Mathematics general

[Anonymous No.8416217]

Feel free to post about mathematics in general, just keep the shitposting about 300k starting, CS, and stuff out of this thread. Let's actually talk about something meaningful this time.

>what are you currently reading?
>any problems?
>any suggestions for cool exercises?

Personally right now I am doing some reading in Kähler manifolds, in particular from the viewpoint of symplectic geometry.

I was wondering if anyone has any good books on differential equations as that has been a very abandoned subject for me since my undergraduate years. Maybe there is a good book in Springer's library? Thanks!

[Anonymous No.8416225]

I wake up in the morning and the first thing I remember is that I have my goddamn doctorate in mathematics, this results in an immediate rush of endorphins and testosterone which will last for the rest of the day. I then get to work writing down the proofs to the theorems that I solved in my head last night. After that I fuck my 10/10 wife while thinking about my research. I then leave without eating breakfast since I no longer require sustenance. Next follows morning lectures gracing plebs with some invaluable insight into my mind and its firm grasp on the most important field of study in human history. The rest of the I spend developing various mathematical structures on blackboards, the results of which will applied to curing all cancers and building thermodynamically efficient cold fusion reactors. As the day winds down I ponder my surreal existence, I have to dig up my degree to remind myself of the reality that I do in fact have a PhD in mathematics. Finally I fall asleep holding my doctorate and 300k salary slip in my arms.

And then I do it all again.

[Anonymous No.8416227]

>>8416217
I'm a mathematics undergraduate and I'm doing a bonus-project on stochastic processes. Right now I have gone over the Brownian motion (properties, Lévy construction, filtrations), and I at least want to go over the stochastic integral after this. Any recommendations for other things I can include? Or recommendations for literature?

[Anonymous No.8416252]

>>8416227
You could take a look at ergodic theory afterwards. It might be a bit off to the side but you never know.

A reference suggestion for that if you wanted to look at it:
"Ergodic Theory, with a view towards Number Theory" - Einsiedler, Manfred, Ward, Thomas

[Anonymous No.8416303]

I said it before and in the cateogry theory thread: I'm gonna buy the Idris book and learn that in 2017, among other things. I'm gonna make threads about it here, or at least I'll start and see how it goes.

Actually, I'm probably moving in the few weeks and gonna have a "new start". I was thinking about doing some youtube videos on math/physics, but maybe that would be taking the avatarfagging too far.
Anyone else motivated to do such a thing? You know, raising the content value of /sci/ in that regard, maybe getting into some proper discussion. Like two people explaining and discussing shit, you'd have the videos and a point for discussions in the thread.

[Anonymous No.8416312]

I've been thinking... I think 0 and infinity are in the wrong place.

I think the order should be:

Infinity, 1, 2, 3, 4, ... , 99999999, ..., 0

It just makes so much more sense

[Anonymous No.8416317]

>>8416303
I would love to see the content value of /sci/ rise, especially in terms of discussion of mathematics. I think that we would need the mods/janitors to really crack down on shitposting threads though to make that happen.

But yeah we could definitely start here and try to make some useful teaching and discussion events. Demo the idea.

[Anonymous No.8416327]

>>8416312
Of course! Since -1/12 is the supremum of the naturals and clearly -1/12 < 0.

[OHP No.8416345]

>>8416303
>youtube series
This is an awesome idea, but I was apprehensive for the same reason when I thought about doing this: people really loathe avatarfagging, namefaggin, tripfagging, and in general any identityfagging. If you start doing it and it is well-received on the whole, I will likely try my hand at it as well, for my own subject matter. Best of luck!

[Anonymous No.8416349]

>>8416217
I'm reading Rudin for my analysis class and Hoffman and Kunze for my linear algebra class. Not reading much else because midterms are near. Hold me anon

[Anonymous No.8416357]

What's n meant to represent here? The pic is from the Wikipedia on Midy's theorem which I just learned about and find interesting.

[Anonymous No.8416372]

>>8416357
It's just the fact that you need the decimal period (the number of decimals until they cycle again) is even. So instead of writing "n, where n is even" they just say "2n". They write out the expansion like that just to show that the pattern continues up to a_n and doesn't change beyond that.

[Anonymous No.8416375]

>>8416345
I agree, I'd love to see more people try this. I was thinking about doing a youtube lecture series on topology. Just recently I was thinking about doing a lecture thread about it.

[OHP No.8416379]

>>8416375
I really think it would enrich the dialogue of the board. Hopefully your lectures make more of a splash than my category theory one.

[Anonymous No.8416380]

>>8416372
Oh. Simple enough. Thanks.

[Anonymous No.8416383]

>>8416379
How did your category theory one end up? I noticed on this board it is not common to find people who have a desire to learn, so much of the community is just here to spread memes and argue about meaningless things.

[Anonymous No.8416386]

>>8416345
Since I'm interested in what you do, you speaking about your stuff would be great for a conversation - assuming you try and explain the basic stuff too. I think I remember you doing an elaboration thread and a series of posts, but it wasn't really basic at all.

>>8416317
Demo the idea? I'm not at my new place yet, but you can do videos and rant quickly, like so
https://youtu.be/eAffVHGjeR8

Well who are you and what can you do?
I'd eventually do "teachy" videos, but to gain some critical mass an interest, one would first want to do very informal discussion or accessible topic stuff, then one can start getting to the roots of things and go towards topics that are more popular in research math.

[OHP No.8416391]

>>8416383
Very few replies. I offered a preliminary set theory lecture which got a lot of replies, but a lot of people decided to be overly pedantic when it was meant to be an overview so I could use the language of functions between collections to define categories. I think it is indicative of what I consider to be a souring of mathematics, wherein people choose to treat theorems as the Truth, and axioms as the basis, even though mathematics is a dynamic body that builds off of our intuitions of the Truth and approximates it via theorems, which are then founded via axioms.

[OHP No.8416395]

>>8416386
I appreciate the feedback. I have been wanting to put my musings on YouTube to initiate some discussion for a while. Maybe I will begin organizing an outline for the stuff I want to go over. And, if the thread you mentioned was the category theory one, it was all material covered in the first five chapters of Mac Lane, so I think the inaccessibility was due primarily to my failure to communicate the ideas cogently. I'll have to work on that!

[Anonymous No.8416398]

>>8416391
Yeah there is a big "axiom meme" thing on this board. It's like babby's first math experience on /sci/ and they take it too far. Majority of professional mathematicians don't care about axioms.

[Anonymous No.8416442]

>>8416391
I see you around a lot, you even helped me once (thanks!). How do you put up with the shit this board spouts trying to give a lecture?

[OHP No.8416444]

>>8416398
Yes, it's largely why the French school has such a successful track record. They basically just redefine things to fit the theorems that "should" be true, and then go from there. The axioms are put in place to give it all a rigorous backbone.

[OHP No.8416446]

>>8416442
Eh, I kind of went in expecting it to flop. I see potential in this board, including the misguided contributors, and my rampant romanticism won't let me throw in the towel. :P

[Anonymous No.8416451]

If [math]P[/math] is a projective generator in an abelian category [math]\textbf{A}[/math] and [math]R = Hom_{ \textbf{A}} (P, P)[/math], I've made a list of questions I will look into:
>if [math]R[/math] is a localized/commutative/noetherian/semilocal ring, what can be said about [math]\textbf{A}[/math]?
>can i import a notion analogous to Morita equivalences into abelian categories using the endomorphism rings like [math]R[/math]?
>what if [math]R-\textbf{mod}[/math] or the subcategory equivalent to [math]\textbf{A}[/math] is a localized category?
These should keep me busy for a while.

[Anonymous No.8416452]

whats 9 plus 10

[Anonymous No.8416454]

>>8416446
What kind of background do you have in mathematics? What's your specialty? You and I seem to share a lot of values in terms of wanting the community to flourish despite evidence against its possibilty.

[Anonymous No.8416458]

>>8416446
Yep this board could certainly profit from having a capable janitor. Would be awesome to see the major-ranking threads and aversion to social sciences fuck off to make place for some actual scientific discussion. But I stick around for the random interesting facts and resources that float by.
Out of curiosity, youre in mathematics right? At what level and in what field, if it doesnt reveal too much?

[OHP No.8416464]

>>8416451
Mmm, nice questions anon! For some odd reason, I get the feeling that the tools Scholze developed for working with perfectoid spaces would be of use. It's a very strange feeling, though, and I wouldn't be surprised if it was just silliness. Anyways, keep me posted! Very cool indeed.

Also, remember that the endomorphism object is the pullback along contravariant and covariant hom functors on the base object (part of the theory of Grothendieck constructions), and you can paste pullbacks of course.

[Anonymous No.8416476]

>>8416452
Twenny-wan

[OHP No.8416480]

>>8416454
I'm a sophomore, but have been independently studying math for five years now. I started doing my own research a couple of years ago. Right now a professor is trying to get me into the graduate program early so that I can focus on my research. It's nice to connect with people that share my ideologies on here!

>>8416458
My field is higher category theory/homotopy theory. My level of education is above^, and I suppose my identity is fairly obvious if anyone here happens to frequent Quora (I'm a top writer in a few areas there). I'm also the guy that did that survey on the fields of /sci/.

[Anonymous No.8416484]

>>8416452
At least three levels above my pay grade

[Anonymous No.8416519]

>>8416480
Category and homotopy theory research as a second year. You mustve done a lot in the years leading up to uni, fuck. Hats off.

