, Anon, that’s it.

>>10672390
>>10672505
Oh and if they're not relatively prime there should be arbitrarily large numbers that cannot be written in such a sum, even if you allow negative multiples you can only get multiples of gcd(m,n) by Bezout.

>>10672390
If m and n are not mutually prime, then any combination am+bn will be a multiple of d = gcd(m,n).

If m and n are coprime, then for any integer s, you can find integers a and b such that am+bn=s, but a and b may be negative.

Group theory won't help much because [math]\mathbb{Z}^+[/math] is not a group and neither its multiples.

Any math books dealing with the concept of chick with a gimmick?

>>10672520
Yeah, but the numbers less than n that are relatively prime to n mod n are.

>>10671959
tibees applying for her math PhD, 201x colorized

Has anyone actually computed a counter example to Euler's conjecture in the exponent three case?

>>10672953
Nevermind, found it.

Let's make a generator of gibberish latex equations. Like, equations that are pages wide an contain exponents inside exponents inside exponents...

>>10672145
Pls respond

>>10673836
>generator of gibberish latex equations
Just open up Mochizuki.

>>10673836
Nobody is stopping you.

>>10672145
>>10673880
Usually in cases like these I prefer to find course notes first. So one set of notes is by Ronald Rivest, the "R" in RSA.
http://courses.csail.mit.edu/6.857/2017/handouts
Next is Scott Aaronson's notes on complexity
http://stellar.mit.edu/S/course/6/sp16/6.045/
http://stellar.mit.edu/S/course/6/sp16/6.045/materials.html
And finally quantum complexity and security
https://www.scottaaronson.com/barbados-2016.pdf
Happy trails anon.

>>10671823

How do you go about determining the coordinate ring of a closed set?

By this, I mean find the generators of the ideal of the polynomial ring

>>10674440
If [math]X\subseteq \mathbb{A}^n[/math] is defined by equations
[math]f_1(x_1,\dots ,x_n)=0,[/math]
[math]f_2(x_1,\dots ,x_n)=0,[/math]
[math]\dots,[/math]
[math]f_m(x_1,\dots ,x_n)=0[/math]
then the coordinate ring is
[eqn]K[X]=\frac{K[x_1,\dots ,x_n]}{(f_1,\dots ,f_m)}[/eqn]

>>10673836
Mathgen.

Do mathematicians have big collaboration projects like physicists have?

>>10674661

See: Classification of the finite simple groups

>>10674661
>Do mathematicians have big collaboration projects like physicists have?
https://en.wikipedia.org/wiki/Polymath_Project

>>10671823
why

What is a good book or website to prep myself for my first two actuary exams, the FM and P exams? Also, I just finished my sophmore year of a math degree, am I too late in starting to take actuarial exams? Where do I even go to sit for actuarial exams? Also most entry-level actuarial jobs I look at only require that I have 1-2 exams done. Is that really all I need to get employed after college? Thanks in advance lads.

>>10671874
are people this egotistical that they'd think this? i never understood this. i always figure when i get into graduate school i'll first pick newer fields were i can contribute more and slowly read up with the research on 'more interesting' fields of research. i actually remember remarking how i didn't think X field had much low hanging fruit to some guy, and he got so mad and he was thinking that i was making fun of him when really i just think that going for low hanging fruit is simply the best thing to do... i dont get it anons

>>10675654
>I just think that going for low hanging fruit is simply the best thing to do.
This desu. Graduate school is a time when you are really free of other responsibilities and a time when you can really devote yourself to becoming (almost) an expert in something, whereby in the end you have a decent result to show for. Picking something really out of reach only jeopardizes the value of those free years.

Once you make yourself known to a few people in that research area, you can sigh in relief for the however short job security and maybe start branching out to closer fields and harder problems. So yeah, going for low hanging fruits are the best thing to do; thanks to your advisor, you can be safe that you aren't doing something really trivial either.

>>10672449
thanks me too

>>10675654
Please proofread your writing before you post

>>10675710
srry anon i just get so emotional interacting w people... it's best to be alone

>>10675728
I feel the same

>https://arxiv.org/abs/1401.0300
>This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.
Is there any big list of unsolved problems in other math area that has the same feat as the Kourovka Notebook in group theory?

>>10675903
Gromov has a list of 91 questions and conjectures related to embedding theorems.
https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/nash-copy-Oct9.pdf
Yau's book has a list of open problems at the end
https://intlpress.com/site/pub/files/preview/bookpubs/00000308.pdf
This site also has quite a few
http://www.openproblemgarden.org/

Stein and Shakarchi say that integration works on sigma finite measure spaces. However, I see lots of people saying infinite sums are just integration with respect to the counting measure on the powerset. This doesn't make sense to me since the counting measure is not sigma finite in these cases.
What gives? Is the point that even though integration doesn't make sense in general for the powerset everything still works for countable sets?

>>10676606
>on the powerset
The interesting question is on the powerset of *what*.

>This doesn't make sense to me since the counting measure is not sigma finite in these cases.
But the counting measure is sigma finite on the Powerset of the natural numbers, right?

If you have a finite sum, you are integrating a function from the natural numbers to the reals, in that case you always have the basic case of a step function.

>>10676319
Thanks friend

>>10676616
>But the counting measure is sigma finite on the Powerset of the natural numbers, right?
The counting measure is only sigma finite on countable sets

>>10676606
>However, I see lots of people saying infinite sums are just integration with respect to the counting measure on the powerset.
Wait, I think there is some confusion on your part. An infinite sum is an integral with respect to the counting measure *on your summation set*. Said measure is indeed a function *defined on* the powerset (or a subset thereof) of that summation set, but the powerset is not the space over which the integral is computed.
The relevant measure space here is the summation set, ie. a subset of the integers which is in particular countable, hence sigma-finite.

