/sci/ - Science & Math

>>10210629
>like our Lord in Heaven
Please, for goddness' sake, tell us where we might go to find this magnificent piece of literature, anon.

>>10210747
Grillet's Abstract Algebra.

>>10210755
This looks like it has exactly the same chapters as Judson's Abstract Algebra

http://abstract.ups.edu/download.html

>>10210766
I've just downloaded Judson's Abstract Algebra, and Grillet's goes quite a bit further.
As in, it finishes everything that Judson covers by half the book, has covered some additional stuff, skipped stupid stuff like cryptography, and keeps going for the other half.
Kinda like Lang's, desu.

>>10210629
This is the dihedral group right? Had an exam on that today.

>>10210629
>Since a and b have finite order...
This hypothesis is irrelevant to the conclusion.

>Christianity
This is garbage for sure.

PS ¿Can you say nigger on 4channel.org?

>>10211182
Great, I saw some propaganda to the contrary but it looks like any censorship is appropriately subtle.

>>10210629
I've always wondered why no textbooks straight up show the correlation between automata and monoids (including groups). That shit makes stuff like OP's picture trivial to understand (not to mention all the other concepts of group theory) and you don't have to go beating around the bush with all these vague-ass weak theorems and appeals to a higher power. You could even do it by defining monoids as categories (in a slightly different way than the usual) if you don't want to talk about formal languages and automatas.

>>10211182
>PS
Technically the board is nsfw so you can get your post deleted for that but really it comes down to whether or not your post is shit enough for someone to report you for it.

>>10211182
>he's never listed an useless fact by habit when giving a proof
>>10212100
The board is sfw.
The post is bad, but it definitely isn't bad enough to report as "Extremely low quality."
Maybe as trolling.
>>10212095
The stuff in the picture is already trivial to understand.
The appeal to a higher power is a joke.

>>10211182
Read Pascal

>>10212103
>The board is sfw.
Sorry, I meant sfw. That was a typo. /sci/ has more reporting options including racism and such. I agree the post isn't bad enough for me to report it.
>>10212103
It can be made even more trivial. Even up through Galois Theory.
http://www.presentica.com/ppt-presentation/a-talk-without-words-visualizing-group-theory

>>10211981
>there is no correlation between serious maths and olympiad problems
Is he right? If so, what makes him right?

Fellas, I'll be far from my desk, so I need some light math subject to study.
Good notation is the pro of pros.
Worst case scenario physics works.
Thank you very much.

>>10212343
olympiad problems can't require too much specialised knowledge so that everyone can participate
serious math requires a lot of specialised knowledge

I am reposting from /sqt/, just to improve my odds a bit.

Ok, here's my question. Rubinstein's bargaining game has been extended by Herrero (1985) and Haller (1986) to an n person game. In particular Haller proves that any allocation of the cake is an SPE. My questions are:
1) Would this hold if players had outside options?
2) Would this hold if the discount factor was 1?

>>10212817
the stupid thing about olympiads is that they really do require a lot of highly specialised knowledge. no one would come close to a decent score on an IMO just by improvising solutions. You need a very big toolbox.

It seems like such a waste to me to round up the smartest high-school age mathematicians in your country and teach them 70 different ways you can rearrange AM-GM instead of teaching them actual mathematics.

>>10211182
>This hypothesis is irrelevant to the conclusion.
No it is not. It is very relevant to the fact that [math]a^{-1}[/math] and [math]b^{-1}[/math] are positive powers of [math]a[/math] and [math]b[/math] respectively, and therefore that elements are products of just a's and b's.

>uni is running a series of talks on the work of the most recent fields medalists
>go to one on Akshay Venkatesh
>person giving the talk doesn't really work in the same field
>clearly doesn't understand his work super well
>mumbles something about convex bounds

>>10212849
Do your own homework.

>>10213099
Literally why would you do that instead of just reading the papers they published.

this is simple number theory

why bother?

>>10213099
It's not homework, I am just genuinely curious about how this would work.

>>10213117
One of them is an hour long talk which gives a broad overview and allows for questions and interaction between me and the person giving the talk. The other is several hundred pages papers on a subject I have little understanding of and potentially little interest in.

>>10213336
Absolutely pleb.

Have you ever had a book insult you?

Brainlet here. On a completely different topic i learned about the relationship between squares and odd numbers. So that was pretty pleasing.

do any anons prefer ring theory over group theory, well at least in the context of stuff covered in abstract algebra textbooks. feel that ring theory is just more elegant

>>10213660
Why do you find it more elegant? I like Ring theory, but I think group theory is more elegant just because it's simpler to a certain extent. I don't know. Rings theory is cool though.

So, I have a problem that I need advice on dealing with. When I study math from a textbook and begin answering the math questions at the end of a chapter, I inevitably hit a few that are essentially time sinks. My question is: What is the optimal period of time I should work on them until I essentially should stop and either look at the solution or get help? I understand that I should make an effort to solve all problems without any assistance, but this stuff on uniform continuity is really eating away time I have to deal with other classes.

Right now I'm thinking of using a 15 minute timer per problem but I have no clue if that's not enough time or too much time.

>>10213697

I feel like ring theory made me friends with all these structures like Z, R, Q, C (and polynomials over these) in a way group theory never really could (or at least did). And idk, ring theory just seems so much cooler. though i must say, i really, really liked Lagrange's theorem for finite groups.

>>10213697

15 minutes seems too little a time. Though it depends on what kind of math you are doing. For upper level math courses, if you are trying out to figure out every problem in under 15 minutes, then either you're a genius or you're not doing it right

>>10213900
It's upper level. I think an hour is too long, but 30 minutes too short, so 45 minutes?

>>10213722
Not enough time. Depends on what you want to get out of it obviously but hard questions are good for you. The most challenging course I took we had to solve 3 or 4 textbook questions a week.

>>10213454
They're not wrong.

>>10213900
>ring theory made me friends with all these structures like Z, R, Q, C (and polynomials over these) in a way group theory never really could
Could it be because polynomial rings are over rings or algebras instead of groups?

>realize I know nothing about complex analysis on several variables
>try to pirate something on libgen
>nothing
>download Barry Simon's book
>also take the real analysis one for good measure
>open up the summary for his Real Analysis book
What the fuck is this.

>>10214323
Post a pic

>>10214323
>What the fuck is this.

>>10214327
It's three pages long, give me a minute.

>>10214331

>>10214333
>hey, let's write a functional analysis book and pu real analysis in the title
>why?
>fuck if I know

>>10214331
>>10214333
>>10214334
>not covering Riesz products in your highschool calculus class
Do Americans really do this?

>>10214334
that's all standard material

>tfw test in an applied class in less than an hour
wish me luck lads
>tfw trying to memorize error formulas instead of doing some comfy algebra reading

>>10214922
I'm glad my semester is over. Now I can finally read what I want.