>I'm also the guy that did that survey on the fields of /sci/.
I probably missed that, Im not around much. Got a link handy?

[Anonymous No.8416545]

>>8416480
Were you studying math in highschool? Or are you older than 19/20?

[Anonymous No.8416547]

>>8416464
>For some odd reason, I get the feeling that the tools Scholze developed for working with perfectoid spaces would be of use. It's a very strange feeling, though, and I wouldn't be surprised if it was just silliness
I shall check these out. If they happen to be useful, I will gladly utilize them, and otherwise I'll just learn something new and maybe cool.

>Also, remember that the endomorphism object is the pullback along contravariant and covariant hom functors on the base object (part of the theory of Grothendieck constructions), and you can paste pullbacks of course.
Yes, this is one of the things I was thinking about using. Thanks for answering.

[Anonymous No.8416576]

Why is Category Theory so popular nowadays? Does it have practical opportunities to solve Number Theory problems, for example? Or does it just like Cantor's Set Theory for patching logical holes or something like that?

[OHP No.8416581]

>>8416519
Here you go: https://docs.google.com/spreadsheets/d/1LYhJtdkcg3-3u9FZ2av-zAjDpjkCav0j_EJ7l89Z6mA/edit?usp=drive_web

Please don't dox, thank you!

>>8416545
I'm twenty. My senior year I started learning category theory, and then during the skip year after graduation I honed my knowledge and started my research.

>>8416547
>learn something new and maybe cool
Yes, quite a few actually think he will earn the next Fields medal for his work! You'll probably like it.

[OHP No.8416591]

>>8416576
It has been very successful in unifying distant concepts, so as to permit the translation of results in one field to another. For example, can you show that the product of two graphs is always a graph? It's a lengthy proof semantically, but synthetically it is evident from the fact that graphs are presheaves on a simple site (parallel arrows), and so they form an elementary topos.

[Anonymous No.8416597]

How can I get better at proving mathematical statements? Actually, how can I learn it? I'm ridiculously bad at proving the most simple concepts.

[OHP No.8416605]

>>8416597
Honestly, reading lots of proofs is the best way to hone that proof intuition. Seeing the tricks of trade, per se. A book like Halmos' Naive Set Theory has many great examples that aren't too conceptually challenging.

[Anonymous No.8416608]

>>8416597
I would say that you need to read more proofs. Also struggling is a good way of learning. Do you read a lot of mathematical literature? Lots of people just jump into math not having read a lot of it and they end up missing the point of proving stuff.

Also you might want to look up a foundations of mathematics course lecture notes or something. Usually universities have a course done in 1st or second year where you learn to do basic proofs in set theory and logic.

[Anonymous No.8416617]

>>8416581
Checking them out is a must!

>>8416597
Look at some simple proofs and try to extract the proof structure out of them. In simplicity, you assume something and either
>deduce it directly from the assumption
>use induction
>derive a contradiction from the assumption that the claim is false (if not constructivist)
I never read any books on how to prove stuff and, honestly saying, have this gut feeling they are pretty much redundant.

[Anonymous No.8416624]

>>8416576
Thank you. I don't understand clearly what you've written but I'm more motivated already.

[Anonymous No.8416628]

>>8416591
Thank you. I don't understand clearly what you've written but I'm more motivated already.

[OHP No.8416640]

>>8416628
Sorry about that, I was just trying to communicate the idea that category theory is a tool for pulling things together and drawing relationships. My example was probably not well-placed. I hope you enjoy what you learn about it!

[Anonymous No.8416692]

>>8416217
I've been working through KhanAcademy, and I'm enjoying it; however, I truly want to understand math. KhanAcademy is great in helping me understand simple rules in mathematics, but are there any good intro books to help me better analyze what I'm doing?

I'm working through Algebra II at the moment.

[Anonymous No.8416704]

Proof by induction - how the fuck do I do this? It's obvious that it's the product of two consecutive integers >0 but I can't make the right side have the form of the left side. It's gotta be something retardedly easy. Help?

[Anonymous No.8416713]

>>8416692
Take a look at these lecture notes and tell me how you manage them. Just take them slowly.

http://alistairsavage.ca/mat2762/notes/Hofstra-MAT2362.pdf

Also, >>8416597 can use these notes as well for learning some basic proofs.

[Anonymous No.8416715]

>>8416704
Show that it holds when n=0, then assume it holds for some n and derive from your assumption that it holds for n+1. Hint: (n+1)(n+2) = n(n+1) + 2(n+1)

[Anonymous No.8416718]

>>8416704
It's not true for all n>0.

[Anonymous No.8416733]

>>8416718
I brought that up to the instructor but he said that the idea is that it's the product of 2 consecutive integers (2)(3), (4)(5) etc and that it's not a mistake. It fails when you just simply plug in 2, for example, into n.

[Anonymous No.8416735]

>>8416217
Thank you for this post, OP. I'm studying differential forms right now, and have this problem:

At the beginning my prof defined differential forms as they should be -- forms acting on the elements of a tangent space. But when we started exterior derivative he switched to define them acting on vector fields.

I know, if there is a smooth partition of unity theorem, then they're equivalent. But we investigate it only after exterior derivative.

Next, I have a problem with vector fields acting on differential forms. What are they? Is this legit? And also the notion of continuity of a differential form (from definition). How can I compare differential form at different points? I suppose continuity means continuity in local coordinates, as I can't formulate it in terms of forms acting on elements of tangent space.

Next, I have just a leisurely question: in differential forms we use largely alternating property, even exterior derivative relies on it largely. But in physics they often use a more general notion of tensor fields. What can you even do with that one? What operations are useful with that object? I just wonder.

[Anonymous No.8416742]

>>8416733
It fails for all n>1.
I don't understand what you or your professor are trying to say.

[Anonymous No.8416744]

>>8416715
Can you throw me another bone? It's that 3 in the denominator of the right hand side that's fucking me up - I have tried several different approaches (including just now with your hint) and just can't get it.

Any help is much appreciated.

[Anonymous No.8416750]

>>8416744
It seems to be false, like these guys are saying here. Try this one on your own first:

[math]1^2 + 2^2 \dots n^2 = frac{n(n+1)(2n+1)}{6}[/math].

[Anonymous No.8416753]

>>8416744
It seems to be false, like these guys are saying here. Try this one on your own first:

[math]1^2+2^2 \dots n^2 = \frac{n(n+1)(2n+1)}{6}[/math]

[Anonymous No.8416754]

>"Use sage to compute the 1234th prime"
>use sage, write down 10,061
>get marked off half a point for not writing down the code I used to find it

Really? You didn't even ask for us to write the code down you asshat professor

[Anonymous No.8416777]

>>8416735
What is your reference text? I have never known vector fields applied to a differential form.
http://math.stackexchange.com/questions/265085/exterior-derivative-well-defined-on-vector-fields

[Anonymous No.8416808]

>>8416777
There's only an older reference text http://www.mi.ras.ru/~kazarian/papers/calc03.pdf, and I've just noticed that in this text he defines it in a usual way, huh.

>I have never known vector fields applied to a differential form
Vector field gives a tangent vector at each point of manifold, therefore k vector fields can be substituted to k-form at each point. Basically, this https://en.wikipedia.org/wiki/Exterior_derivative#In_terms_of_invariant_formula
This is the way he defines exterior derivative now.

[Anonymous No.8416832]

>>8416808
Oh you meant vector fields acting /as/ a differential form. I see. So your problem is that he is defining the exterior derivative on vector fields rather than differential forms? And what?

[Anonymous No.8416846]

>>8416735
>But in physics they often use a more general notion of tensor fields. What can you even do with that one?
For example, it allows one to define a metric on a manifold: rather than being a field of alternating multilinear forms like differential forms, it's a field of symmetric bilinear forms

[Anonymous No.8416854]

>>8416832
I mean differential forms acting /on/ vector fields (instead of elements of tangent space, as usual).

Although I almost get it now, I was confused about this way of defining them and was in doubt how do vector fields act on differential forms in this formula https://en.wikipedia.org/wiki/Exterior_derivative#In_terms_of_invariant_formula

[Anonymous No.8416871]

>>8416735
>How can I compare differential form at different points?
There is a notion of "vector bundle" that allows you to "glue" together the spaces of k-forms on each tangent space into one space (denoted by [math]\Lambda^k(T^*M)[/math]) in such a way that a differential k-form can be seen as a smooth function from [math]M[/math] to [math]\Lambda^k(T^*M)[/math]. But basically, it does mean that it is described by continuous functions in each set of local coordinates (the definition of the space of k-forms is set so that the two coincide).

[Anonymous No.8416878]

>>8416846
wow, that's interesting, thank you. so there are metric tensors, and therefore things from differential geometry (first/second form) might be looked as acting on them. Thank you, that's cool, I'll think about that.

And what about integration? Does integration work the same on tensor fields? Sorry if my questions are stupid, I'm just curious.

[Anonymous No.8417048]

>>8416581
>https://docs.google.com/spreadsheets/d/1LYhJtdkcg3-3u9FZ2av-zAjDpjkCav0j_EJ7l89Z6mA/edit?usp=drive_web
What does it mean? How did you measure it and why do measures on the same subject differ?

[Anonymous No.8417054]

>>8416217

[Anonymous No.8417162]

>>8416713
I really appreciate this. I will very slowly work my way through this.