>>10676858
>The relevant measure space here is the summation set, ie. a subset of the integers which is in particular countable, hence sigma-finite.
This won't actually be a measure space though because there's no sigma algebra.
I guess the point is that it doesn't need to be one though since everything needed to integrate will still go through?

>>10676858
Wait I think what I'm saying makes no sense.

>>10676858
Thanks.
I was getting confused because the sigma algebra associated with the natural numbers is uncountable. What actually matters is that the natural numbers themselves are countable. The counting measure is obviously sigma finite.

>>10676894
Yeah, typical sigma algebras on infinite sets will be uncountable (most of the usual examples are) although many examples are sigma finite, eg. Z, or R and other reasonably sized geometric spaces (fd manifolds etc.) with the Borel algebra

>>10676854
>The counting measure is only sigma finite on countable sets
Yes, countable sets like the natural numbers for example, that is exactly my point.

>>10676894
>I was getting confused because the sigma algebra associated with the natural numbers is uncountable.
Interesting fact: every sigma algebra is either uncountable or finite (a power of 2)

>>10677070
Assuming the Continuum hypothesis?

>>10677076
Nope. It's elementary but somewhat non trivial to prove

>>10677070
Fun, I posted >>10676995 so I was just asking myself this exact question. I'll think about it

>>10677076
It's essentially the same statement as "the power set of X either has finite or ouncountable cardinality."

>>10677113
Oh wait no I think I get it... If you have an infinite sequence of pairwise distinct subsets in your sigma algebra A, then you can form an infinite sequence of pairwise disjoint subsets [math]S_1, S_2, \dots[/math] in your sigma algebra and then you have an injection [math]f: 2^{\mathbb N} \to A[/math] by setting [math]\displaystyle f(X \subset N) = \bigcup_{i \in X} S_i[/math]

>>10677118
Yeah, that's also what I though, but naively I thought that maybe there could be some more interesting cases. But I don't think there was any good reason, CH or not, for me to believe that.

>>10677124
>then you can form an infinite sequence of pairwise disjoint subsets
How would you do that? When you take relative complements, you might end up with only a finite number of non-empty sets.

>>10677127
Yeah that's what I am wondering now heh... I think this is how it should go, but I'm not so sure now

>>10677129
Use the general idea behind compactness.

>>10677136
How would you use compactness here?

>>10677138
Not literal compactness, the idea lad.
Consider coverings by measurable sets. If they all admit finite subcoverings, we have a finite sigma-algebra. If they don't, we have an uncountable sigma-algebra.

>>10677140
That's not at all clear. Coverings of what? Arbitrary subsets? And by which measurable sets?

>>10671823
Why is he so sleepy? He seems very strange looking desu

>>10671959
god I miss xtranormal

if you do math come hang out https://discord.gg/39wsGv

>>10677308
>discord
Sorry, I'm not subhuman.

>>10677320
cool, do you do math?
irc / MSE / MO chats are too inactive, discord is the only place math people hang out online afaik

>>10677367
He just said he's not subhuman, of course he does math.

>>10677382
what do you mean of course
95% of posters here don't do math lol

>>10677367
>discord is the only place math people hang out online afaik
What kind of people do you surround yourself with?
We have this thread, why would we need some kind of chat?

>>10677408
I'm not creating a server from scratch for this thread
I just figured I'd invite people from here, cuz I used to post in /sci/ some years back and I remember there's a couple math people
it's just to talk math. talking math here is almost impossible, it ends up being 50 posts to discuss something elementary

>>10677423
Your kind isn't welcome here. Take the hint already, fucktard.

>>10677308
third time joining this server... you guys are so mean...

>>10677429
I'm not talking to you, but the math people lol

>>10677308
if u wanna advertise the math discord at least mention the secret hentai channel cobi

>>10677455
Animals do not get to decide the parties they address.

>>10677484
no I don't want more degenerates you're too much to deal with already

>>10677497
say that to my face and see what happens punk

>>10671823
>Geometric Langlands Edition
Can't prove number theory statements, doesn't make predictions about physics
Useful as philosophy? In any case, metaplectic and quantum Langlands are the hotter topics now.

>>10673836
I'll make the logo.

>>10677502
>math server
>hentai
Just as I thought, probably a place full of degens.

>tfw math anxiety because I'm afraid I won't be able to do the exercises

Anyone know this feel?

>>10677601
Do more exercises. Fear, in general, is overcome with repeated exposure to thing feared.

>>10672558

>>10677510
>>10677423
also you do realize there is a official /sci/ server right? it's on the wiki... no one reads the wiki...

Would you bully me if I started asking questions about theoretical computer science?

>>10677799
Only if you're cute

>>10677799
Computer science isn't maths.
It's in the "almost pure maths but definitely not science" group with microeconomics and mathematical physics.
Unless you mean the concrete parts of computer science, in which case you'll want to make your own thread.

>>10677840
Well, yes it isn't pure maths.
But some fields such as computational number theory for example are interdisciplinary and I'm wondering if discussions about such topics are welcome in these thread of if this is essentially a pure maths only thread.
I frankly haven't been lurking for long.

>>10677150
Right, right. Have an easy proof:
A sigma-algebra is closed under countable intersections. A subset of a countable set is countable. Assume the sigma algebra is countable. Take a point. Take the countable intersection of all measurable sets that contain the point. Repeat this for every point, until you have a nice, clean partition. Apply the previous argument.