>>10214922
went pretty well I think. Sort of had to bullshit some stuff on error analysis but beyond that it wasn't too bad

>>10214922
>mfw the introductory optimization class at my university always fills up because it has a reputation for being easy
>mfw I have to study way more than my theory courses because literally everything in this class is just an asspull algorithm which makes no sense and you have no choice but to memorize them all
fuck applied courses
give me theoretical autism or give me death

>>10214922
>taking any applied classes at all
Cuck

>open up analysis book
>"prove this property"
>ayy lmao one line
>prove this identity
>three lines
>prove this limit
>half the page
>evaluate this
>two pages
>prove this inequality
>fifteen applications of Cauchy-Schwarz, four of Holder's, two of Young's, and about half of your normal measure theory book later it looks done

>>10215174
>not understanding the motivation behind optimization
not gonna make it.

>>10215174
>his tests don't have the formulas at the end
It's like your uni thinks you're a memorizing monkey.

>>10215197
it's a requirement
I do go to a meme school though, so you may be right regardless

I have decided this is the year I teach my cat how to integrate.

Wish me luck.

>>10215669
Godspeed.

>>10214922
Aaaaah the semester if finally over and I should be an approved math major any moment now, I can finally resume my comfy read through of Mathematical Reasoning . Feels good being a new fag.

>>10214323
>>10214331
>>10214333
>>10214334
I don't see the problem here, if you read Folland's book many of the same topics are covered. Also, there's a decent amount of topics in real analysis that intersect with harmonic and functional analysis as well, you have to talk about things like function spaces and norms in order to ask questions about whether this sequence of functions converges to this function and in what sense and blah blah blah. All of these are necessary for analysis of real functions.

>>10215669
>>>/toy/

>>10215669
riemann or lebesgue

>>10215669
I've taught my cat how to do RG analysis, integration should be easy.

>>10215965
I'm not supposed to tell anyone but my cat is actually the one behind IUT. That why no one can understand Mochizuki, he's trying to translate fucking cat.

>>10215669
>tfw galois theory exam allows us to use one cat
>tfw grabbed my measure theory and topology cat instead of my fields, rings, and groups cat

>>10215995
>tfw am allowed to bring dog to topology test but not cat
>tfw just continuously deform cat until it looks like a dog
dose this count as a violation of the academic integrity code?

>>10210629
Analysts please help. How do I show [math]\int_0^\infty x^{-x}dx=2[/math]?

>>10216171
>How do I show ∫∞0x−xdx=2?
What have you tried?

>>10216177
calling the suicide hotline

>>10213454
What book is this?

>>10216171
Uh I don't know dude probably need complex analysis or numerical methods. Why are you still solving integrals, are you an accountant or something?

>>10216250
paul zeitz - art and craft of problem solving

How important is spatial intelligence for a mathematician working on geometry?

>discrete math
>after a few weeks all we do is analysis
what kind of ruse is this

>>10217488
Somewhat important but I think it's more important to just be intelligent in general.
Relying on spatial intuition too much is sketchy once you get to more abstract geometry because your only trick is to solve the problem in dimension <=3 and hope it's the same for all n. But very often things are not the same in higher dimensions and then you're fucked if you don't have non-visual intuition available.

Help me out guys. As far as I understand, the last use of Euler's formula in pic related near the end also relies on the fact that we assumed [math]\text{gcd}(c, pq) = 1[/math] at the beginning of the proof. How would I prove the uniqueness of the solution in the case where [math]\text{gcd}(c, pq) \neq 1[/math]?

>>10217634
Fuck it, I may have figured it out. How does this look: if we prove that
[eqn]
u \equiv c^{d} \ (\text{mod} \ p)
\\
u \equiv c^{d} \ (\text{mod} \ q)
[/eqn]
then by Chinese Remainder Theorem [math]u \equiv c^{d} \ (\text{mod} \ pq)[/math] since [math]\text{gcd}(p,q)=1[/math].

Since [math]\text{gcd}(c,pq) \neq 1[/math], [math]\text{gcd}(u,pq)[/math] must be either [math]p[/math] or [math]q[/math]. Let's assume that it's [math]p[/math]. Now [math]p | u[/math] so [math]u \equiv 0 \ (\text{mod} \ p)[/math] and likewise [math]c^{d} \equiv (u^{e})^{d} \equiv 0 \ (\text{mod} \ p)[/math] since [math](u^{e})^{d}[/math] is a multiple of [math]p[/math]. Therefore
[eqn]
u \equiv c^{d} \ (\text{mod} \ p)
[/eqn]
is proven.

Now [math]\text{gcd}(c,pq)=p[/math] implies that [math]\text{gcd}(c,q)=1[/math], so we can use Fermat's little theorem to show that
[eqn]
u \equiv u^{de-k(p-1)(q-1)} \ (\text{mod} \ q)
\\
\equiv (u^{e})^{d} \cdot (u^{(q-1)})^{-k(p-1)} \ (\text{mod} \ q)
\\
\equiv (u^{e})^{d} \cdot 1^{-k(p-1)} \ (\text{mod} \ q)
\\
\equiv c^{d} \ (\text{mod} \ q)
[/eqn]
which (hopefully) completes the proof. Does this look sane at all or did I screw up somehow?

There was a thread a while back of some anons working through Atiyah's commutative algebra book. Whatever happened to it? Is the project still ongoing?

>>10217626
Reminder everyone thought the 7-sphere had only one smooth structure.
>>10217927
I think it literally just finished, but it was 99% one guy carrying everyone, and that sort of shit is unhealthy.

>>10210629
Hey lads, im interested in logic and mathematical logic but more fom the philosophical side, should i study philosophy(logic classes but no actual math) or math (logic but also algebra and calculus wich i don't like because i have to memorise things wich i don't enjoy)?, thanks

>>10216254
Taking probability currently. I need to show 1/2x^(-x) is a distribution, then find its moments.

>>10216177
Integration by parts, holder's inequality, Jensen's inequality, bounding the integrand above and below. Nothing is working.

>>10217962
Did you try complex analysis or using transforms?

>>10217961
>the philosophical side
What do you mean?

>>10217973
abstraction as opposed to number noodling i think

>>10217979
Math is "axiomatic abstraction", philosophy is a "repeatedly self grounding and ungrounding abstraction", if that makes sense to you.

>>10217979
Abstraction is the mathematical side though.

>>10217927
I was hosting it. As the other anon said, I was basically in an echo chamber so eventually I just dropped it. I only got up to chapter 3's exercises, and it's been about 1.5 months since I stopped posting solutions

You know, I get the impression that a reading group with tight schedules, discord groups and serious learning is bound not to work here, but something that's lighter, stupid on purpose or historical might actually work.
Say, an Euclid's elements reading group could work. Everyone here already knows the whole geometry in it, people would join up for the sake of Euclid's approach and writing being neat, or historical interest, and wouldn't lose much for skipping or joining midway.