[Anonymous No.8417228]

>>8417054
This is the correct answer, yes

[OHP No.8417283]

>>8417048
The Big Five is a personality assessment; I started a thread wherein people (anonymously) gave their test results from a fixed test and also their field of study. I was just collecting data. It was interesting. Make some plots and examine how different factors change for different fields.

[Anonymous No.8417529]

this is fucking awful
i'm a math major and i'm doing relatively well in my classes, but only because i spent the whole summer studying beforehand
one proof i proved during my summer studying took me almost a whole day
several other proofs took me multiple hours
and monday i was supposed to do 8 proofs for homework, due wednesday
if i hadn't studied over the summer i would not have been able to recognize how to work those proofs
and even with prior experience, it still takes me a very long time to complete proofs
it's been this way since the start of the semester
i'm terrified that i'm going to run into a few problems i can't solve
i've also been unable to leverage study time with work, and end up falling behind on one or the other

to top it off, my GPA is already pretty low because i did so bad with some humanities classes

how do you deal with being mentally deficient
i'm seriously considering suicide

[Anonymous No.8417531]

>>8417529
what year are you in?

[Anonymous No.8417538]

>>8417529
You just have to do that extra work to make up for it. If you can't work hard enough to make up for it do something else and if you love math it's a hobby. You don't need to be smart to be happy, so you don't need to kill yourself if your not smart.

[Anonymous No.8417584]

Anons, I have a midterm coming up next week for my first ever real analysis class. We're using the johnsonbaugh/pfaffenberger foundations of mathematical analysis text.

What's the best way to prepare for a midterm for an proof class? I'm afraid I won't know anything when exam time comes.

[Anonymous No.8417600]

I see no common theme in the posts posted

[Anonymous No.8417672]

>>8417600
thanks for sharing

[Anonymous No.8417700]

>>8417584
Know the theorems you did in class and how to prove them. Avoid memorizing stuff, associate each proof with some key ideas so you can derive the argument on command.

Do lots of exercises.

Go to your professors office hours, or send an email, and ask what you should focus on and recommendations for extra practice.

[Anonymous No.8417730]

>>8417529
>>how do you deal with being mentally deficient
>i'm seriously considering suicide
yeah it sucks When other people are naturally better than you. but most people are just as brainlet as you. the only thing to do is to continue until you fail to get a degree.at the phd level, you can relax. also, the main issue is to understand why you do such or such proof, why this theorem matters [what do you want when you study such field]

[Anonymous No.8418041]

>>8417529
You are so down on yourself over nothing. Math takes dedication, and as long as you stay dedicated and up to a beat, there is nothing else to it. Eventually the quick-to-prove ability comes, but not if you keep on insisting it won't. Suicide is a wild response to your particular insecurities.

>>8417584
The best way to know how to prepare the midterm is to ask your professor what s/he will be testing you on. Some professors make you regurgitate proofs from class. Some make you regurgitate proofs from homework. Some make you prove new results. Ask your professor if the midterm is gong to be like the assignments, if you are going to have to know how to prove every theorem done in class, etc.

[Anonymous No.8418092]

>>8417529
> tfw /suicide/ here.
> have algebra mid terms this week
> no idea what the fuck is going on in class
> behind three homeworks.

[Anonymous No.8418110]

>>8416386
So coming back to the video idea, I figured out a book which should gather people from various fields of interests. "Probabilistic robotics" by Thurn et. al., which ought to be downloadable on google.
Shall I initiate a sort of reading group as a moderator, with a video later? Any takers?

[Anonymous No.8418127]

>>8416217
This question occurred to me recently: how do mathematicians discover new (ie previously unknown to them) identities? Like, if you have some sum or product or whatever, and you have no idea what it could be equal to at first, how do you eventually figure out an equivalent representation? It's easier in school, when the identity is already given and you just have to prove it, but what if the right-hand-side is unknown?
Like, say, "figure out what 1^n + 2^(n-1) + 3^(n-2) + ... + n is equal to". Is there a process for this, or do mathematicians try whatever formulas occur to them until they find something that, once summed, multiplied and divided in various ways, forms the relevant identity?

[Anonymous No.8418146]

>>8418127
You do examples, try to seek a pattern, try to rewrite it or connect it to something already known.

For your sum, I'd first try to write down what you really want to compute.

It's

[math] \sum_{k=1}^n k^{n-(k-1)} [/math]

Sanity check: it's of the form a^b and for the lower bound k=1 you get 1^(n-(1-1)) = 1^n and for the upper bound k=n you get n^(n-(n-1)) = n^(-(-1)) = n.

The sum is also

[math] \sum_{k=1}^n k^{n+1-k} [/math]

or

[math] \sum_{k=1}^n k^{n+1} \cdot k^{-k} [/math]

Now just

[math] \sum_{k=1}^n k^{n+1} [/math]

is something well studied, namely under the name Generalized Harmonic numbers:
https://en.wikipedia.org/wiki/Harmonic_number#Generalization
That sum is [math] H_{n, \, -n-1} [/math]
It's some integer, but not trivial to compute.

What you have is something rather more complicated. Each term of the above sum, you weight with a small number 1/k^k. I.e. your sum is bound by that H, except the higher numbers are strongly suppressed.

The sum over 1/k^k alone is also known and goes by
https://en.wikipedia.org/wiki/Sophomore%27s_dream

Given the variations that are already soft or tricky, there will likely be no simple form for your sum.

[Anonymous No.8418167]

>>8417228
I had to check, but I've noticed it is the correct answer, but how? I've always followed PEMDAS.

[Anonymous No.8418178]

>>8418146
I've never heard of the sophomore's dream before... so you have to know all known identities beforehand, otherwise you might miss out on the patterns?

[Anonymous No.8418205]

>>.8418178
I don't know what you mean by "have to". If you recognize something, it helps of course. If you need to rediscover something, it's more work. It might not be necessary though. And there are proof techniques. Like, for such problems, induction. E.g. if you define S(n) to be your sum, does it fulfill some relation.. Trial and error.

[Anonymous No.8418212]

>>8418110
Do it fgt.

>>8418205
>Screen Shot
Why not 'Bildschirmfoto' anymore?

[Anonymous No.8418245]

>>8418212
New laptop.

>>8416217
Not sure if anyone responded to your question, actually.
So the book I take as reference to stuff like symplectic forms and Hamiltonian flows is not English, but it itself references this fat book
http://s3-eu-west-1.amazonaws.com/flooved-v2-books-jquery/3586/Book.pdf

[Anonymous No.8418250]

>>8418205
>I don't know what you mean by "have to". If you recognize something, it helps of course. If you need to rediscover something, it's more work.
"have to" as in "necessary in order to figure out a solution". I guess that means you don't have to know them all, but at the cost of setting off to solve a separate problem whenever an unknown formula occurs.

As for noting patterns, I can see a bit of that in your screenshot (down the columns), but I don't see much that the original formula didn't tell me already.

>Trial and error
I suppose in the end, there is no overall process except trying things and seeing what sticks. Thanks for the explanations.

[Anonymous No.8418264]

>>8416754
It's implied. Throughout school you should have been taught to show your working. This helps your professor to understand your process, check if you just looked up the solution or developed it yourself, and check you did it using the correct method.
Do you need to be told to do everything?

[OHP No.8418272]

>>8418146
Emma Stone Guy? Is it you?

[Anonymous No.8418281]

>Var()
>Corr()
>Cov()
>Exp()
>Hom()
>Succ()
Gotta give it to you math/statfags, you're pretty creative. Where'e the Ass() tho?

[Anonymous No.8418304]

>>8418250
>"have to" as in "necessary in order to figure out a solution".
How could you "have to" know something, you could derive all of math yourself :^)
You don't have to know any relation, you can consider finding out those relations part of the problem - a lemma to the theorem you want to get at

>I suppose in the end, there is no overall process except trying things and seeing what sticks.
https://en.wikipedia.org/wiki/Entscheidungsproblem

>>8418272
Overhead press? Is it you?

>>8418281
https://en.wikipedia.org/wiki/Associator

[Anonymous No.8418464]

>>8418212 [math] \to [/math] >>8418462

[Anonymous No.8418520]

Where is the hardest place to do mathematics at undergrad level?
t. attending Cambridge

[Anonymous No.8418544]

>>8418520
Caltech maybe?

[OHP No.8418558]

>>8418304
>Overhead Press
It's-a me. How is life? What have you been studying?

[Anonymous No.8418566]

>>8416391

>my view of mathematics is more true than your view of mathematics

wow, thanks for the insight! and here I thought /sci/ was just for shitposting :)

[OHP No.8418574]

>>8418566
If you would like to debate, then please make a case for why you hold your view of mathematics.

[Anonymous No.8418576]

>>8416753
Base Case for n=1: sum i=1 to 1 for i^2 = 1
and 1(1+1)(2*1+1)/6 = 6/6 = 1 so we gucci

Proof 1
suppose this holds for 1<n<k

then sum i=1 to k for i^2 = k^2 + sum i=1 to k-1 for i^2 = k^2+ k(k-1)(2k-1)/6
= (6k^2+k(k-1)(2k-1))/6 = k(6k+2k^2-3k+1)/6
= k(2k^2+3k+1)/6 = k(k+1)(2k+1)/6 QED

Another Proof
Suppose this holds for 1<n<=k

sum i=1 to k+1 for i^2 = (k+1)^2 + sum i=1 to k for i^2 = (k+1)^2 + k(k+1)(2k+1)/6
= (k+1)[6(k+1)+k(2k+1)]/6
= (k+1)(2k^2+7k+6)/6 = (k+1)(k+2)(2k+3)/6
= (k+1)[(k+1)+1][2(k+1)+1]/6 QED

[Anonymous No.8418586]

>>8418574

this thread is literally just you complaining about /sci/, acting like you're above the shitposting, and acting like the community here is in need of your guidance and insight to truly understand math. circlejerking is just another form of shitposting, and hence this is a shitposting thread.