What is the newest development in math? Does it seem like it's stagnating to you guys? It just feels like no new fields have gone through major development since like group theory and knot theory in the 70s and 80s.

>>10677601
Yes, and still after 5 years of university math. The problem is anxiety or not being prepared, not math.

>>10677992
There have been lots of interesting results in "derived" geometry and algebra since the 90s.
Certainly not stagnant yet either.

Hey lads.
>Let X be uncountable. Then there is Y such that P(Y) bijects with X
Is that equivalent to the GCH?

>>10677993
My anxiety is much worse. I can't get through any textbook without feeling too anxious to continue.

>>10676995
Actually sigma algebras can't be countable.

>>10673836
Ok. It's still a simple prototype, but it should work!

[eqn]
\uparrow y \mathbb{Z} \left(\sum_{i=\beth}^{\beta} \left(\frac{\left(\left(\Omega \beta \right)_{\cdot}\right)_{\alpha}}{\sum_{i\in\mathcal{F}(\omega,\Phi,\mu)} \left(\Lambda-\left(\eta\right)_{\varepsilon}\right)}\right)\right)^{\sigma \times \sqrt[\sharp]{{\left(\eta \right)_{\left(\sum_{i\in\mathcal{F}(\sigma,\mathbb{Z},\kappa)} \left(\aleph\right)\right)^{\triangle}}}}\times x }\left(\clubsuit \right)_{\sqrt[\lambda]{{\eta}}}\cdot \sum_{i\in\mathcal{F}(\omega,\heartsuit,\mathbb{Q})} \left(\cdot \right)-\heartsuit -\left(\left(\mathbb{Q} \right)^\mbox{It follows that}\right)^\mbox{It follows that}\sum_{i\in\mathcal{F}(y,\sharp,\aleph)} \left(\kappa \right)\sigma \sqrt{{\left(\left(\left(\sigma \right)_{\sqrt[\Omega]{{\sum_{i=y}^{\lambda} \left(\mathbb{R} \right)}}}\right)_{\heartsuit}\right)^\mbox{It follows that}}}\times \sqrt{{\left(y \right)_{\Phi -\left(\left(\left(\left(\sqrt{{\Omega }}\right)^{\mathbb{C}}\right)^{\mathbb{C} }\right)_{\sum_{i\in\mathcal{F}(\sigma,\mathbb{Z},\lambda)} \left(\zeta\right)}\right)_{\beta}}-\eta }}
[/eqn]

If it doesn't, it's /sci/'s fault (I have tested this one).

>>10678022
>Is that equivalent to the GCH?
What have you tried?

>>10678239
The way back is trivial.
Suppose that there is a set X with cardinality between aleph 0 and aleph 1. Then there is Y such that P(Y) bijects with X. But Y needs to have a cardinality smaller than aleph 0, otherwise P(Y) would have a cardinality greater or equal to Y. But any set with cardinality smaller than aleph 0 is finite.
The individual statements are essentially proved through Cantor-Bernstein and extending functions from X to Y to ones from P(X) to P(Y) the obvious way.
I'm asking mostly because I unbelievably suck at messing with finicky set theory stuff.

>>10678195
Holy shit it's Mochizuki, I'm a huge fan.

>>10677601
Try to get ahead of your courses. Skim through an introductory book before taking the course, and once you're taking it, study advanced topics. Does wonders for me.

>>10677601
No? That never even occured to me that such a thing was possible.

I just get angry at myself when I can't do it.

>>10678195
Post code

>>10671832
>Curve
>It's made of straight lines

Is the answer "all points with x /= 0"?

It seems too easy...

>>10679634
Regular means an element of the coordinate field, i.e. it can be expressed as a polynomial. Regular at a point means regular in a neighborhood. That function doesn't seem to be anywhere regular, unless you are supposed to consider the circle as in a projective plane.

>>10677923
this is a thread for mathfags to commiserate with each other over their cognitive diseases. Go on >>>/g/ if you would like to talk about computard

>>10679664
>Regular means an element of the coordinate field, i.e. it can be expressed as a polynomial.
No, it does mean an element of the coordinate ring but it does not necessarily mean that it has to be expressed as a polynomial. For example, 1/x is regular on A^1 \{0}...
>>10679634
You are right, there is a slight trick. Clearly, the function is regular whenever x is nonzero, as you have noted.
Now, there are two points where x is 0, ie. (0,1) and (0,-1). The function is regular at one of them, can you see why ?

>>10677799
Yes. Fuck off elsewhere.

>>10677799
you are anonymous, do whatever the fuck you want

>>10679664

It's trivially not a regular function because it's undefined at a point, but how do you show that it's not regular at any point in the set?

>>10678820
>Post code
>code
>>>/g/.

>>10679692

Are you saying that in order to be regular at a point, a rational function doesn't need to be defined at that point?

>>10679728
My point is that there is one of these extra points at which our function is defined (and regular), although it is not obvious from this representation.
A "rational function" is not actually a function, rather an equivalence class of functions. One representative not being defined at a point does not entail that the rational function cannot be made sense of or regular at that point

>>10675379
p-pls respond
I promise I will do grad school after I suffer through CAS certifications and make some money.

>>10677752
there are literally no math people in the /sci/ discord

Who proved the uniqueness / existence of the tensor product of vector spaces?