>tfw sub 100 non-verbal IQ
I'll never make it mathbros.

What's some good semi-light reading for an undergraduate? I'm looking for something to work on over the winter break.

>>10218975
if you're asking to read math over a holiday, then you're autistic enough to read a first year graduate level book.

>>10219005
Any particular suggestion, or should I just pick something that looks interesting? I only have around a month.

>>10219060
depends. if you're planning on graduate school, look at qualifying exam syllabi of the institution you're interested in and read those books.

If not, then either and analysis or an algebra book. I say Folland for analysis, or Dummit and Foote for algebra (skip the'yre chapter on modules and rep theory )

>>10218975
>What's some good semi-light reading for an undergraduate? I'm looking for something to work on over the winter break.

>>10217634
>>10217777
>No replies in over 15 hours

>>10219060
Don't listen to that other guy. You can't do anything interesting with a graduate book in a month. Last Christmas break I read Rotman's Galois theory in about a week or two, a very concise intro to Galois theory. Requires knowing some basic group theory. It has an appendix with all the group theory you need to understand the book, but I personally found it very unhelpful since I hadn't seen composition series before, but any other source to complement it is probably good.

If you say what background you have and what topics you like, then I can recommend something better

>>10217777
hopelessly wrong

how can [math]\gcd(u,pq)[/math] be [math]p[/math] or [math]q[/math] if [math]0\not\equiv u\equiv c^d\pmod p[/math] and [math]0\not\equiv u\equiv c^d\pmod q[/math]

>>10218975
Hilbert & Von Cossen - Geometry and the Imagination

>>10219627
>hopelessly wrong
Honestly not surprised

What would a correct proof look like? Help out a brainlet here because I have no clue what to do.

>>10219631
I think you kinda had the right idea until you decided to declare gcd(u,pq)=p or q. If anything, it should have been gcd(c,pq)=p or q or pq, with the latter case being trivial, and with emphasis on c, not u.

>>10218975
Berger - Geometry Revealed
Godement - Analysis
Hilbert & Cohn-Vossen - Geometry and the Imagination [a favorite]
Klein - Development of Mathematics in the 19th Century

>>10219635
Okay, so let's remove the wrong declaration. Is there anything else wrong? If [math]\text{gcd}(c,pq)=p[/math] doesn't that still mean that [math]p | u[/math] since [math]u^{e} \equiv c \ (\text{mod} \ pq)[/math]? Or is that wrong too?

I want to learn more about categorical logic. I've taken a grad level course in category theory and did my undergrad as a double degree in pure math and comp sci as my background. I've also done a bunch of logic and have done a little bit of work (projects and studying into categorical logic and a bit on lattices, frames, and logic).
That said I don't feel I really "get it" when it comes to the internal logic of a category. I mean I get that you can take a category with certain limits and give it some logical semantics but when it comes to saying anything meaningful about logic or the category or about anything really I just don't see it. I'm not even sure I get the point of internal logic.
Are there any resources you guys can point me to or can someone give me some basic intuition about what it's all about and how to actually use it?

>>10219648
I'll check in a couple hours, busy now

>>10219351
If you haven't taken topology yet then this is a great resource that will make your life easier overall (especially if you also haven't taken analysis yet). It's basically a crash course on topology meant to help fill in the gaps for undergrad math students.
https://www.dpmms.cam.ac.uk/~twk/Top.pdf

For a fun popmath book I'd recommend Art of the Infinite.

>>10219652
By the way, even though those topology notes say they're 109 pages they're actually much shorter. All of the real content is covered in the first 40 pages and the rest is extra exercises and solutions to exercises/proofs to theorems (in case you got stuck trying to prove them in the main text). The notes are meant to be used for self study. If you've already taken analysis then you can probably skip to page 15 where it starts doing topology proper (you aren't really missing much by skipping to page 15 even if you haven't done analysis).

Are there any intro proof books that focus on a specific topic in maths? Should they be avoided? I have an intro to proof writing course coming up and I am also reading a book about proof writing that quickly surveys multiple topics. But I really really want to get a better taste of number thoery. If I could find one can I just read an intro to proofs book about number theory and then wait for my course to give me a broader taste of different topics come next semester?

>>10220075
>intro to proof writing course
do Americans really do this?

>>10220087
Well it's more of a... well yeah fuck you. Just answer my question.

>>10220087
>pathetic yuropoor always grasping at straws for his superiority complex
Most intro to proof classes are actually Foundations classes that introduce things like fields, the formal definition for convergence of a sequence and Cauchy sequences, the construction of the integers, rationals, and reals along the way for preparation for real analysis and algebra. It's not an entire class on induction, contradiction, direct proof, etc.

>>10220075
I know the other nigger is being a faggot, but you are being a niggerfaggot with that question.

Just introduce yourself to the methods of proof and then pick up a book on elementary number theory, and stop fucking around looking for a number theory specific intro to proofs book.

>>10220075
Hey man Introduction to Mathematical Thinking by Gilbert and Vanstone is an intro to proofs book that's mainly about number theory.

>>10220087
Most universities everywhere do this. Thing is it's typically integrated into another course like discrete math or abstract algebra so you don't usually see it called intro to proofs.

>>10220075
There are a lot of discrete math textbooks and a few abstract algebra textbooks that act as intro to proofs books. I'm not aware of any number theory books but that doesn't mean there aren't any.
There are also a lot of intro to proofs books that are just proofs How well they'll work will depend on how much work you put into them.
The intro to proofs course will have a huge effect on your ability to do mathematics going forward (make sure to get a good prof, do not take it with a shitty prof). Our university has an honors version of the course that's taught by a shitty prof and all the students who take it end up struggling a lot more than the ones who take the "easier" course taught by a brilliant prof.
You may also opt to take a formal logic course or to work through a book like The Logic Book by Bergmann (specifically the chapters on Fitch style proof systems).
The idea with proofs is that you have a few basic patterns (see pic related) and that every logical statement, no matter how complicated, can be broken down into a combination of these patterns. So a proofs course will teach you how to deal with each of these patterns. That way, no matter what you're asked to prove you will always be able to break it down and figure out a few strategies to attack it. A number theory book may teach you how to do some proofs but it's really better to actually do an intro to proofs course or text.

That said, Dudley Underwood's 1978 "Elementary Number Theory" second edition book (he has two books with similar names) is a great Number Theory book to work through if you want a straightforward methodical approach.

>>10220134
>>pathetic yuropoor always grasping at straws for his superiority complex
I'm not European.