[Anonymous No.8418588]

>>8418576
or I guess the most correct bounds would be to prove the base case as n=0 and then use 0<=n<k and 0<=n<=k respectively

[Anonymous No.8418607]

>>8418558
Probably nothing that would really interest you.
I write a Python GUI and API that controls a lab with superconducting qbits and I do the statistical evaluation ("data science") of accident track record of the Viennese police.
I think about physical units and try to rewrite foundations avoiding Joules or any mention of "energy".
A friend is teaching me some 20th century philosophy and another one is doing a space start-up.
For this and other reasons I want to get more into optimization in numerics and a bit mechanics.
Now I want to read the Probabilistic Robotics book (>>8418462) and then the Idris one.
My foundations are more constructive than yours. Then again, this thread was mine (>>8403989) too.

>>8416383
Go here (>>8418462) and learn/teach/discuss. Thanks!
Some probability basics are up next, then I'll see where to go.
The book actually says in chapter one that jumping from chapter 2 (Probability theory and other math necessities) to chapter 7 (Mobile Robot Localization: Markov and Gaussian) is a valid option, at least for teaching

[Anonymous No.8418633]

>>8418520
ENS Ulm

[Anonymous No.8418651]

>>8417529
It's me when studying for IMC :(

[Anonymous No.8418661]

HELP ME /SCI/ I HAVE A NUMBER THEORY MIDTERM IN AN HOUR

HOW THE FUCK DOES THIS STEP WORK

Solving x≡2mod3, x≡1mod5, ≡3mod29

So x=2+3a, ok that makes sense

then 2+3a≡1mod5, ok easy now subtract 2 from each side

3a≡-1mod5 then my professor does this

a≡3mod5 ??? How does this work? Any help would be appreciated

[Anonymous No.8418675]

>>8418661

Messed up the third equation, it's supposed to be x≡3mod29

Again, the step in question is this:

3a≡-1mod5 -> a≡3mod5

[OHP No.8418678]

>>8418586
How is it "literally me just complaining?" This thread has had multiple lines of discussion. Why are you so sour? I'm not even the only one in this thread that has mentioned the deteriorating quality of this board.

[Anonymous No.8418689]

>>8418661
Use the SQT please >>8417305

(Hint: multiplicative inverse modulo n)

[Anonymous No.8418699]

>>8418586
But he IS above the shitposting because he commonly posts about interesting university-level mathematics, which isn't shitposting but exactly what this board needs more of.

[Anonymous No.8418714]

>>8418661
What you want to find is a form for a to substitute it back in x=2+3a, thus giving you a solution to both equations. If a = b mod c, then ad = bd mod c. Naturally this holds if d is an inverse modulo c. As 5 is prime, gcd(3,5) = 1 so 3 is invertible. You can easily see 3^-1 = 2 mod 5, as 3*2 = 6 = 1 mod 5. 3a*2 = -1*2 mod 5 => a = -2 = 3 mod 5.

[Anonymous No.8418731]

>>8418714

Ah so x^-1 != 1/x with regards to modulos but rather you're trying to multiply by something to get a remainder of 1 with regards to the modulo. Thank you

[Anonymous No.8418733]

Any recommendations on math fiction?

[Anonymous No.8418739]

>>8418731
Yes, division generally has no meaning in modular arithmetic. But if the inverse x^-1 of x exists, you may write x^-1 = 1/x.

The inverse of a number x mod c is defined as the x^-1 so that x * x^-1 = 1 mod c. An inverse exists if and only if gcd(x, c) = 1. This implies that for c prime every residue class is invertible.

I dont like number theory much so I am trying to coach you in my stead, but this is extremely elementary. You should pay more attention during lectures.

[Anonymous No.8418792]

>>8418733
Stewart: Letters to a Mathematician

[OHP No.8418802]

>>8418607
Well that's awesome man. It sounds like you are enjoying your intellectual endeavors, and that's really all that matters. Keep on keeping on! The operad thread slipped under my radar; I'll take a closer look when I get home.

[Anonymous No.8418825]

most recent research i did involved transforming sets of binary trees into objects in euclidian space and studying their properties. im working on a phd now and i constantly feel like a fucking brainlet but i still like math.

[Anonymous No.8418840]

Don't know if this is the correct thread to ask but whatever. Let's say that I want to become a professional in computer graphics, apart form opengl,c,c++ that it doesn't seem the hard part to me, what books would I need to achieve greatness getting into the field ? I would like to compose a self-study curiculum more or less so I can be the best I can be.

[OHP No.8418844]

>>8418825
That sounds cool! I have a sweet spot for geometric combinatorics. Can I read your thesis, or would that reveal your identity too much? If not, shoot me an email at this throwaway: [e-mail]

[Anonymous No.8418854]

>>8418840
I dont know much about it but what I do know is you will need linear algebra. Loads of it.
Learning abut linear optimalisation and algorithm complexity wont hurt you either.

[Anonymous No.8418855]

>>8416581
how did you start your own research out of school?

[Anonymous No.8418859]

hey /sci/ i'm a foggy old man trying to help his granddaughter with her math how do i solve a quadratic formula problem? in roughly the amount of time it takes to get a cup of coffee?

[OHP No.8418871]

>>8418855
Asking lots and lots of questions and then looking for answers that actually illuminate what the questions mean in the grand scheme of things. You'd be surprised how little of category theory has actually been tamed, especially higher category theory. They are new fields, so that helps.

[Anonymous No.8418903]

>>8418520
maybe Harvard.

[Anonymous No.8418905]

>>8418859
I baught more time by saying i had g-pa toots, Here is the problem x2+4x+4=0

[Anonymous No.8418906]

>>8418905
>>8418871
Use the quadratic formula.

[Anonymous No.8418908]

>>8418906
the what now? son i'm going on 60 years old i didn't attend high school

[Anonymous No.8418913]

>>8418908
I'm sure you can Google it. My dad's older than you and he can Google things.

[Anonymous No.8418916]

>>8418913
ah well, thanks for the help. I'll ask someone who knows

[Anonymous No.8418952]

>>8418245
Thanks for the book!

[Anonymous No.8418974]

>>8418903
this must be bait

[Anonymous No.8418987]

>>8418906
Don't use the quadratic formula, the solution is obvious.
You should know that [math](a+b)^2 = a^2 + 2ab + b^2[/math].
So now you plug in a=x and b=2 and you get that [math]x^2 + 4x + 4 = (x+2)^2[/math].
So [math]x^2 + 4x + 4 = 0\ iff x=-2[/math]

[Anonymous No.8419026]

>>8418739

Thanks again, I think I aced the mid term. (proofs were pretty easy, like if a | b and c | d then ac | bd.)

[Anonymous No.8419028]

>>8416217
>Let's actually talk about something meaningful this time
>this time

[Anonymous No.8419040]

>>8419028
hi anon, what sort of mathematics do you like?

[Anonymous No.8419072]

>>8418792
Is this actually good? Looks like something I'd like based on the Amazon description.

[Anonymous No.8419077]

>>8418678
That guy is being rude to you but I can agree on some of his points.

Just because your favourite themes don't receive too much attention and just because you _think_ that other discussions aren't deep enough (or maybe you don't get why they are) doesn't mean they're of poor quality / shit posting.

In other words, some people like your stuff and others don't, and those won't appreciate your attempts to fix the board, so just keep it in mind

[OHP No.8419106]

>>8419077
I apologize if it came across that way. I'm fine with discussions that I am not personally interested in; I thought he was attacking my take on mathematics going beyond the deterministic mashing of axioms. I have no problem with topics that don't interest me or that I cannot contribute to, however I am unabashedly opposed to homework threads which clearly don't enrich the conversation of the board.

[Anonymous No.8419108]

doing research on quantum walks and making some dank visualizations

https://www.youtube.com/watch?v=oA3bsl9oCBE
https://www.youtube.com/watch?v=Pm5e0qMKOk8
https://www.youtube.com/watch?v=9QMzRruku2U

[Anonymous No.8419161]

>>8419106
homework threads are often the truest examples of genuine mathematical conversation. They really are the only place "science and math" get done. True, they are in the guise of "solving problems" but that is the essence of mathematics.

I suppose it helps that I like basic linear algebra and vector calculus, but I have found that the sqt threads are not seldom the most germane to the whole science and math theme.

[OHP No.8419256]

>>8419161
I actually like stupid questions threads, because everyone needs to learn and having different topics next to one another is cause for the making of new connections. Very good point. Questions like, "how do I calculate this integral?" just don't seem to be discussion based at all though.