>>10675379
>>10679850
https://www.soa.org/education/exam-req/edu-exam-p-detail/
https://www.soa.org/education/exam-req/edu-exam-fm-detail/
If you're satisfied post cute doggos, this is a new rule for all of /mg/ from here on out.

>>10679868
I did. Or, actually, I did that for modules. It was a problem in the Algebra II exam.

>>10679885

can't decide whether I'm jealous that you got to learn about this shit in a college course or if I'm laughing at you because you had to take a college course to learn about this shit

>>10678820
https://github.com/dewanai/ranodm-crap

It still uses the built-in random generator of python, which isn't optimal, and the decisional tree could be improved.

Also, I have added matrices:
[eqn]
\left[\begin{array}{cc}x & \left[\begin{array}{cc}\left[\begin{array}{cc}\Theta & x \\ x & \left(y \right)^{\sqrt{{\left[\begin{array}{cc}y & y \\ \left[\begin{array}{cc}\frac{\mathbb{Q} }{\uparrow } & \left(\kappa \right)^{y } \\ \sigma & \left[\begin{array}{cc}\mathbb{Z} & \nabla \\ \Lambda & \left(\sum_{i\in\mathcal{F}(\eta,\omega,\partial)} \left(\uparrow \right)\right)^{\left(\left[\begin{array}{cc}\diamondsuit & \sqrt{{\left[\begin{array}{cc}\Theta & \Lambda \times \beth \\ \clubsuit & \left[\begin{array}{cc}\left[\begin{array}{cc}\left(\left[\begin{array}{cc}\left[\begin{array}{cc}\mathbb{R} & \aleph \\ \partial & \left[\begin{array}{cc}\left[\begin{array}{cc}\natural & \Theta \\ \left[\begin{array}{cc}\beth & \left[\begin{array}{cc}\zeta & \left[\begin{array}{cc}\left(\eta\right)_{\zeta} & \spadesuit \\ \spadesuit & \sigma\end{array}\right] \\ \Omega & \mu \end{array}\right] \\ \mu & \alpha \end{array}\right] & x \end{array}\right] & \omega \\ \eta & y \end{array}\right]\end{array}\right] & \sigma \\ y & \mho \end{array}\right]\right)^\mbox{It follows that} & \diamondsuit \\ \Theta & \sum_{i=\hbar}^{x} \left(\hbar \right)\end{array}\right] & \varepsilon \\ \beta & \downarrow \end{array}\right]\end{array}\right]}} \\ \eta & \partial \end{array}\right]\right)^\mbox{It follows that}}\end{array}\right]\end{array}\right] & \mathbb{R} \end{array}\right]}}}\end{array}\right] & \Theta \\ \sigma & \mathbb{C} \end{array}\right] \\ \downarrow & \mathbb{R} \end{array}\right] \eta \alpha \zeta
[/eqn]

>>10680021
I've not seen "It follows that" inside a matrix before. Did you fuck that tex up somehow?

>>10680021

>>10680039
No, I have added \mbox{It follows that} among the allowed operations.

>>10680066
I never post on math pages because you guys are serious NERDS
Put your nerdiness to good use and learn physics

>>10680067
>good use
>learn physics
But physics it's just boring repetitive calculations around a small set of equations that somehow have a connection with the real world, and whoever manages to summarize the largest number of equations into a single one wins. How is that good use?

>>10671876
You're just lazy and making rationalizations for your laziness

>>10679634
everyone that replied to you (including you) are retarded

Clearly the circle is regular everywhere, as you should know because you have fucking eyes.

Now, (1-y)/x = (1-y)x/x^2 = (1-y)x/(1-y^2)=x/(1+y). So clearly it is regular at x=0, because it has a different representation there that is in fact regular.

>>10679634
As you know from general trig, the tangent line to (a, b) in the unit circle has linear coefficient -a/b. This gives us the direction in which we differentiate the function to find it's partial derivative along the circle.
The rest is left to the reader.

anyone willing to help me out with a spivak problem?

discord: rogalik#3582

>>10680097
>everyone that replied to you (including you) are retarded
no u
>Clearly the circle is regular everywhere, as you should know because you have fucking eyes.
neither here nor there
>Now, (1-y)/x = (1-y)x/x^2 = (1-y)x/(1-y^2)=x/(1+y). So clearly it is regular at x=0, because it has a different representation there that is in fact regular.
No. There are two points on the circle such that x = 0: (0,1) and (0,-1). The function is regular at (0,1) because the second representation you give is regular at (0,1) but the first representation clearly shows a pole at (0,-1).

>>10677601
actually, I love the feel of opening a new book of maths and staring at a bunch of incomprehensible shit: it's like entering unexplored ground, it's fun

>>10680021

this looks like something I would see hanged on the CEO's room

>>10680085

>You're just lazy and making rationalizations for your laziness

We brainlets have no chance to lean math. It's like saying someone's German is bad because they're lazy -- if you didn't learn math rigorously as a child and/or teen your brain will be literally wired differently than all those jewish and asian kids doing calculus at 6th grade.

>>10680509
>anyone willing to help me out with a spivak problem?
Sure.
>discord
Oh, never mind.

>>10680127
Is there even an algebraic analog of trig functions when considering geometry over general fields?

>Inb4 wildberger rational trig

I'm learning baby level math and enjoying the shit out of proving trig identities and solving trig equations. Are there similar activities in higher math?

>>10681990
>trig identities and solving trig equations
Please, tell me what you're using for this.
>similar activities in higher math?
Yes. Personally, I love doing this in abstract and linear algebra.