>>10219079
>skip the'yre chapter on modules
But that's the best chapter in the whole book

>>10220138
It's funny that you say that because this >>10220087 , made me think of doing just that. I just realized it would be no diffrent than reading through one of those other proof books. And I already finished the intro part of my current book so fuck it.
>>10220164
This is not the answer I thought I was looking for, thanks.
>no matter how complicated, can be broken down into a combination of these patterns
This was a big problem for me when I was trying to prove shit outisde of my proof book. Thanks again.

Where my /moron who's good at pretending to be smart/ niggas at?

> When the prof says "uniqueness" instead of "unicity".
> When the prof says "cor-roll-a-ree" instead of "corra-larry".
STOP

>>10221161
>unicity

>>10221161
>unicity

>>10221161
>unicity

>>10221161
>unicity
what?

>>10221161
>unicity

>>10221192
>>10221191
>>10221189
>>10221177
>>10221164
X-ness just sounds lazy

>>10221197
And x-ity doesn't.

>>10221161
>> When the prof says "cor-roll-a-ree" instead of "corra-larry".
Only burgers say "cora-larry"

\The theory of Numbers have an end is science and worship because there is a harmony between Mind, Logic and scientific truth.
-Mohamed Ababou-\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Mathematical physics is the ultimate and best field of math.

>>10221254
b8

What should I start reading and fail follow through on this winter break /mg/?
>topology (munkres)
>algebraic topology (hatcher)
>some number theory book

>>10221325
Topology, get started NOW

>>10221325
>What should I start reading and fail follow through on this winter break /mg/?

>>10221325
>learning alg top before top
just no

Also munkres. Skip most of the final set theory chapters (still skim through it) and come back to it later when you get to such considerations, but I'd suggest doing the first couple ones because they are pretty useful. I still occasionally refer back to the exercises on the chapter of functions because it contains all the interactions of set theoretic operations with functions that you might need.

I say this because if you don't get to the topology fast, you will get pretty bored and quit.

>>10221325
Don't read Munkres read Willard or Engelking if you want topology (Engelking might be a bit much depending on your background). Munkres gives you a perspective that does not translate well into a later graduate general topology course at all.

>>10221346
>meme picture
hilarious

>>10221385
>>meme picture
What do you mean?

>>10221387
Go look at the archive to see countless people calling it retarded and stop trying to derail this thread.

>>10221395
>Go look at the archive to see countless people calling it retarded and stop trying to derail this thread.
How can a curriculum be retarded?

>>10221402
Ask your childhood.

>>10221422
>Ask your childhood.
What did he/she mean by this?

>>10221426
wew lad

>>10221373
kek

math guys, wolfram always tells me "cum putation time has run out"

what software can I use to use my own computation and solve algebra stuff?

please

>>10221499
Mathematica. Or just solve it numerically in matlab or something.

>>10221524
>Mathematica
that looks like just what I need, thanks

>>10210629
you are my last hope, I need help with the last one >>10220636

>>10221346
I'm glad it's only a picture now and not a wall of text.

>>10221499
maple works too and you can torrent

>>10221200
>Uniqueity
Do you have brain probs?

I wanna learn math, where do I start? I'm an absolute pleb.

>>10222315
Either Calculus or Linear Algebra.

>>10222193
>hhahahhah look at me misunderstand on purpose hillarious isn't it my fellow redditors?

>>10222316
How does this look? https://github.com/rossant/awesome-math

>>10222358
Eh.
Just download the book on libgen with the highest number of editions.

>>10222319
So yes.

define 'mathematics'

>>10222403
the study of logical abstraction

Does an answer key for Lovasz's Discrete Math: Elementary and Beyond or Hammack's the Book of Proofs? I can't find any complete solutions, only scraps here or there.

>>10222403
All I know is actuaries aren't mathematicians

>>10222618
Why do you need one? Just post a few of your proofs here for feedback.

Additionally there are often multiple sensible ways to approach a proof, so an answer key would be difficult to produce.

>>10222633
I have a final coming up and it would be easier to study if I had the solutions next to me.

Is it worth trying for a masters if you don't think you'll do a PhD?

undergrad here, completed calc through 3 (multivariate), ordinary differential equations, and linear algebra.

where do I go from here? cant take a math course this semester but i want to continue learning math on my own. i figured partial diff equ would be a logical step forward but the material i've found seems to include concepts from some sort of analysis field that are not unlearnable to me but definitely seem to imply there is an intermediate course I should take.

>>10222705
Flip a coin, and study real analysis if it falls on heads, or complex functions if it falls on tails.

>>10222705
Don't listen to the other guy, pick up an abstract algebra text and ignore the siren song of analysis

>>10222705
Do you like calculation or theory more? You sound like you'd like analysis but it's fucking gay compared to algebra or number theory

>>10222709
>>10222714
>S E E T H I N G A L G E B R A I S T S

>>10222715
you can't even construct the field you work in lmao

>>10222719
>he thinks I can't even construct the completion of the smallest field with char(0)

>>10222720
Do it then

>>10222714

I don't think I've had enough experience at this point to say, but I really enjoyed ODEs (hence going to PDEs). im just confused about wtf comes next after lower division math

>>10221161
>unicity

>>10222726
>he hasn't realised I literally just did
I forgot to specify, but the completion is under the ordering imposed on said field, which is, of course, unique.

>>10221435
based spivak

>>10222737
you've just taken it as fact that this structure exists, which is the approach I personally accept but it is NOT a construction.

I have a proof that P = NP but I'm not going to share it

>>10222746
>wanting me to spend fifteen minutes repeating the results you can read in any algebra book
Smh nigga.

>>10222732
If you liked ODEs, then I'd highly recommend going through Pugh's Real Analysis book. Pugh's focus is dynamical systems, so his book is written with a very geometric dynamical mindset. It's an excellent book, especially for self study because the exercises are absolutely golden.
Plenty of pdfs online for you to check it out.
After you've done your fill (I'd say you'd want to get through Riemann integrals, the rest of the book is fine but you don't need it yet) you could move to a proper differential equations / dynamical systems book like Hirsch Smale Devaney or to a PDEs book like Strauss. Both excellent but you'll want the real analysis intermediate.
Do not listen to algebraists, they will lead you away from diff eqs and pdes.

>>10222749
I can't read the result because it doesn't exist. The reals are not constructible and their existence relies on extending ZF(C)

>>10222747
based, good on you for saving the jobs of all those complexity theorists

>>10222747
I also have a proof but it can't be explained in polynomial time

brainlet here can somebody pls quickly show how integrating x*x' gives (1/2)x^2

>>10223191
by nutting on ur grandmas shed

>>10223221
nice rhyme

>>10223191
Integration by parts.
[math] D^{-1}(xDx)=x^2-D^{-1}(xDx)[/math]
[math]2D^{-1}(xDx)=x^2[/math]

>>10223257
right, that makes sense, thanks anon

>>10217961
Unless you're doing something very wrong, math courses should have the lowest memory-load of all.