[Anonymous No.8419267]

>>8419106

>enrich the conversation of the board

I'm attacking your arrogance and belief that unless the board conforms to your ideal of what a science and math community should look like, then the board is inadequate.

every time you post you reference how low quality the board is, while either not acknowledging or not realizing that a significant portion of the shitposting is done by people at the same level of education as you that use the board for entertainment purposes that may not entirely align with what you think the board is. if you want stackeexchange, then go to stackexchnage

and the fact that you feel the need to distinguish yourself from the anonymous community only makes the problem worse. you're some kid that acts like he's superior to the community here and like you're trying to guide us towards how intellectual, educated discussions should be

tl;dr you're a tripfag with an undeserved sense of being distinguished that claims to want an "enriched discussion" while choosing to continue to browse a board with content that's contrary to what you claim to want, so you should kys

[OHP No.8419291]

>>8419267
Alright, to each their own. The evolution of the board is subject to the individual ideals that everyone brings to the board, but you're of course right: it's solely my vocality that sours things enough to make you post about it, and I'm just here to try to project my ideals on everyone and bitch when nothing changes.

No, this is not the case, and you are really just doing that which you claim to detest: because I am not using the board the way you want me to, I am the root of the problem and it seems all you are posting about is why I am somehow bringing the board down. I don't complain nearly as much as you describe, and often times it is in agreement with the sentiments of others. You can keep your warped view of my place on this board, but I don't think you are fooling anyone into thinking that my place here is an issue.

On the bright side, your incessant whining is adding to the shitposting that you seem to be entertained by!

[Anonymous No.8419303]

>>8419108
I like it. Is it actual quantum walks? Are you a physicist?

[Anonymous No.8419305]

>>8419291

I've never noticed your name outside of this thread, so I have no opinion of "your place on this board." you're just another obnoxious namefag that thinks what you have to say is worthy of being distinguished from the anonymous posts.

if you sincerely wanted detailed discussion about math without shitposting, you'd go to one of the several websites that offer such discussions, but instead you're going to stay here, namefag, and act above the ironic shitposting so that you can continue to hold the belief that you are superior and distinct from the community that needs you to guide it.

[OHP No.8419331]

>>8419305
Say what you will, but I have had good discussion here and there is far less disdain for musings that escape the mathematical norm and contradict its culture (I post regularly on MathOverflow as well, and the difference is apparent). There is nothing you can do about me namefagging here, so I don't understand what you expect to get out of this. Why is my behaviour of any concern to you? I'm not even vocal or judgemental toward individuals, since I understand that the board will do as it pleases on the whole. Regardless, if the conversation moves to the topic of the state of the board, I do post my opinions. You aren't obliged to read. Sorry if my existence disturbs you enough for you to tell me to kill myself, but only you can affect your temperament.

[Anonymous No.8419378]

>>8418699
>exactly what this board needs more of

If you want serious math discussion go to stackexchange or mathoverflow or (shudder) quora. Those sites are designed to foster serious and constructive discussion. Why try to change the culture of this place when there are already other sites that provide what you want?

[Anonymous No.8419381]

>>8418733
Saunders MacLane: Categories for the Working Mathematician.

[Anonymous No.8419413]

>>8418586
Can't tell if shitpost about shitposting about shitposting is going too far but you have the shit, man.

[Anonymous No.8419426]

>>8416217
can anyone in this friggin board help me with fields, specifically finite fields, specifically GF(2^8) like what the hell is happening

[Anonymous No.8419447]

>>8419426
>, specifically GF(2^8) like what the hell is happening
Polynomials with coefficients in GF2) and degree <8.

[Anonymous No.8419563]

how do I prove de morgan's theorem for propositional calculus without writing a truth table?

[Anonymous No.8419566]

What's a good textbook for Introductory Linear Algebra?

[Anonymous No.8419619]

>>8419566
Grade 10 math

[Anonymous No.8419721]

>>8416217
So I won a book voucher and I'd like to spend it on maths, CS or programming books.
Any recomendations?
I'm in my last year of highschool so it can't be anything really advanced.
I think game theory (and to some extent game ai) is pretty cool, so it'd be nice if there are any good books on that.

[Anonymous No.8419875]

>>8419072
It's OK. Just snatch a copy and see if you like it.

It's just that it's hard to come up with anything when you say 'math fiction'.

[Anonymous No.8419878]

>>8418974
No? Harvard has a super-tough introductory sequence for pure math majors. So does U of M. Chicago is probably another.

[Anonymous No.8419880]

> reach partial differential equations
> feel like a complete retard
> consider dropping maths entirely

feelsbadman. i was never that great at math to begin with and my lack of natural ability is really starting to show

[Anonymous No.8419885]

>>8419563
Depends on your definition of 'propositional calculus' and whether you assume the excluded middle. In the usual classical setting, where you define the interpretation of connectives on a kind of intuitive level, I don't think there's any other way than truth tables.

[Anonymous No.8419890]

>>8419885
It depends on your inference rules and axioms. This is a stupid question though; not for this thread.

[Anonymous No.8419891]

>>8419566
Shilov

>>8419721
What's your level of knowledge? Last year of high school is pretty vague since kids these days are curious af. (There's one extreme of the spectrum in this thread.)

BTW, there's a reading club on 'Probabilistic Robotics' right now, but there are some probability prerequisites that might baffle you if you aren't familiar with the theory: >>8418462

[Anonymous No.8419940]

>>8419378
>Why try to change the culture of this place when there are already other sites that provide what you want?
What exactly is the culture of /sci/?

[Anonymous No.8420052]

>>8419563
Natural deduction is useful.

[Anonymous No.8420136]

>>8419880
Don't worry, you don't necessarily need to handle partial differential equations as a mathematician. Math is so diverse that there's bound to be an area where you have natural ability.

[Anonymous No.8420155]

>>8419721
game theory and game ai are totally different topics

>>8419426
What do you need help with?

>>8419940
shitposting

[Anonymous No.8420159]

>>8420155
im the finite field guy, so when we multiply polynomials, we have to first find an irreducible polynomial and work mod that, but how do we determine if a polynomial is irreducible? i know the few claims such as the constant term must be 1, must have an odd number of terms. And how do i find inverses because the extended euclidean algorithm sees nightmarish with polynomials

[Anonymous No.8420161]

>>8420159
>i know the few claims such as the constant term must be 1, must have an odd number of terms.

[Anonymous No.8420162]

>>8416217
>>8416217
>tfw regret not working in mathematics back in hghschool

Fuck everything.

One of my mates who actualy gave a fuck about school even got a ride to the MIT.

Worst thing is that I find math interesting now.

REEEE!

[Anonymous No.8420164]

>>8420161
find an irreducible polynomial that has even terms and a constant term of 0 for me then u fucking faggot retard kill urself

[Anonymous No.8420168]

>>8420164
>irreducible polynomial that has even terms and a constant term of 0
That's not the negation of the statement. That should be
>irreducible polynomial that has even number of terms or a constant term different from 1

[Anonymous No.8420171]

>>8420168
find me an irreducible polynomial over z2[x] that has an even number of terms AND a constant term of 0, and kill urself u retard

[Anonymous No.8420173]

>>8420159
There are a bunch of things. So say you are trying to find one of degree 3 over Z_2. Then we look at a few things.

First, you are right, you need x not to divide p(x) (where p(x) is the polynomial we are trying to find). So it needs a constant term. Second, you need that there are no roots of it. Note that if p(x) were reducible, we could factor it. So it has a linear factor (x-a) for some a and then possibly a quadratic factor or two other linear factors.

Hence it suffices to show that p(a) != 0 for all a in Z_2. So you plug in 0 and 1 and if it is 0 for one of them, that is not your polynomial.

For tools you can use to find irreducible polynomials, look at these:
https://en.wikipedia.org/wiki/Eisenstein%27s_criterion
https://math.dartmouth.edu/archive/m31w05/public_html/Reducibility.pdf
http://math.stackexchange.com/questions/1935/methods-to-see-if-a-polynomial-is-irreducible

[Anonymous No.8420201]

>>8420173
How does Eisenstein's criterion work over finite fields?

[Anonymous No.8420204]

>>8420171
Do your homework yourself.

[Anonymous No.8420427]

A little follow up, I am still thinking about doing a lecture on topology. So keep your eyes peeled as I will mention it in this thread if and when I do it.

[Anonymous No.8420474]

Hugely appreciate it if someone can help me with this question.

[Anonymous No.8420475]

>>8420474
>>stupid questions thread

[Anonymous No.8420558]

>>8416597
First you need a mathematical foundation in logic. From there you need to read a bunch of proofs and practice doing them. There's no gimmick really you just have to keep grinding until you git gud. I was never really good at proof writing as a math undergrad (lel) because I never practiced or took hw that seriously. But I was just good enough to shit them out on exams and such, so don't be discouraged if u suck cuz math is hard man.

[pt. 2 of "Probabilistic Roboti(...) No.8420578]

(>>8420519)

>>8416375
>>8420427
Regarding the background which you want to introduce there, you can do this right there in the link, generate some momentum for such lectures and get the basics out of the way.
I've made the second video linked above, and it's about the start of section 2 (after the introduction) where they talk about probability theory - but they do it only for distributions over R^n.
If you would, for example, his the thread with basic set discussions - introducing necessary stuff like [math] \cup [/math] and so on, then I'll interact with you right there.
E.g. the formulation of Bayes theorem for sets like here
https://proofwiki.org/wiki/Bayes%27_Theorem

[Anonymous No.8420711]

How do you determine if an integer is a prime or not, _by hand_?

Let's say I want to prime factorize the integer 11055:
Since it ends with a 5, it is divisible by 5. Now 11055=5*2211.
The sum of every digit in 2211 is 6, thus I can try to divide it by every integer which is a multiple of 6. 2211 is divisble by 3. Now 11055=5*3*737.