>>10682036
>Please, tell me what you're using for this.
Just the basic stuff they teach in high school like sin^2 x + cos^2 x = 1, double/half angle formulas, etc.
Do I get more options later?

>Yes. Personally, I love doing this in abstract and linear algebra.
I guess I have some things to look forward to.

>>10679931
I was never in "college".

>>10682059
Thanks. Also, if you're into proofs, look for some intro to number theory books.

>>10682059
>Do I get more options later?
Sorry, didn't read the question. So, I asked becuase I'm in dire need to learn trig once and for all.
So, since you're having fun with trigs, I guess you could see some other geometry things like congruence of triangles, if you haven't already that is.

>>10681225
Nah, I know many people that were lazy until senior years and became very good with some effort. One of them topped 2nd year math at our uni (which is arguably one of the best in the country).

Hi guys, can someone help me to prove this? It's about superior and inferior sums (of Riemann sums). The 'log' is neperian log, actually. I know that there is a property that only one number exists between a inferior and a superior sum and that (by logic) should be log 2, so the problem would be to prove that the left side of the inequality is a inferior sum and the right side is a superior, but I'm struggling to do that. Can any mathfag help me? Thanks in advance.

>>10682139
is your country a shithole?

>>10682200
Aussie land, so not that shit.

>>10682232
lol what's your 2nd year of math, Lang's Basic Mathematics

>>10682172
De donde eres, joto?

What is best for a 2nd course in linear algebra?

>>10681738
What do you mean by analog of trig functions ? What property would you like to emulate ?

>>10681738
You can define them as formal power series over any field, but you need some notion of convergence (e.g. a normed field) if you want them to actually be functions. I suppose you could work backwards and try to interpret the inputs of these as "angles" somehow.

How doable is it to self-study graduate level mathematics on your own? I’m about to graduate with my bachelors in math and I want to keep learning, but I have no desire to stay in academia. I’m not looking to get to the level where I can understand cutting edge research, but I would like to learn more advanced topics in algebra, analysis, topology, differential geometry, etc. Any advice appreciated

>>10682635
Are you close to your favorite subject's professor? If yes, just straight up ask him to guide you through your studies.

Good question, I wonder if there are any online tutoring services for high level math.

>>10682657
I am not however I do have a good friend going through a math PhD program that I would be able to bounce things off of, I just don’t want to rely on him too much

grothendieck

>>10682657

Terrible idea

It's doable. The only thing limiting you is your motivation to do it.

>>10682751
>Grothendieck

>>10682635
Just give it a try. Of course it is *doable*, you just have to put in the effort.

Pick some field, then some textbook on a part of that field which is somewhat more advanced then what you know and get working.

For me, there is a real benefit in having a university type system with lectures and exercises, but not having that isn't a real barrier if you are disciplined.

what are the prerequisites for category theory? I have solid understanding of everything but only advanced in discrete math

How would I prove that the intersection of finitely many Borel algebras is itself a Borel algebra? Haven't done any maths in about 8 months now and clearly I'm regressed to being terrible at it.

>>10679852
and who's fault is that?

>>10682916
Prerequisites almost none, except maybe examples. Also, it gets messy soon.

>>10682916
>what are the prerequisites for category theory?
Autism.

>>10682917
>How would I prove that the intersection of finitely many Borel algebras is itself a Borel algebra?
What have you tried?

>>10682635
>How doable is it to self-study graduate level mathematics on your own?
Why don't you try it and find out?

>>10682917
the intersection of arbitrarily many borel algebras is a borel algebra.
come on now. if you have sets in all of them and do a countable operation on them, then the result is in each algebra, so it is in the intersection.

Threadly reminder to work with physicists.

>>10682916
commie algebra is a good place to see category theory at work,
not that you need it

>>10682801
Was Grothendieck a 2hu?

>>10682751
It’s grothendick anon

>>10682916
Category theory is very self referential, so you meed to figure out your motivation early. In other words, the best way to study category thy is by using it to better understand something else (usually algebraic objects)

>>10682751

>find out some problem related to my thesis was "solved" by a physishit a while back
>go read his paper
>his solution to a previously unsolved problem is just writing down the identity with no trace of justification offered at all
>this was formally published

>>10671823
>good object
>very good object

>>10682172
Somebody?

>>10671823
Guys I am gonna lose my shit, help me understand this. He finds the correlation coefficient to be /4 at the end.

can any topologist anons give me some advice? I am going to start preparing for my masters dissertation in symplectic geometry next academic year but I might need to learn some machinery from algebraic topology

is there a good way of getting familiar with homology, chain complexes and singular homology without having to read Hatcher's from cover to cover or end up spending half the year enrolling in like 3 different topology courses just to get to the useful part right at the end?

if anyone knows some skeleton chapter numbers from Hatcher or some sort of crash course anywhere online that would be much appreciated

>>10682172first use the mean value theorem to show that 1+1/2+...+1/n=ln n +C+something smaller then 1/n, then go from there.

>>10682172
>It's about superior and inferior sums (of Riemann sums).
Yeah, I mean, that's it. So you know that [math]\int_{1}^{2}\frac{1}{t} dt=\ln2-\ln1=\ln2 [/math]. Partition the interval [math][1,2][/math] into n equal intervals [math][1,1+\frac{1}{n}],[1+\frac{1}{n},1+\frac{2}{n}],\dots,[2-\frac{1}{n},2][/math].
Now, all you need to do is construct the upper and lower sums, which by definition serve as the upper and lower bounds of your integral. The lower bound would be
[eqn]\frac{1}{n}\left(\sum_{i=1}^{n} \frac{1}{1+\frac{i}{n}} \right)[/eqn]
And the upper bound would be
[eqn]\frac{1}{n}\left(\sum_{i=1}^{n} \frac{1}{1+\frac{i-1}{n}} \right)[/eqn]
This makes sense since [math]\frac{1}{t}[/math] is a decreasing function so the left sums should be greater than the right sums.