I'd say go with math and suppliment with your own philosophical readings. You'll end up less of an asshole than the guy who does 4 years of french obscurantists + college algebra.

>>10222747
I proved it ages ago but the logic can't be expressed in mere symbols or explained in human language, it exists at too high a degree of abstraction.

Can someone spot a way to factorize this monster?

[eqn]f(a,b,c)=2(abc^2+ab^2c+a^2bc+ab^2+b^2a+ac^2+a^2c+bc^2+b^2c+ab+bc+ac)-3(a^2b^2+a^2c^2+b^2c^2+a^2+b^2+c^2)-6abc[/eqn]

>>10223560
it doesn't have to be necessarily into a single product btw, but it would be extremely helpful if it were

>>10223560
Wolfram alpha gives alternate forms of polynomials.
I tried copy pasting your post to see if it worked, and I think it read it correctly, but it didn't give anything useful.

>>10223579
yeah, it's usually terrible at doing most of the things I want it to do. Hopefully someone with access to some computer algebra program can do it.

>>10223560
(a + b + c)(a + b + c)
(ab + bc + ac)(a + b + c)
(ab + bc + ac)(ab + bc + ac)
^ After analysing the polynomial I've come up with this that would be kind of an approximation I think

Maybe something like
(a + b + c + ab + ac + bc)^2

Also for you who are wandering how this is done, stop trying to find an answer and start observing the patterns in the data.. (listen that is)

>>10223780
My simple solution works almost perfectly it just has a +6abc instead of -6abc and +1 as the second coefficient instead of -3

>>10223783
I have some ideas about how to solve this
first in order to make 6abc negative a possible solution would be that each opposite has reversed signs, for example a is positive and bc is negative or a is negative and bc is positive, however, I sincerely doubt it's possible to make the second coefficient negative, based on the arrangement I'm proposing here... and I don't think there is one where you could because they are squares, and negative times negative is positive (so unless it's imaginary or something I doubt you can make the second coefficient negative...

>>10223794
except if it's not a square, say a multiplication of unequal factors... anyway, what do you think of my solution so far? >>10223780

>>10223813
[math](K_1ab+K_2ac+K_3bc+K_4a+K_5b+K_6c)(L_1ab+L_2ac+L_3ac+L_4a+L_5b+L_6c) = L_1K_1a^2b^2+L_2K_1ab^2c+L_3K_1a^2bc+L_4K_1a^2b+L_5K_1ab^2+L_6K_1abc+L_1K_2ab^2+L_2K_2b^2c^2+L_3K_2abc^2+L_4K_2abc+L_5K_2b^2c+L_6K_2bc^2+L_1K_3a^2bc+L_2K_3abc^2+L_3K_3a^2c^2+L_4K_3a^2c+L_5K_3abc+L_6K_3ac^2+L_1K_4a^2b+L_2K_4abc+L_3K_4a^2c+L_4K_4a^2+L_5K_4ab+L_6K_4ac+L_1K_5ab^2+L_2K_5b^2c+L_3K_5abc+L_4K_5ab+L_5K_5b^2+L_6K_5bc+L_1K_6abc+L_2K_6bc^2+L_3K_6ac^2+L_4K_6ac+L_5K_6bc+L_6K_6c^2[/math]
Whenever the L subindex is equal to the K subindex then it should yield 3, whenever it's 6 4 5 3 2 1 L for 1 2 3 4 5 6 K it should correspond to the abc's of 6abc and the rest should have a 2 as coefficient...

>>10223858
However the one of two is not exactly true, after all each term is repeated twice, so the coefficients combined should equal to two. (such that you can switch L index with K index)
Anyway there is a typo: L_1K_2ab^2c instead of L_1K_2ab^2

>>10223858
It's kind of a mess, and it should read instead 6 5 4 3 2 1 L for 1 2 3 4 5 6 K
and the equation should be L_1ab +L_2ac + L_3bc the solution has a different order so that's why the L doesn't descend nicely

Anyway these are the equations:

[math]L_1K_1=L_2K_2=L_3K_3=L_4K_4=L_5K_5=L_6K_6=-3 [/math]
[math]L_1K_2+L_2K_1=L_1K_3+L_3K_1=L_1K_4+L_4K_1=L_1K_5+L_5K_1=L_2K_3+L_3K_2=L_2K_4+L_4K_2=L_2K_6+L_6K_2=L_3K_5+L_5K_3=L_3K_6+L_6K_3=L_4K_5+L_5K_4=L_4K_6+L_6K_4=L_5K_6+L_6K_5=2[/math]
[math]L_1K_6+L_2K_5+L_3K_4+L_4K_3+L_5K_2+L_6K_1=6[/math]

>>10223916
All the multiplications in the second kind of equations of L1K2 + L2K1.... are positive because L1 is opposite to K1 and L2 is opposite to K2 (because of the first kind of equations has to be negative), therefore L1K2 and L2K1 have the same sign but 2 is positive therefore all of such equations are positive

>>10223927
however this would imply that L_1 has the same sign as K_2 in order to make it positive therefore:
sgn(L_1) = sgn(K_2) = sgn(K_3) = sgn(K_4) = sgn(K_5) = sgn(L_2)
yet sgn(K_2) by the first kind of equations should be different from sgn(L_2) therefore by contradiction it is not possible to factorize the equation >>10223560 with exactly those coefficients

>>10223953
not even in the imaginary case

>>10223953
might be cleaner to write it as
sgn(L_1) = sgn(K_2)
sgn(L_1) = sgn(K_3)
sgn(K_2) = sgn(K_3)
sgn(K_3) = sgn(L_2)
therefore sign(K_2) = sgn(L_2) (a contradiction)

Is it worth trying to teach myself functional analysis over winter break if I haven't taken a course in analysis yet? I have the Griffel text on applied functional analysis which claims a background in real analysis isn't necessary, but am unsure if this is a waste of my time, since each page has been taking me upwards of 15 minutes to decode. Tbqh as a physishit I'm mainly interested in order to bolster my understanding of Hilbert spaces and Fourier transforms/PDE for quantum.

>>10224324
kreyszig maybe

>>10224324
15 minutes per page is not at all an unusual amount of time as long as you are eventually grasping what you're reading.

>>10210629
it's just the dihedral group
stop memeing omg omg so fun
that's basic as fuck and the author is trying so hard it's embarrassing

>>10220087
I bet you are that favela monkey brazilian poster. Go die you beaner scum

>>10223953
>>10223958
>>10223869
>>10223858
>>10223813
FUCK thanks a lot guys, I'll have to double check my work... It would be extremely painful if it isn't a square

>>10224436
>I bet you are that favela monkey brazilian poster.
Mathematicians use "we", not "I".

Tell me something nice about groups that are isomorphic to their group of endomorphisms.