How do you continue? The sum of its digits are 17, which is a prime, making it harder to find a divisor. sqrt(737) is somewhere between 20 and 30 (~27 if you have access to a calculator). Since I'm doing this by hand, I don't feel like performing trial division up to ~30 divisions by hand.

Is there a way to determine whether or not it's divisible without having to resort to trial division?

[Anonymous No.8420716]

>>8420711
no, unless you're willing to learn an algorithm line GNFS or AKS.

[Anonymous No.8420723]

>>8420711
That's a research field. The general number sieve is the fast one for large numbers.
https://en.wikipedia.org/wiki/Integer_factorization#Factoring_algorithms

>Is there a way to determine whether or not it's divisible without having to resort to trial division?
If you don't care about what the primes are, just the knowledge IF there are any factors, there are more fast algorithms for that
https://en.wikipedia.org/wiki/Primality_test
https://en.wikipedia.org/wiki/AKS_primality_test

[Anonymous No.8420724]

>>8420716
>>8420723
Ok, thanks!

[Anonymous No.8421236]

>>8416391
>dynamic body that builds off of our intuitions of the Truth and approximates it via theorems, which are then founded via axioms
>dynamic body
Amen. Change the "body" change the math.

[Anonymous No.8421240]

>>8421236
This is "CS" OP, Punk Ass Bitch.

[Anonymous No.8421277]

>>8416597
THE BEST WAY TO GET BETTER AT PROOFS IS TO READ MORE PROOFS.

[Anonymous No.8421854]

Reminder that there is no truth in math, not even in logic

[Anonymous No.8421900]

>>8421240
Ah just looking at this I can taste the 500,000 dollars that was paid to the consultancy firm to create it

[Anonymous No.8422025]

>>8416217
I feel that I lack the insight on many things that I study. I don't know what to ask and get stuck at the problems. I've started with quite deep questions and interest but as I have to sit many hours on each problem and still tackle only technical side I stopped asking any questions to myself and to anyone else. I don't know how do I continue.

[Anonymous No.8422061]

>>8422025
the main issue is to understand why you do such or such proof, why this theorem matters [what do you want when you study such field]

[Anonymous No.8422083]

Could someone suggest some onlne sources for numerical analysis? Youtube channels, etc. ? It's easy to find stuff more Analysis and Linear Algebra, but not too much about this stuff.

[Anonymous No.8422190]

Can someone post one of those shapes with circles in the corners with numbers in them?

[Anonymous No.8422221]

>>8416312
Look into number circles (as opposed to number lines)

[Anonymous No.8423401]

bump

[Anonymous No.8423450]

Has anyone here studied any of Scholze's work on p-adic geometry? Been skimming through some notes but my algebraic geometry is weak

[Anonymous No.8424104]

Masters, i come with some newblulz questions.

For variance, if I define it as such:

sum((x_i -E[X])p_i) for all i

Is because such expression would always yield 0, correct? I have this vague idea from some concept in calculus where the integral of the average value of f(x) minus the integral of f(x) would yield zero.

Please senpais help this brainlet.

[Anonymous No.8424183]

CS major going into Pure Math for grad school. Currently reading Baby Rudin, Adkins, and some Munkres to prep up. I guess it's some basic stuff, but it's pretty cool. Excited to see what Papa Rudin has to offer.

[Anonymous No.8424572]

Got a question. Is there such a thing called non-linear algebra?

[M No.8424832]

>>8424572
Yes, "Algebra". Much of it is motivated by solving equations like
[math] a+bx+cx^2+dx^3=0 [/math]

[Anonymous No.8424838]

>>8416704
What the fuck am I looking at. Are you saying that n(n+1) = n(n+1)(n+2)/3 for all n? That's just not true. A polynomial of the third order isn't equal to a polynomial of the second order...?
What are you trying to say?

[Anonymous No.8424840]

>>8416753
Try this one.

1^4+2^4+3^4 + ... + n^4 = n(n+1)(2n+1)(3n^2-3n+1)/30

or the easier one
1^3 + 2^3 + ... + n^3 = (n(n+1)/2)^2

[Anonymous No.8424851]

No one answers at SQT so I ask it there:

My prof said that you can check if C[x,y]/f integrally closed just by looking at f's real plane graph.

What's the reason it's true? Why is that enough to look at reals? Why is that enough to check elements of the form (y-c0)/(x-c1) for real c0, c1 if they're integral?

[Anonymous No.8425065]

>>8416225
Kek

[Anonymous No.8425594]

Anything that anyone wants to talk about mathematics-wise?

[Anonymous No.8425604]

>>8425594
How many are there?

[Anonymous No.8425606]

>>8425604
How many what?

[Anonymous No.8425607]

>>8416704
for n>0 you can safely divide by n*(n+1) and you are left with the equivalent equation 3=n+2

there is exactly one value of n for which this equation is true. Can you guess which one it is?
(You wont need induction on this one, thats a bit too advanced for you)

[Trinibro No.8425609]

tfw doing Actuarial but would much rather be pursuing Mathematics or Physics

Fuck business subjects. Boring ass shit.

[Anonymous No.8425612]

>>8425609
Statistics might be a good area for you, then. Can do pure mathematics and also physics easily via applications. Meanwhile, you have all the info you need to pass the actuarial exams easily.

[Trinibro No.8425619]

>>8425612
If switching was easy I'd have done it already. I'm contracted to study Actuarial via the scholarship I'm one. Switching would render it void. On the plus side my tuition is paid for and I get a monthly allowance. Hey, I'm making money though, maybe I'll start enjoying it soon.

I just really didn't expect it to be this much work.

[Anonymous No.8425621]

>>8425606
Nonisomorphic abelian toposes.

[Anonymous No.8425627]

>>8425621
>how many abelian toposes up to isomorphism
None, Toposes cannot be abelian.

[Anonymous No.8425658]

Is there a name to describe a property similar to the following:

Taking element Xi from a set Y, and a function F that take elements from Y to another element from Y.
X1 is any element of Y, X2 = F(X1), X3 = F(X2), X4 = F(X3)...

If Y has a finite size then it takes you in a loop through every element of the set. No smaller loops.

Or if Y has an infinite size then it never takes you in a loop, Xj always different to Xk if j different to k.

[Anonymous No.8425669]

>>8425658
"F is a permutation of Y".

[Anonymous No.8425684]

>>8425658
sounds like you're talking about cyclic groups and their generators

[Anonymous No.8425688]

>>8425669
no, and the guy asking the question is a retard anyway, but first to why you are wrong: the "loop" the retard is describing is just the orbit under repeated application of the function F.

even if F were an element of the symmetric group S_n,, that does in no way imply that the orbit of every element from {1,....,n} under F is the entire set. In fact, that would be equivalent to F being a pure cycle. (do you know what cycle notation is?)

as for the retard who asked the question: of what is that supposed to be a property of? the set? the function?
F can only ever be defined on one set. If you change the set the function necessarily has to change too.

so it simply doesnt make sense in my opinion. repeat: what is this supposed to be a property of?

[Anonymous No.8425694]

>>8425688
Why are you so aggravated? People are just trying to suggest him names. He wasn't rigorous in his definition and so people don't treat him rigorously.

[Anonymous No.8425699]

>>8425669
Elabourating (to appeal to the angry-pants pedant who replied), You described an injection on a set. This is a permutation when restricted to its image.

[Anonymous No.8425701]

This is what anon meant
>>>/g/57161092

[Anonymous No.8425705]

I'm in a math modeling class (senior undergrad), and for our term project we have to do work on a model, either novel or an improvement over one that already exists

I've been leaning towards stochastic models, possibly brownian motion, or something involving monte carlo simulations. any other nifty ideas that I could look into?

[Anonymous No.8425712]

>>8425658
also interested in this, please answer him /sci/

[Anonymous No.8425724]

>>8425712
It's called an F is called an injection from Y to Y. This is a function F:Y-->Y such that for all x,y in Y, F(x)=F(y) implies x=y.

If Y is a finite set, it is surjective, and hence a bijection (called a permutation, see previous answers to the guy's question). On an infinite set, an injection will continue forever, but might not necessarily be bijective (that is, F might not hit all elements of Y).

[Anonymous No.8425725]

>>8425701
Oh no we've been exposed!

[Anonymous No.8425729]

>>8425725
>we

[Anonymous No.8425736]

>>8425701
he should have copied his post verbatim here. or linked to it.
his reformulation ITT makes no sense.

>Is there encryption OR hashing algorithms that are guaranteed to cycle through all possible options without repetition if you repeatedly pass the output of the algorithm as input?
What do you computer science nerds call this property (in the general case, not just pertaining to hashing)?
a function that cycles through all members of a finite set is called a cyclic permutation
https://en.wikipedia.org/wiki/Cyclic_permutation

[Anonymous No.8425743]

>>8425701
he should have copied his post verbatim here. or linked to it.
his reformulation ITT makes no sense.

>Is there encryption OR hashing algorithms that are guaranteed to cycle through all possible options without repetition if you repeatedly pass the output of the algorithm as input?
>What do you computer science nerds call this property (in the general case, not just pertaining to hashing)?
a function that cycles through all members of a finite set is called a cyclic permutation
https://en.wikipedia.org/wiki/Cyclic_permutation

[Anonymous No.8425754]

>>8425743
I'm not the person who posted here initially. I was asking /g/.
But thanks that's exactly what I was asking for.

[Anonymous No.8425758]

>>8425743
Fucking kibibytes. That shit triggers me so much.

