/sci/ - Science & Math

First for functional analysis and combinatorics

>>11603154
Blessed.

>>11603154
>First for functional analysis and combinatorics
literal boomer math

What is the cursed area of maths that you are not supposed to touch?

>>11603284
IUT

Threadly reminder to work with physicists.

>>11603287
He just wants an apology since he was proved right.

I'm trying to show path-connectedness in R^n implies chainability. Here is what I have so far. Fix [math] \epsilon >0, a,b \in E[/math]. Since E is path-connected there exists [math]f: [0,1] \to E [/math] s.t. [math]f(0) = a, f(1) = b[/math]. I need to show there exists a finite number of points [math]x_k[/math] s.t. [math] \| x_k - x_{k-1} \| < \epsilon [/math]. Intuitively, it makes sense that if there exists a path between two points, I can glue together the path by taking pieces of the path that are less than epsilon. I know I need to use the continuity of f in some way but I don't really know how.

why did teichmuller hate the jews?

>>11603608
It was 1930s Germany. Everybody who wasn't a Jew hated the Jews.
The only thing different about Teichmuller is that he died before the war ended so he never had to pretend he didn't.

The lower semicontinuous hull of a function [math]f : X \to \mathbb{R}[/math] where [math]X[/math] is a normed vector space is defined as
[eqn]\sup_{N \in \mathcal{N}(x)} \inf_{y \in N} f(y)[/eqn]where [math]\mathcal{N}(x)[/math] denotes the set of neighbourhoods of [math]x[/math]. Is this equivalent to taking
[eqn]\lim_{\epsilon \to 0} \inf_{\|y-x\| \leq \epsilon} f(y)[/eqn]? I can't quite get an intuition for the original definition.

Finally covering analysis after finishing my BSc by reading Rudin. Dedekind cuts are one of those objects in mathematics that make me wonder how anyone came up with it at all and know it would have the power to be the object you were looking for to finish a construction, as in constructing the reals in this case. List more things like this. Please stray away from equations, they are boring and pleb tier.

What's the steepest introductory analysis book? I want a difficult book that doesn't really assume much.

>>11601108
Hey, didnt really do much today, hope ill finish set theory problems tomorrow through out the day and then read on mappings and do their exercises before next upload, maybe I could do more but I think thats the start of the stuff I dont know

>>11604030
Rudin

>>11603866
Dedekind cuts seem like a very natural construction to me. You want to build the reals from the rationals and you realise there's stuff missing. How do you talk about the missing stuff in terms of the rationals? Well you have sets defined in terms of rationals that are bounded by a real number (like the set of numbers whose square is less than 2). Then you play around with sets like this for a little while (maybe working out how to represent non-algebraic numbers like pi), then start trying to see if you come up with a unique representation using these sets. Before you know it you have Dedekind cuts.

A lot of maths is like this. If you look at the definition of a scheme it will look bizarre and unmotivated, but if you follow the history of studying curves then surfaces, then algebraic varieties it's a fairly natural construction. A topological space seem weird (and indeed took a long time for mathematicians to define) but there is a clear path to it's discovery (via metric spaces).

>>11604063
Ops forgot to link the channel: https://youtu.be/jPuQQSJ89Cg
I dont really mess with youtube or video editing but Ill get better over time

>>11603866
>and know it would have the power to be the object you were looking for to finish a construction
You don't. You come up with ideas that look promising and try them out. Most of them fail, but you don't publish the ones that get 1/4 of the way in and break down.

>>11604077
>steep

Where does this meme even come from? Literally you're just applying the definitions of the terms. There's nothing to "think" about. Getting the answers is almost mechanical. This is more true in analysis than in more "spatial" problems which is why I don't get where the whole "analysis is so hard" comes from. Like do people actually try to visualize the fucking terms when they're solving the problems? Because if they are, then yeah, I feel bad for you- that sounds hard. But you don't need to. Analysis is just semantics. Not a lot of "visualizing" or grey area.

>>11603866
>Please stray away from equations, they are boring and pleb tier.

You need to read some math history then get your head checked kiddo.

>>11604078

Topological spaces make a whole lot more sense when you approach them from graphs than from functions. I don't know who in their right mind thought trying to explain connectedness by talking about something most people conceptualize as continuous (even though they aren't necessarily continuous as we all learn) was a "good starting point." Like, c'mon.