>>10210629
So I know you can define a group object [math]G[/math] in any category in terms of morphisms [math]m:G \times G \rightarrow G[/math], [math](-)^{-1}:G \rightarrow G[/math], [math]e:\bullet \rightarrow G[/math] with diagrams encoding your axioms for the operations. I'm assuming you can similarly define an abelian group object, but I can't figure out how to encode commutativity of [math]m[/math]. Is there a canonical "switching" morphism [math]G \times G \rightarrow G \times G[/math] in a category with finite products?

>>10224633
The morphisms literally commute.

>>10224642
The other kind of commutativity smartass

>>10224633
define the category of group objects and define a group object there

>>10224633
> Is there a canonical "switching" morphism [math]G\times G\to G\times G[/math] in a category with finite products?
Can you not use Yoneda to define this ? It is easy enough to define for sets, so you can define a compatible family of flips [math]Hom(A, G\times G) \to Hom(A, G \times G)[/math] using the fact that [math]Hom(A, G\times G) \simeq Hom(A, G) \times Hom(A, G)[/math] naturally in A.

>>10210629
Hello math bros,
Engi fag here, how do you study theorems? Till now it was enough for me to just understand exercises, but now i'm expected to know theorems and its picking my brain apart, needing to know every little detail. Do you bruteforce memorize, or use some tricks?
I need to learn around 30articles so like 100+theorems.

>>10225099
Read and understand the proofs

>>10225099
The exact same way you memorize formulas for engineering.

Quick question. How to define unique morphism from pushout in category of sets? Picrelated's a picture from Wiki and I'll proceed with using naming from it.

So, [math]P[/math] is defined as a disjoint union of [math]X[/math] and [math]Y[/math] with [math]x \in X'[/math] being equivalent to [math]y \in Y'[/math] if [math]x = f(z)[/math] and [math]y = g(z)[/math] for some [math]z \in Z[/math] where [math]X', Y'[/math] are isomorphic copies in disjoint union. Now, I have a question on how to define the unique morphism [math]u: P \rightarrow Q[/math]. Can I just say that [math]u([p]) = j_1([p])[/math] if the representative of an equivalence class happens to be from [math]X'[/math] and [math]u([p]) = j_2([p])[/math] otherwise? I don't know how properly define [math]u[/math] so that the choice of representative of an equivalence class doesn't matter. Moreover, should I provide a concrete construction of a disjoint union (like [math]X \times \{0\} \bigcup X \times \{1\}[/math])? Because if I don't do this, I technically cannot define [math]u[/math] by using [math]f, g[/math] because I work with isomorphic copies and cannot extract element from original sets.

>>10225336
>So, P is defined as a disjoint union of X and Y with x∈X being equivalent to y∈Y if x=f(z) and y=g(z) for some z∈Z where X,Y are isomorphic copies in disjoint union.
>something something equivalence classes something something
If you're going to use such an autistic construction for P you might as well just make a morphism from Z to P and build u from that tbqhwyfam.

>>10225336
>So, P is defined as a disjoint union of X and Y
no, it's defined as the object in the pushout of the diagram and, for sets, turns out to be the disjoint union. But okay, I don't wanna be that guy.

> Now, I have a question on how to define the unique morphism u:PQ.
The morphism depends on the X,Y,Z,f,g you start with, just saying.

I think what you try to do is right, but I think it's a mistake to even start to argue in terms of elements of the sets.
In particular
>u([p])=j1([p])
is quite the abuse of notation, if it makes sense, since the two arrows u and j1 don't share a domain.

> I don't know how properly define u so that the choice of representative of an equivalence class doesn't matter
Again, why do you want to deifne it in the first place? The existance of such an u is what grants that a certain construction is a pushout

If you look at
>X×{0}⋃Y×{1}
(I think you wrote X×{1} there by mistake)
then the i's are the obvious injections (e.g. for i1, mapping v to the pair <v,0> for v in X) and the unique u is given, for elements stemming from X, by proj_left followed by j1

>Moreover, should I provide a concrete construction of a disjoint union (like X×{0}⋃Y×{1})?
It's a free country, what do you mean by that question? That is, in any case, a possibility.

>Because if I don't do this, I technically cannot define u by using f,g because I work with isomorphic copies and cannot extract element from original sets.
why do you feel the need to construct u?

Again, for any given X,Y,Z,f,g, a diagram with the other elements (in non-dashed lines) is a pushout, IF you can construct an u. You can search for an u if you want to check wether some concrete situation actually makes for a pushout, but there are arbitrarily many pushouts in the category of sets, even with the same starting X,Y,Z,f,g.
And if you eventually drop logic+set theory and do raw cateogry theory, there are no standalone elements anymore anyway, only arrows from singletons

That said, if you pin down your generic equivalence construction, maybe there's a roundabout way of defining u in terms of that construction - I don't quite see it yet but I don't want to discourage you to set this up.
Might be easier to tackle first the question to prove that a category of all sets has all pushouts.

could you please help me convert

a*s^3 + b*s^2 + c*s + 1

to

(xs + 1) (ys+1) (zs + 1) (ws+1)

is there a generic name for this so i can look it up?losing my mind a little

>>10225481
never mind im a retard i was lookint at it all wrong i won't even delete my post just to show u that there are, indeed, retards. good night

>>10225393
Thanks, anon, the answer was educative.
I know, the concrete construction shouldn't matter as the object is defined up to a unique isomorphism.
>IF you can construct an u
My problem is that I didn't know if I could construct it and needed these set-theoretic arguments to cope...

>>10225481
>>10225561
https://www.wolframalpha.com/widgets/view.jsp?id=15b128aa42d812ef9f3a9640bf1fb3fa

Guys I was reading this old book to refresh my high school math and suddenly it starts to address groups, "mathematical structures", isomorphisms, cyclic groups, etc, and I search it all up and found out it's a bunch of stuff from group theory and abstract algebra. Did people actually learned that in high school back in the 70s? I'm kind of having fun with that, but I had to slow down a little bit to understand the proofs.

>>10225925
It's still in the curriculum in some exam boards in the UK...

>>10225925
>US Math before Sputnik: Now, the real numbers is what we call a field. A field is a set that is a group under one operation and a ...

>US Math After Sputnik: Multiplication GOOOD, Division BAAAD, just use the calculator brah lol

what did he/she mean by this?

https://kerodon.net/
>an online resource for homotopy-coherent mathematics

>Forage
>Part I: Higher Category Theory
>Chapter 1: The Language of ∞-Categories
>Chapter 2: Examples of ∞-Categories

What's an effective way to self study out of textbooks?

Should I just write down definitions/theorems and then do problems? Some people tell me I should stop and try to prove the definitions myself, but that really slows down my progress in self learning. Some of the proofs seem to require genius level insights.