FUCK YOU FOR POSTING THAT.

[Anonymous No.8425782]

>>8425724
>F(x)=F(y)
are you sure about this? for example

F(x) = y
F(y) = z
F(z) = x

here F(x) != F(Y) despite F being cyclic (or whatever you call that)

[Anonymous No.8425786]

>>8425782
are you autistic?

[Anonymous No.8425788]

>>8425786
almost (sperg)

[Anonymous No.8425809]

>>8425782
Jesus fucking christ. It's the definition of an injection.
For elements x and y in the domain of F, if F(x) just happens to equal F(y), then x=y.

Examples of injections:
f(x)=x on R
f(x)=x^k, k an odd natural number, on R
f(x)=e^x on R
Examples of non injective functions:
f(x) = x^k, k an even natural number, on R
f(x) = |x|

[Anonymous No.8425816]

>>8425782
So, continuing, look at the elements of the codomain of F.
What elements map to y?
What elements map to z?
What elements map to x?

[Anonymous No.8425819]

>>8425724
>>8425809
being injective is wholly unrelated to being cyclic. the identity is injective, but is it cyclic? in fact, the original /g/uy asked about hash functions, which are almost never injective

where is your god now?

[Anonymous No.8425821]

>>8425809
>For elements x and y in the domain of F, if F(x) just happens to equal F(y), then x=y.
then explain this

f(x) = x % 2

f(2) = 0
f(4) = 0

2 != 4

[Anonymous No.8425822]

>>8425821
that's a non-injective function dumbshit

[Anonymous No.8425824]

>>8425822
k, make sense

[Anonymous No.8425826]

>>8425824
why are these high school brainlets not banned from this board? worse than deliberate shitposters

[Anonymous No.8425830]

>>8425826
wow, sorry for being useless but aren't you scientists forced to spread knowledge by some code like doctors are forced to heal?

[Anonymous No.8425832]

>>8425830
> but aren't you scientists forced to spread knowledge
says who? why do you make things up like this?

[Anonymous No.8425840]

>>8425832
whatever. i think you have failed to give a decent answer to >>8425701

[Anonymous No.8425843]

>>8425840
the question is terribly worded and has nothing to do with my posts

you could even consider encryption algorithms that just go through every option without repeatedly passing the output, try posing a question that isn't so stupid next time

[Anonymous No.8425849]

>>8425821
Goddamn. Your reading comprehension is shit.

>>8425819
Notice I never said a single thing about cyclic functions :^)

[Anonymous No.8425850]

>>8425843
why are you so mean? not everyone is a digital nerd. How do you call a function that reach _all_ its possible outputs if you feed back the previous ouput as input, again and again.

[Anonymous No.8425861]

>>8425850
https://en.wikipedia.org/wiki/Derangement

is google too hard for you to use? you're supposed to be at least 18 to use this website not in middle school

[Anonymous No.8425864]

>>8425861
enough, i leave this. sci confirmed for being the worst board of 4chan.

[Anonymous No.8425866]

>>8425864
try not wording your questions in a way that make you out to obviously not have a clue what you're talking about brainlet

[Anonymous No.8425920]

>>8425864
Thanks!

[Anonymous No.8425923]

can anyone solve pic related?

[Anonymous No.8425929]

>>8425923
You don't "solve" for antiderivatives. Also, that antiderivative is really elementary. Any automatic antiderivative calculator can easily compute it and give you the procedure. No need to ask here unless your function is actually special.

[Anonymous No.8425938]

>>8425929
>that antiderivative is really elementary
I see what you did there :^)

[Anonymous No.8426249]

Anyone ever have a topic that seems really interesting, but then once you get into it it's a complete slog to get through? Experiencing that with statistics right now. Topology is an example for me of the opposite, though.

[Anonymous No.8426289]

Fuck measure theory, god damn, what a horribly unintuitive subject.

Anyway, for the more educated in this thread, I finished Milnor's Morse Theory and Topology from The Dif. Viewpoint, and Hatcher's Algebraic Topology.

Does anyone have any recommendations on books for someone looking to study low dimensional topology (3,4 manifolds) as well as some further books in the field of differential topology?

[Anonymous No.8426345]

>>8426249
>took an actuarial math course as an elective because I thought finance might be an interesting application and a possible career path
>95% of the course is DUDE GEOMETRIC SERIES over and over
>actuarial exam problems are piss-easy questions masked in deliberately confusing wording and the most autistic contrived sets of starting information imaginable

[Anonymous No.8426420]

>>8419040
Abstract and Linear Algebra
Vectors in any respect
Spectral Theory

[Anonymous No.8426423]

>>8421900
Posted it. I feel ya.

[OHP No.8426469]

>>8425621
>>8425627
Precisely one, the terminal category. This object is a categorified version of [math] \mathbb{F}_{1} [/math].

[Anonymous No.8426551]

>>8417529
Question time.

1: Do american Universities allow you to do how ever many subjects you want per semester? Like, 2 per sem, 1 over summer, 1 over winter? Cause that work load will probably alleviate a lot of pressure for you bro.

2: If you're in such a rush to finish your degree / also willing to end it all and hence you're fine with just not finishes at all in a sense, why don't you just do less subjects at a time and take much longer to finish your degree? What's the rush? Just do part time...

>>8418041 is right, dedication is all that's needed, patients is a fucking virtue of for a reason, chill the fuck out isn't just an expression meant to be ignored, your just a chemical stewing pot that needs to learn how to breathe and meditate because it's lacking oxygen, JUST RELAX MAN AND FINISH YOUR DEGREE IN AT YOUR OWN PACE NOT THE PACE YOU 'THINK' YOU'RE MEANT TO FINISH IN!.

Jesus.

[Anonymous No.8426615]

How does one find the roots of an arbitrary function of degree greater than 4?

[Anonymous No.8426732]

>>8426615
numerically

[Anonymous No.8426757]

>>8426732
How?

[Anonymous No.8426760]

>>8426757
please learn to use google
https://en.wikipedia.org/wiki/Root-finding_algorithm

[Anonymous No.8426779]

>>8426760
Thanks anon! I wanted to know if there is a much better to way find roots.

[Anonymous No.8426808]

>>8416312
Do you have reason for this to make sense, or is this just "look at it. Just look at it" kind of making sense. Regardless look up the extended real/complex numbers.

[Anonymous No.8426815]

Well, I'm currently taking Topology and it's awesome. I'm also learning about matrix groups and linear operators and shit with linear algebra for my thesis. It's been particularly interesting using quaternions. I'm also learning about mathematical methods in physics which is pretty cool and regression analysis which is more interesting than expected.

[Anonymous No.8427055]

How do you do, guys? Tell me something amazing you've realized recently

[Anonymous No.8427113]

>>8427055
All functions are continuous.

[Anonymous No.8427117]

>>8427113
How so?

[Anonymous No.8427129]

>>8427117
Not that guy but I will entertain your question for a second since I think I know what he is referring to:

Exercise: /without/ using the law of excluded middle, give an example of a discontinuous function from the reals to itself. That is, a discontinuous function f : R --> R.

[Anonymous No.8427131]

>>8416217
>kahler manifolds
>symplectic geometry

My fucking nigger, my thesis is on classical integrable systems. I haven't actually read much into Kahler manifolds any good reviews you know?

For DE's it depends what sort you mean? Are we talking of the basics or more rigorous analysis? Or intractability? Maybe pertubation theory and the like?

I dont' have a good one for the basics, but for our functional analysis course we are using Rudin which has a section on DE's and transforms.

For approximate methods I used Bender's book:
Advanced Mathematical Methods for Scientists and Engineers I

For integrability theres a nice one on solitons that introduces you to some of the basic ideas I think its called: Solitons and infinite dimenionsal lie algebars.

I'm currently using intro to classical integrable systems by Babelon.

I'm currently studying for my exams, so just rattling through functional analysis, homological algebra and category theory, and some conformal field theory. The stress is real

[Anonymous No.8427141]

>>8427131
Look at the Kahler manifold section of Ana's notes (https://www.math.tecnico.ulisboa.pt/~acannas/Books/lsg.pdf)) and tell me how hard that stuff is for you right off the bat. If it is unfamiliar, read it.

After that take a look at:
http://moroianu.perso.math.cnrs.fr/tex/kg.pdf
http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/kaehler0609.pdf
and Huybrechts's Complex Geometry definitely since it rox.

[Anonymous No.8427153]

>>8427141
hahaha yeah I've used ana's notes a bit for my research already.

Those other ones look rad man, thank you so much.

[Anonymous No.8427154]

>>8427153
No problem, mate. Any time. :)

[Anonymous No.8427220]

>>8427129
Can you define cos/sin?
Series? (It mustn't be different from defining continuity)
Then the Fourier series of say x/pi is discontinuous

[Anonymous No.8427239]

>>8426760
going through that page, I like what's happening here:
https://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method

[Anonymous No.8427264]

>>8427220
Any "discontinuity" which relies on being a before x and b after x is automatically not possible to make.

[Anonymous No.8427337]

>>8427264
I don't see this kind of discontinuity in this case.

[Anonymous No.8427368]

>>8427337
Try proving that it is discontinuous and let us know how that turns out.

[Anonymous No.8427381]

>>8417054
thats a really fucking wonky way to do it, but it works

[Anonymous No.8427458]

>>8416704
Am I doing this wrong?
>>8416733
Doesn't this only work when n = n^(3)
(When n = 0 or 1)

[Anonymous No.8427464]

>>8426289

[Anonymous No.8427467]

>>8427458
The equation holds for
n = -1, 0 , 1 exactly.