>well anon, we spend three months in this chapter, I don't think I'm going to find a lot of errors
>whole chapter is filled with more corrections than the last ones
whyyyyyyyyyy

Israel M. Gelfand

Can someone help me solve this problem? I've spent way too long on it. I'm just starting calc III so excuse my retardation.

>>10686341
First off, you're on /sci/, we're all retards. Second, this is a /sqt/ question, not an /mg/ one, take it there first. Third, it's literally the Pythagorean theorem dude.

[eqn] \dot x \;=\; f\left(x\right), \qquad x\left(0\right) \;=\; x_0 [/eqn]
>[math]f[/math] is chaotic iff [math] x_0 \;\longmapsto\; x [/math] is not Lipschitz-continuous for the uniform norm
Is that how chaotic systems are defined?

I want to get into the field of Computer Graphics but it's been many years since I did anything math related and I feel like I forgot everything but most importantly my ability to have a plan to solve things and that makes me scared when I try to get into linear algebra,geometry and I always postpone it thus hurting my attempt getting into the field since those things are vital.

I have this idea of "relearning math" going from highschool to university stuff. How would I go about it ? Would Khan Academy suffice or are there any books that would provide a more complete experience ?

Mainly I would like to have the ability to actually really know what I'm talking about a model,rotating,translating,quaternions and having the ability to picture them perfectly in my head while doing so.

>>10686341
What have you tried?

>>10685377
>physics paper
>math argument
hoo_boy.jpg
>"lol it just werks"
every time

>>10685752
Don't read Hatcher. That's my first tip. Read Rotman instead. 2 tips given'sed.

Is there a typo at the end? It should be Aij instead of Aik right?

Holly shit, it's so frustrating when a book has literally a typo on every section.

>>10671823
I've gotten a quite large increase of deja vu lately, sometimes it's as frequent as once per day, but it happens at least every other week.

Is this bad?

>>10686830
Math is in large parts just rote learning and identifying what you have already seen, multiple déjà vus is a sign of progress. Too much déjà vus is a sign of too redundant material.

>>10686834
Shit thought i posted in /med/, thanks for answer though.

>>10685667
lol what’s this from?

>>10686834
>Math is in large parts just rote learning
What? If you have to rote you're doing math wrong.

>>10686867
Of course you have to deeply understand the math you are learning but if you can't remember the theorem you read you will never be able use them to solve problems or deduce harder theorems.

>goldfish memory
>if you don't write things down I can't even solve a multiplication of two three digits memory


Does this get better with practise or am I going to have to take meds?

>>10686871
do you use anki or something

>>10686871
It's no more than the amount of memorization required in any other field of study, especially medicine

>>10686404
No
There's more than one definition
Devaney's is common:
1.) Topological transitivity
2.) Periodic points are dense in domain
These imply the third
3.) Sensitive dependence on initial conditions

>>10679542
A line is literally a curve. A curve is the image of a continuous map from an interval onto R^n.

Who /wildberger/ here?

Am I right guys?

>>10687139
Sure

>>10677308
pls give new invitation, this one expired

>>10671823
Why Jews love mathematics and physics so much?

>>10687181
Jews are ugly in average so they need to cope with STEM fields.

>>10685377
>physishit
>related to my thesis
You're not much better than the physishit.

>>10686968
But lines aren't curved

>>10685752
>http://geometry.ma.ic.ac.uk/acorti/wp-content/uploads/2016/07/AlgTop.pdf
but no cohomology

>http://www.mit.edu/~sanathd/18-905-notes.pdf
everything you need

>>10687139
Yes, this is right

Both motors will spin at 280 rpm.
Is the resultant output rpm 280(280)rpm?

>>10688805
>280(280)
You fucking what m8? It would be roughly 280+280.
I am assuming the extra load/extra drag at higher speed doesn't slow them down. It would in real life but this is vidya so idk.

>>10686667
I figured it out. I originally assumed O was some random point and started doing some futile algebra (which obviously didn't lead anywhere) until I realised that O is the origin. It took me like 2 seconds to figure it out once I realised that.

>Working on unsolved problem in combinatorics
>getting real discouraged cuz it's so fucking hard and I'm an amateur

>uni doesn't post required material for class until a few weeks before the semester starts
I just want to learn.

>>10689227
>not looking up the syllabus from the last semester the course was taught

do you guys have friends?

>>10689765
I don't

>>10689082
What problem? I'm working on polyominoes

>>10689082
What's your problem anon? I teach math to pre-schoolers, but I enjoy small combinatorics questions, wanna see what's out there.

>>10690008

have you ever encountered any child prodigies? I find it very inquiring how someone with so little living can just pick up concepts on a whim like that so easily.

>>10689765
I have 3.

Hey, help me BTFO a tripfag from /vg/ you big brain'd math-heads.

>>10690250

Give me something stylish so I can shit all over this mongoloid retard.

In the future, would you need to know common core math if you want to be a professor in freedomsland?

>>10679877
ty lad

I think everyone here is a crank we should sll kill ourselves

>>10686968
A curve is a one dimensional variety you fucking cretin

>>10690638
>A curve is a one dimensional variety you fucking cretin
Do you really need to swear?