>Start self-studying topology from munkres
>Start with the preliminary
>Chapter on functions implies that the inverse of a function only exists if it's already bijective
>Proceeds to ask a question in the exercise section that uses the inverse of f notation while implying f may or maynot be bijective
This is why no one likes topologists

>>10226343
>>Start self-studying topology from munkres
Munkres is a meme.

>>10226327
A textbook is disposed in the milimetric order which makes each theorem's required lemmas the closest possible to it. Ultimately, proving anything in a textbook is easy, because it's all pre-packaged.
Then again, sometimes absolutely basic proofs just don't come to you. Happens to the best of us. Erdos couldn't comprehend the Monty Hall problem without literally seeing a simulation.

But yes, you should try to prove stuff on your own, absolutely. Solving exercises isn't for training, it's to help you memorize.

>>10226351
Okay, that's sound advice.

What if I really get stuck? Should I then just copy down the proof?

> Solving exercises isn't for training, it's to help you memorize

What helps me train then? Just doing the proofs out?

>>10226419
If you mean training for an exam, doing exercises is marvelous. If you mean training for being a mathematician, I'd recommend picking up problem books on the subject, trying to prove ahead, or trying to give different proofs, maybe even with the methods of different subjects. A lot of the problems in functional analysis can be solved through differential equations techniques, and vice versa, group theory has a similar relation to combinatorics, synthetic and analytic geometry with each other, etc. If you find you have no such basis, try to do proofs straight from the definitions.

>>10226343
you shouldn't associate inverse function notation with bijectivity. It's most important use is as preimage. It is inconvenient that they share the same notation, but it is obvious from the context which one is which, and there is zero ambiguity.

>>10226343
You can talk about [math]f^{-1}[/math] for any function [math]f:X\to Y[/math]. For any [math]y \in Y[/math], [math]f^{-1}(y)=\{x \in X : f(x)=y\}[/math]. If the function is bijective, then you know that the set I just described will contain exactly one element of X for each element of Y.

>>10226438
>You can talk about f−1 for any function f:XY. For any y∈Y, f−1(y)={x∈X:f(x)=y}.
Not well defined.

>>10226445
>no reading comprehension

>>10226447
>>no reading comprehension
That's not why it's not well-defined.

>>10226451
You mean how he made a function that maps points of Y to sets of X and implied that if f is bijective that's the normal inverse?
Yeah, I noticed, but shitposting's nice.

>>10226445
>Not well defined.
it is well defined. it's a SET

>>10226459
Your mom's a set

>>10226458
>if f is bijective that's the normal inverse?
This is false.

>>10226465
>unironically no reading comprehension

>>10226458
>implied that if f is bijective that's the normal inverse
I never said anything about a "normal" inverse. The other guy going about "badly defined" is just trolling.

>>10226473
No, no, the problem is the beginning of the sentence. You make it sound like you're extending the definition of a function's inverse.
It's also bad notation, which is why people usually only use inverses like that for sets on Y.

>>10226481
Oh, I see. Next time I talk about preimages I'll specify that f^-1 is just notation and that it is usually clear from context what the symbol refers to. It would have been better to define [math]f^{-1}:P(X)\to P(Y)[/math] and then [math]f^{-1}(B) = \{x\in X : f(x) \in B\}[/math] for B a subset of Y, but I thought this was too pedantic.

>>10226438
>>10226447
>>10226458
>>10226459
>>10226467
>>10226473
>>10226481
Retards. I'm not that >guy, but you should know how the axiom of choice makes such things possible, and why the other >guy, who probably ironically rejects it, says it's not well-defined.

>>10226481
Also I agree about bad notation and I feel it would be clearer to have "f with an arrow pointing left at the bottom" for preimage, something like [math]f_{\leftarrow}[/math] (hoping my LaTeX skills are good enough).

>>10226487
Oops, [math]f^{-1}[/math] goes from P(Y) to P(X) of course.

>>10226490
Absolutely swell notation, 10/10.

>>10226487
>f−1:P(X)P(Y)
Assuming the axiom of power set is true.

>>10226497
>complaining about the axiom of power set
A new low for /mg/.

>>10226289
Why do you think that happened? One would think that having a deeper undestanding of the math being things would help in creating great scientists in that generation.
I'm brazilian and so is the book, by the way, so I don't think Sputnik had this much of an impact on us.

>>10226537
>Why do you think that happened?
A few things concerning the US's reasons.
1. Government High School diploma mills:
Back during the time of Sputnik, having a High School diploma actually meant something. You might be able to get away with shoving marbles up your ass in grade and middle school, but back then, you actually had to apply to get accepted into a high school. However with a sizable number of adults uncomfortable with putting 13 year olds to work if they could not make the cut, laws were passed that forced mandatory high schooling for all kids younger then 18. The sudden flux of mandatory kids who gave 0 fucks about any kind of learning outside the body of the opposite sex has not only ruined the high school system, but made the degree worthless as they had to dumb down all subjects to meet minimum graduation requirements.

2. Sputnik scared the US government so shitless that it tried to ram down advanced mathematics into elementary school kid's throats in order to compete with the soviets( https://en.wikipedia.org/wiki/New_Math
) . What the US government failed to account for is that you have to be able to count before you can ask what the cardinality of a finite set is. This set kids up for failure, leading to the next section

3. Since the 1960's, advertising has forced the "math is hard and is for gay losers so buy our shit to look cool" meme pretty hard, and given the amount of continued success it has enjoyed, it set up people learning math to think they are just going to fail. In order to try and entice them to continue, we continue to dumb down the content.

>>10226537
>math behind things
Fixed.
>>10226589
Ok, is there any talk of another math education reform in the US? How's the Common Core thing going?
>we continue to dumb down the content.
At this pace you guys will eventually stall entire generations when it comes to math skills.

>>10211159
yes

>>10211159
You had an exam on the dihedral group?

>>10212095
>automatas
Are you retarded in the head?

>>10226589
The logic behind teaching sets and shit to young kids is an example of human hubris gone mad
>I was a maladjusted kid with connections that set me up for success
>I had to earn a degree to do this
>it took 6+ years AS A GROWN FUCKING ADULT to cope with this material
>I bet children could understand it as well as me
?????

>>10226747
You took 6 years as a grown man to cope with basic abstract algebra?

>>10226751
New Math tried to teach set theory and axiomatic methods to kids. Not one person thought "I was 20+ when I could finally do this shit correctly." Not one.

>>10226747
>mfw Mathematicans are still expecting year 1 graduate students to know Homological Algebra even though they themselves have spent 15+ years researching and mastering the material
I mean it's just Cohen-Macaulay rings duh doi

this thread in a nutshell

Are there any links between Category Theory and Number Theory?

>>10226789
Yes, it's called autism

>>10226438
If the function isn't surjective, some of the sets will be empty.