[Anonymous No.8427469]

>>8427129

Are you trying to say that for any f(x) in R, it is continous for some interval?

[Anonymous No.8427483]

>>8427469
No. Without using the law of excluded middle, you cannot come up with a discontinuous function.

[Anonymous No.8427487]

>>8425821

The idea is that every input must result in a unique output. If you randomly take 2 inputs, a and b from the range of input and you know that f(a) = f(b) then you are sure that a is actually the same as b.

[Anonymous No.8427499]

>>8425658
You could say that the trajectories of F = <f> are of maximum size.

[Anonymous No.8427529]

I read a survey of Milnor's construction of exotic 7-spheres yesterday, and I have been pleasantly surprised (and entertained)

[Anonymous No.8427620]

>>8427368
Okay, so the function is
[math]f(x)=\sum_{n=1}^{\infty} \sin(nx) \frac{(-1)^{n+1}}{\pi n}[/math]

[math]f(\pi)=\sum_{n=1}^{\infty} \sin(n \pi) \frac{(-1)^{n+1}}{\pi n}=0[/math]
let's consider the sequence [math]x_k = \pi(1+ \frac 1 {2k})[/math] which converges to [math]\pi[/math].
[math]f(x_k)=\sum_{n=1}^{\infty} \sin(n \pi + n \frac {\pi}{2k} ) \frac{(-1)^{n+1}}{\pi n}=\sum_{n=1}^{\infty} (-1)^{n}\sin(n \frac {\pi}{2k} ) \frac{(-1)^{n+1}}{\pi n}=\\ =-\frac {1}{\pi}\sum_{n=1}^{\infty} \sin(n \frac {\pi}{2k} ) \frac{1}{n}[/math]
It mustn't be hard to verify that for [math]k \to \infty[/math] this approaches to -1, which isn't 0.

[Anonymous No.8427623]

>>8427483

What about reciprocal?

[Anonymous No.8427625]

>>8427620
Is that by chance a proof by contradiction you use?

[Anonymous No.8427629]

>>8427625
where did he assume continuity?

[Anonymous No.8427634]

>>8427629
Continuity preserves limits. So assume it is continuous, then it obviously should preserve limits. It doesn't preserve this limit so it must not be continuous.

He used double negation.

[Anonymous No.8427636]

>>8427625
Uhm... Well, I show that [math]f(x_k)[/math] converges to some number other than [math]f(\lim x_k)[/math]. This seem to be the definition of discontinuity.

[Anonymous No.8427641]

>>8427634
where did he assume it was continuous? he just showed it doesn't preserve limits, there's no assumption made

go brush up on your logic

[Anonymous No.8427646]

>>8427641
It was implicit by his "definition" of continuity.

[Anonymous No.8427648]

>>8427646
what was implicit?

[Anonymous No.8427655]

>>8427648
http://math.stackexchange.com/questions/176279/all-real-functions-are-continuous

[Anonymous No.8427657]

>>8427655
>http://math.stackexchange.com/questions/176279/all-real-functions-are-continuous
what does this have to do with the discontinuous function the other posted gave?

[Anonymous No.8427695]

>>8427657
The "definition" he uses is based on the excluded middle.

[Anonymous No.8427701]

What are the prerequisites for Lie theory? Is group theory and basic differential topology enough?

[Anonymous No.8427709]

>>8427701
Yes. For Lie algebras, all you need is abstract algebra & linear algebra. For Lie groups, your differential geometry and lie algebra will work well.

[Anonymous No.8427748]

>>8427709
How much abstract algebra approximately? I'm two third into the group theory section of Herstein, do I need all of the undergraduate algebra sequence or perhaps more? My interest lies in mathematical physics and it seems the representation of Lie groups is a really important topic.

[Anonymous No.8427768]

>>8427748
Basics. Nothing special. At my university, a second course in abstract geometry is recommended. I would say that probably any Lie algebra book would be self-contained. For Lie groups, you might want to keep a Lie algebra book nearby, but it should also be relatively self-contained.

[Anonymous No.8427786]

>>8427748
It could be sufficient to know what kind of a structure an algebra is. This would atleast be a good start, if not enough. It was pretty funny to notice how the canonical commutator in quantum mechanics is just the commutator of a Lie algebra.

[OHP No.8427829]

>>8427055
Right now I am looking at how to describe dynamical systems entirely in terms of a de Rham cocycle on a specific sort of Lie groupoid (one which looks like an orbit space). But, pulling this back further, it seems one can describe a dynamical system on a manifold by simply describing an [math] \mathbb{R} [/math]-bundle on one of these spaces. I think I can apply the gizmos of Pontryagin duality to furthermore describe this as a singular cocycle on the orbit space (maybe I just need de Rham's theorem, but we'll see), in which case I can just define dynamics on a space by finding a [math] \mathbb{Z} [/math]-bundle on the orbit space. I think that deep down what is going on is that the bundle information supplies cohesive/descent data for how the orbit space is glued together from local pieces, which my collaborators and I would like to characterize in terms of Conley theory. Using this sheaf-theoretic approach, we may be able to examine properties of dynamical systems using tools from algebraic geometry (anything synthetically living inside of Grothendieck toposes, really). This would be really cool, and would allow for generalization away from smooth dynamics to dynamics with arbitrary flow groups and living in arbitrary topoi.

tl;dr Dynamical systems are determined up to some bisimulation-like equivalence by entirely algebraic data.

[Anonymous No.8428007]

>>8427695
and so what "definition" are you using that we were supposed to realize you're working with?

[Anonymous No.8428010]

>>8427701
make sure your linear algebra is solid

[Anonymous No.8428014]

I am nearly done with my bachelor's in pure mathematics, but there is something that is killing my love for the field. After these last few years I am still extremely bad at writing proofs. It's really strange, as all of the material seems to come very easy to me, yet I seem to lack some sort of creativity to find the right train of thought that leads to a proof.

I have written a fuckton of proofs over the years, so it's not lack of practice. I make sure I understand the theorems I can use, I know the definitions and I make sure to thoroughly understand what it is I want to prove.

I am probably going into an applied field because I can't deal with the proofs anymore. But if I'm honest, my passion is still maths. If anyone has any tips, advice or experiences I'd love to hear them.

[Anonymous No.8428020]

>>8428014
>I am probably going into an applied field because I can't deal with the proofs anymore
Tell us more precisely what's wrong, and you may be able to avoid this!

[Anonymous No.8428059]

I've been working on some cool shit recently.

It's a way to model contractions in continuous space of n dimensions. The contractions are called "Point distortions" and are caused by bending the space over a higher dimension, thereby changing it's density.

[Anonymous No.8428074]

>>8428020
If I am asked to prove a statement I will try to understand what is asked, rehash what may be useful (lemmas, theorems, definitions) and then get nowhere. If someone gives me a slight hint the proof becomes trivial. But by myself I have to keep trying loads until I find something that works.

This means I spend an insane amount of time on homework and I score very average on exams because of the limited amount of time I have.

[Anonymous No.8428093]

What does an 'equivariant map' mean in the context of differential equations?

[Anonymous No.8428114]

>>8428007
https://en.wikipedia.org/wiki/Continuous_function#Definition_in_terms_of_neighborhoods

[Anonymous No.8428125]

>>8428074
Could you have a break for a week or so? Like, don't do math at all, that kind of a break. It seems a bit like you were a quality controller in a factory and have been watching the products go past you for 5 hours straight. You no longer see the small things, but if someone points out there's a rat inside a soda bottle, you notice it immediately.

>>8428059
Sounds cool!

[Anonymous No.8428136]

>>8428125
I understand what you mean, it's a good idea. I tried doing that by taking a few days off but it didn't help much. I don't think I could catch up with a week's worth of material if it didn't help, so it's a bit of a gamble.

[Anonymous No.8428200]

>>8428125
please stop avatarfagging

[Anonymous No.8428508]

>>8427768
>>8427786
>>8428010
Thanks for the advice, anons.

[Anonymous No.8428644]

>>8416303
DESU I'd like to see more YT vids on Real Analysis. The Francis Su series is shitty.

[Anonymous No.8428955]

>>8427829
Although I don't know much about sheaf-theoretic approach/topoi/Conley theory this looks quite interesting.

So you might get some technical tools for the actual dynamical systems? Do you usually work with dynamical systems too?

[OHP No.8428980]

>>8428955
I'm pretty new to dynamical systems, and I am learning a lot. The professor keeps describing his vision of obtaining a dynamical system by patching together pieces saying how the flow is about a point, and I am realizing that there may be a controllable or easily described site on which we can examine the sheaves (a site of local flow models, if you will). If such a site exists then it surely cannot be finite since I can describe infinite families of flows about an equilibrium point, but these are well-behaved.

Regarding technical tools, an example would be the description of certain asymptotic behaviours based on properties of a certain category we have associated to the flow. For example, a set is an attractor for some basin if it corresponds to a colimit in this category over a diagram determined by the basin. I'm hoping to see some really cool stuff come out of the de Rham cohomology bit, too.

I'm starting to see connections to the theory of orbifolds as well, so we stumble upon something deeper than anticipated.

[OHP No.8428984]

(>>8428980)
so we may stumble*

Apologies, I'm on mobile.

[Anonymous No.8429348]

>>8419303
Nope math PhD student.