>>10678195
>"it follows that"

kek

How do you decide what is going to be your (next/first) research topic?

>>10690638
>variety
Bonjour mio amigo.

>>10690984
doesn't variety has different context in non-commutative algebra?

Can anyone recommend a textbook or online resource for learning about quaternions? I'm reading a novel (Against the Day by Thomas Pynchon) where they feature prominently in the plot and themes, and I want to get solid understanding of the subject so I can fully appreciate what's going on in the book.

>>10691052

Your book may have been written at a time when the literature of physics used quaternions instead of vector analysis. So, the terminology would be archaic.

http://fexpr.blogspot.com/2014/03/the-great-vectors-versus-quaternions.html?m=1

Anyway, there is a short proof that quaternions describe rotation on wikipedia. The wikipedia article has everything you want to know.

>>10691048
Depends on the language.
In most romance languages, variety and manifold are used interchangeably.

>>10691095
Thanks for the info. Actually the book from 2006, but it's set in the historical period talked about in the introduction in your link. Funny enough I go to WPI, which is the university that paper was written for, which is exactly the kind of coincidence that often happens in Pynchon novels...

>>10691052
>I'm reading a novel (Against the Day by Thomas Pynchon)
you're not actually supposed to read pynchon, he's just a /lit/ meme

>>10691127
Au contraire mon frère. Pynchon is my favorite author, and once I finish this novel I will have read everything he's ever published. Give him a try, you might like it.

>>10691052
Based. Against the Day is Pinecone's best book imo. You don't need to much about quaternions other than that they were used to represents rotations in 3d space.

>>10689765
as of right now i have 127 friends

How do I prove that a meromorphic function over all C with a limit at infinity (either infinite or finite) is a rational function?

Do I use the residue at infinity by treating infinity as a pole of the function?

lads honestly fuck dividing rational numbers am I right

>>10691688
Something something Riemann sphere.

>>10691714
L'Hospital's might also work.

Solve this

>>10690638
>a curve is a one-dimensional variety
THATS A FUCKING ALGEBRAIC CURVE!!!!!! A FUCKING CURVE IS THE GOD DAMN IMAGE OF A CONTINUOUS MAP FROM AN OPEN INTERVAL TO R^n!!!!!! FUCK YOU!!!! FUCK YOU SO FUCKING MUCH!!!!!!

I don't know shit about topology.

I'm reading the book "Basic algebraic geometry" by Shafarevich and it doesn't seem to mention anything about topology. But people have told me "Anon you retard, you'll never learn algebraic geometry without topology!"

Why do people say that? How is algebraic geometry related to topology?

For example, the book says nothing about Zariski topologies in the part about quasiprojective varieties. But if you look at the page for quasiprojective varieties on wikipedia, it makes it sound like they are very important in the theory of quasiprojective varieties

>>10691688
Consider the rational function with the same zeroes and poles with multiplicity as the function (why are there only finitely many of them ?). Then the ratio of the two functions is meromorphic but has no zeres or poles, hence it is constant (same argument as for the fundamental theorem of algebra)

>>10691052
https://eater.net/quaternions

>>10691052
I read that very recently, heres my advice: don't study quaternions, study History of maths and science in the 19th century. In particular
-Maxwell and electromagnetism
-optics
-railroads
-maths scene at Göttingen

The technical details of the mathematical objects arent important to the story. The mood about the progress of science in the turn of that century, however, is important. At that time, anything must have seemed possible, before Gödel shattered Hilbert's dream, before the World Wars

>>10691865
>GOD DAMN
Cool it down with the swearing.

any thoughts on mathematical linguistics?

>>10691986
When you start having to extract invariants (coherent cohomology, Hodge theory), you'll probably find your motivation to understand algebraic topology.

Follow your nose

Is there a way to assign factorization values to the negative integers? Gamma function dosent work. I need factorials of negative natural numbers, but not necessarily all reals.

>>10693733
Define [math]n![/math] to be the product of all integers less or equal than [math]n[/math].
So [math]n![/math] is indefinite for all negative numbers, and 0 otherwise.

>>10693756
well i know that, i was just wondering if anyone has assigned some meaningful values to the negatives by some function?

>>10693733
>gamma function doesn't work
Why? I can't really suggest a way of assigning values unless I know the general rules you want the function to follow.

>>10693834
It diverges at n <= -1
im more interested to know if anyone has done this, and in what context.

>>10693841
https://en.wikipedia.org/wiki/Factorial#Factorial_of_non-integer_values

>>10693733
There isn't a single, clear way to go about it, here's some options that might make sense depending on what you need it for: https://mathoverflow.net/questions/10124/the-factorial-of-1-2-3

>>10691986
>read shit book for undergrads
>doesn't introduce material at an appropriate level of generality and rigor
huh no shit

>>10676606
/r/ book? I started reading Halmos Measure Theory but it seems kind of archaic.

>>10686829
Use Friedberg

Is there a fucking charter detailing which mathjax commands are legal in /sci/'s LaTeX implementation? If there isn't, I'm gonna make one for you guys.
Also, are there any permissions I must obtain before I can add stuff like that to our wiki?

>>10691986
Have you actually been reading the book ? Of course it talks about the Zariski topology.
Not that you need a lot of topology to read Shafarevich but you need to be somewhat comfortable with point-set topology because it is the language in which one talks about local properties and closeness in math in general.

>>10694726

there is nothing about it in that chapter or any mention of topology before then.