>>10226438
Also that's f^-1({y}), not f^-1(y)

I WANT TO LEARN ALL THE MATH AND ALL THE SCIENCE. WHY IS IT SO HARD

>>10226793
Nice meme.

>>10226793
based

>>10226796
>some of the sets will be empty
Yes, what exactly is your point?

>>10226801
Yeah, I fucking love science too!

>>10226881
Then unless f^-1 is defined from B to P(A) (assuming f is defined from A to B), it isn't well defined, and usually when people talk about f^-1, they define it from B to A. So if f is bijective, then f^-1 takes an image and gives you it's unique antecedent. f^-1 also denotes something else; let C be a subset of B then f^-1(C)={x∈A / f(x)∈C}

>>10227434
unless your f^-1*

>>10226970
Fuck you

>>10227434

>>10226306
>what did he/she mean by this?
It says in the about section that it's by Lurie.
Which is good and bad. Bad because I doubt he knows how to write an accessible text.
But the whole book as a web page concept is interesting, strangely I never saw it realized.

https://kerodon.net/tag/0008

There's no reason to worry if a model for something exists before studying it, prove me wrong

>>10227480
Well arguments of that kind always end up boiling down to
>why do I do what I do
>what is the meaning of life

It's worst when arguing with people who unreflected base their worldview on notions like
>technological progress is important
>it would be tragic if time would freeze from one moment to the next
etc.

>>10227480
Godel's completeness theorem.

>>10227500
exactly, think of all the consistent mathematics that can't be explored because you're bogged down with models

>>10227471
>Bad because I doubt he knows how to write an accessible text.
>falling for the "accessibility" meme

>>10227440
Sorry man, my week has been so shit and at the time I posted this I was throwing shit at several people at the same time. Sorry again.

>>10225336
Let (P,i1,i2) be any pushout of (f,g) in Set. Given a pair of commuting maps from X, Y, Z (determined by k1, k2 from X and Y), it is easy to construct u. For any element of P, choose any element of X or Y mapping to it under i1 or i2, call it z. Prove it is well-defined to send this element of P to the image of z under the appropriate k-map, and that this map is forced to meet the splitting conditions. You will use the fact that the maps k1, k2 commute with f,g.

Does anybody have a good recommendation for a survey article on the Kähler-Ricci flow?

>>10228191
...............

>>10227913
You have a nice heart anon, get out from this shithole.

Is Q16 dumb or am I dumb?
Clearly rotating the plane has no nontrivial fixed elements, and a simple computation shows that indeed the fixed points of the rotation matrix is [math]\pm i[/math], so that the fixed "field" is just 0.

>>10228937
fuck im retarded, I needed to find the fixed field of [math]\mathbb R(x)[/math], not [math]\mathbb R[/math]

>>10228937
>>10228987
But no WAIT. My problem is that the function [math\varphi[/math] does NOT leave [math]\mathbb R[/math] fixed, so it cannot be an element of [math]\text{Gal}(\mathbb R(x)/\mathbb R)[/math]

>>10228937
Something something normal extension over x^3=1 and x^3=-1, something something fixes the reals, something something swaps the two third roots of -1 and the two of 1.

>>10228995
fuck never mind im double retarded, i forgot constants do not have [math]x[/math] terms!!

>>10229000
yeah, but this book introduces a Galois extension as a fixed field of a Galois group. In fact, it hasn't introduced separable or normal extensions yet

>>10229003
The Galois extension isn't the fixed field, the Galois extension is the original field plus the part that isn't fixed.

>>10229006

>>10229009
You read the definition backwards. That says "F is the fixed field of the Galois Group of K over F".

>>10229013
Where K is the Galois extension, of course.

>>10229013
Well, I meant that, whatever

Any nice topics for bachelor thesis in mathematics? I will do a master in computer science afterwards so it would nice to do something applicable, e.g. something in discrete math.

Anything cool and cutting edge? I heard about matroids

explain this

>>10229164
You can add the first structure to the real plane without any caveats. The second one cannot be added to the real number line without doing esoteric bullshit.

[math]
\epsilon \ne 0 \implies \frac{\epsilon}{\epsilon}=1 \implies \frac{\epsilon^2}{\epsilon}=\epsilon \implies 0=\epsilon
[/math]
hmmm

>>10229181
I was gonna type that down and say that he just un-fielded the rationals but that works too.

>>10229171
I hope you're only pretending to be retarded. R[i] = R[x]/(x^2 + 1) and R[e] = R[x]/(x^2)

>>10229213
>R[e]=R[x]/(x^2)

>>10229213
embarrasing

>>10229221
What's wrong with that?

>>10229224
>R[x]/(x^2)=R

>>10229226
Are you autistic? Obviously by e I mean epsilon

>>10229228
I give up, someone else explain.

>>10229231
You're literally retarded if you think that's wrong.

I ran out of good Grillet screenshots.
Whoever wants to make the next thread, feel free to do whatever.

How much calculus and linear algebra I need to understand for diff geometry? The introductory book I have says in the preface that all the shit there is trivial and anyone with basic knowledge of multivariable calculus and linear algebra is good to go. I got the calc part covered, but not the LA. I tried reading some of the stuff there but didn't understand shit. Send help.

>>10229259
You need around the first fifty pages of any LA text.

>>10229259
If you think you understand multivariable calc, but you don't understand l.a. you probably don't underrstand either. These courses are basic so plenty of topics will assume you command them. But to understand diff geometry is better to have knowledge from topolgy.

>>10229269
>>10229270
Thanks, frens.

>>10229231
before i continue, i just want you to know you're a retard

open any abstract algebra textbook to its first page

brainlet

>>10229351
>there are two people who are that stupid out there

>>10229367
>>there are two people who are that stupid out there
Who are you quoting?

Why does grad school exist now again?

The fourier transform of this would just be two rectangles centered around -10 and 10, right?
Am I wrong in thinking that multiplication in the frequency domain is convolution in the time domain and vice versa?

>>10229645
FUG

>>10229181
Idk if you're pretending to be retarded but 1/e is not defined.
>>10229226
Idk if you're pretending to be retarded but that's false

The top is a Galois extension with lots of nice properties.
The bottom isn't even a field.

>>10229920
No, U

>>10229920
>e is non zero
>e also has no multiplicative inverse
why the fuck would anyone use such a structure

>>10230154
The virgin mathematician vs the chad machine learner: https://en.wikipedia.org/wiki/Automatic_differentiation

>>10230154
Are you suggesting we only study fields?

anyone know a site where it gives addition or subtraction problems?

i need to improve my mental math

>>10231037
Like adding/multiplying two numbers?
Wouldn't be hard to do in a Python script.

>>10231047
yea 2 number or 3 or whatever
>Wouldn't be hard to do in a Python script.

just link me a site ffs.

>>10231049
Here I made one for you
https://onlinegdb.com/SJxHxuqgN