Daily Putnam Thread - Part A >>10694987

Part B >>10695011

I am procrastinating again

>>10694954
I have it on my desk rn and yes there is. It does not use the term "Zariski topology" but the Zariski topology is defined and used throughout the book. The second section of the first chapter is literally named "closed subsets of affine space".
Because the book aims for minimal formalism and focuses on enlightening examples and problems (as russian books tend to do), there is little discussion of topology since it only serves as a language, but it is clear that the reader is assumed to be familiar with basic topology

>>10695073

Fair enough, I guess I actually did learn some stuff about topology last year without even realizing it

P.s., If a quasiprojective variety is an open subset of a closed projective set, why are closed projective sets "obviously" quasiprojective?

>>10695119
Open sets of themselves, lad.

>>10695037
someone stop me I have autism

>>10695129

So it is an implication of the fact that the empty set is closed?

>>10695138
Yes.

If I have R2 and R with their usual topologies, how do I show that multiplication (described by (x,y)->xy) is a continuous function?

>>10695138
Don't know any algebraic geometry, but the underlying set is open with respect to any topology, hence the empty set is closed.

>>10695149
epsilon delta

what is the fourier transform of sin(x)?

i've looked at several tables of transformations and all of the give variations of a combination of delta functions, but I don't know how to get the right coefficients

for the example in the pic, what determines the pi/j?

for the problem im working on, the omega_0 is just 1

>>10695214
j is maymay electrician notation for the imaginary unit. Delta is Kronecker delta.

What is the transfer homomorphism in cohomology?

>>10695214
The pi is there because you're calculating the transform using angular frequency (radians/sec) rather than frequency (cycles/sec).

The thing about the Dirac delta is that its value is either 0 or infinity. Its "scale" is such that its integral is 1. The coefficient in the Fourier transform of a periodic function (which will consist entirely of deltas) is what's required so that the inverse transform returns the original function.

When the function is periodic with period T and f is an integer multiple of 1/T, the Fourier integral over a single cycle is a finite value, so the integral over [-∞,∞] is infinite. For periodic functions, it might be more useful to think in terms of the Fourier series than the Fourier transform.

>>10693733
Exploit linearity of factorial function to generalize as follows:
[eqn](-n)! = (n - 2n)! = n! - (2n)![/eqn]

>>10693733
What property do you want to have/preserve ? If you cannot answer this, then no choice of values is better than any other

>>10695495

> linearity of the factorial function

boi

What's the rigorous definition of two numbers being equal?
[math]\forall\, \epsilon > 0; x,y \text{ equal if }|x-y| < \epsilon[/math]?

>>10695650
What? Equality is a relation defined in the setting of the formal language itself.
If you just want to use the order, with the <, which you are anyway, x and y are equal if neither x < y nor y < x. But that's stupid and not the definition.

What are the "rules" for messing around with power/laurent series? Is there any theorem or anything about when you can substitute certain things or just divide/whatever to make it work? I realized I never learned what kind of valid manipulations can be done to power series even though I took 2 semesters of honors analysis.

>>10695650
something like [math] x=y\leftrightarrow (x,y)\in\{(x,x):x\in R\} [/math] maybe?

>>10695650
that definition would only work in a complete metric space

>>10695746
First of all, the power series operator is linear. Second of all, you can rearrange as long as you have absolute convergence. Further, you can rearrange finite amounts of terms, and Riemann has a theorem that says that, if a power series converges, but doesn't absolutely, you can rearrange it to get whatever real. You can also do some hard to explain stuff like "sum n and n+1 and set it as the n/2 of another sequence" and it will converge to the same value as the original series as long as that one converges.
Also other stuff I rememberen't. It's relatively jntuitive.

>>10695750
why complete? that definition is literally equivalent to d(x, y) = 0 which by definition means x = y.

>>10695650
[eqn]x=y \;\; \Leftrightarrow \;\; x \in \{y\}[/eqn]

>>10695650
The traditional second order definition is [for all predicates P, P(a) <=> P(b) ].

>>10695650
[math]x=y[/math] if and only if the succession [math]x, x, x, \dots[/math] converges to [math]y.[/math]

>>10696207
*Provided the space is Hausdorff

>>10696207
>converges
in which topology? moron.

>>10696225
all of them


moron

>>10691688
a meromorphic function f can be interpreted as a holomorphic (in the sense of Riemann surfaces) function from the Riemann sphere into itself. Since the Riemann sphere is compact, and the poles are discrete, there are only finitely many zeroes and poles. Take a rational function g that has the same amount of poles and zeroes. Then f/g will be bounded on all of C (since there are no poles). By Liouville, it is constant. Hence f is rational.

what do i study to improve as a gambler?

>>10693578
What is it?

>>10696191
that would make x=x equivalent to
x an element of x
which contradicts the axiom of regularity

but to be fair, regularity is the worst axiom

>>10696229
ah, so in the discrete topology in particular? so you mean that they're the same fucking point? go fuck yourself you stupid sack of shit.

>>10697008

>>10696229
>>10697008
Please friends, treat one another with kindness.

>>10695750
Why complete?

Actually I was interested for a rigorous proof of 0.999...=1 but there's one on wikipedia.

I'm still interested in what it really means for 2 numbers (in [math]\mathbb R[/math] and [math]\mathbb Q[/math]) to be equal

the so-called "real" numbers

what fields in applied mathematics/mathematical physics are good to go into for PhD?

something with low hanging fruit

>>10697470
Dude just learn some logic and all these questions will dissapear themselves. You just need to define what a real/rational numbers is then you will know when to count them equal. Similar with 0.99 and 1 you need a fomral definiton of what a decimal is then you can formally prove this.