>>11604154
what

>>11604077
Isn't that the one prescribed to everyone? I'm looking for a fairly challenging read, something maybe akin to A Course in Arithmetic (but I don't think that's an introduction).
>>11604132
What's a steep intro analysis book then?

Given the equation [math]A\vec{x} = b[/math], where A is a square matrix, x is a nx1 vector an b a nx1 vector, is it true that if A is invertible then this system has an unique solution?

>>11604200
>something maybe akin to A Course in Arithmetic
Why do you want something akin to a book you haven't read?

>>11604204
stupid questions go in /sqt/ >>11601289

>>11604200
Rudin is very steep basically throwing you into the deep end right away. Often it's only recommended as a meme.

>>11604204
Yes. Suppose Ax=b. If A has an inverse, what vector must x be?

>>11603598
Continuous functions map compact sets to compact sets, so the image of [0,1] under f must be compact. Then consider the open cover {B(x,d/2): x is in f([0,1])}, where B denotes the open ball with center x and radius d/2, for any fixed d. Since f is continuous, f([0,1]) is compact, so there will exist finitely many of these balls, each characterized by their center. All of these centers will constitute a collection of finitely many points satisfying the desired condition. Hope this helps.

>>11604154
>I don't know who in their right mind thought trying to explain connectedness by talking about something most people conceptualize as continuous (even though they aren't necessarily continuous as we all learn) was a "good starting point." Like, c'mon.
What do you mean?

>>11603598
[math]f^{-1} B_{ \epsilon} (x)[/math], for all [math]x \in E[/math], gives an open covering of the unit interval. Since the unit interval is compact, choose a finite subcovering.

FUCK MATRIX NOTATION
FUCK MATRIX NOTATION
FUCK MATRICES
FUCK PEOPLE THAT USE MATRICS
FUCK MATRIX NOTATION
NO I am NOT butthurt at not being able to use it or not understand it. of course following the steps is easy as fuck. it's just pointless and makes no fucking SENSE to do it that way instead of just doing it like a NORMAL PERSON. I bet if you go to 100,000 ayy civilizations in the universe who have more math advancement than us, at MOST, 17 of them will still be DUMB enough to use MATRIX NOTATION. Get fucked matrixkikes.
PROVE ME WRONG PROTIP YOU CANT

>>11604206
I've read through some bits of the algebra part, and am aware of the general consensus.
>>11604220
Oh, I didn't realize. Thanks!

>>11604302
Based. Matrices exist just because they are convenient, but it doesn't feel like they are. Fuck matrices.

>>11604302
Imagine being this butthurt over not being able to understand matrices.
It's literally middle school difficulty.

>>11604302
Mega based. You're not alone, anon.

>>11604324
>>11604345
Absolutely based. Happy as fuck to know I am not alone.
>>11604337
Oh I understand them, I just think the notation is dumb as fuck.

>>11604364
Trust me, you don't understand them.
What's an application you ask?
Ever used google?

>>11604302
>instead of just doing it like a NORMAL PERSON.
how do normal people represent linear transformations?

>>11604369
Algebra :^)

>>11604388
>Algebra
>Without matrices
The only part of algebra that doesn't involve matrices is group theory, and even a large portion of that is matrices.

>>11604159
>>11604248

Do you guys really not read math history? Do you literally just jump into the textbook without any curiosity about the origin of the concepts?

>>11604200
>steep

I really don't think like that. I either do or don't know what the terms mean. If I don't, I look it up. It's a very mechanical process for me. I don't use a lot of "intuition" to answer questions.

>>11604397
Let me dumb this down for you, little guy. We could write x-2y-4z=7 and 2x-3y-6z=5 as a matrix equation
|1 -2 4 ||x| = |7|
|2 -3 -6||y| |5|
|0 0 0 ||z| |0|
Yes the syntax is bad cause I am doing it as plaintext but you get the idea. We can either write it like that, or we can write it as
x-2y-4z=7
2x-3y-6z=5
See? That's non-matrix form.
Now take that concept and apply it to all the more advanced applications where matrices are used. Same concept.

>>11604200

But to be clear, I also am not a student. I literally do this shit for fun in my spare time so naturally I'm not under any time pressure to solve the problems and can answer them in whatever way is coziest for me.

>>11604410
I don't know why you're trying so hard to look smart, but it's not working. You just sound like a fag

>>11604410
So what if you want to just deal with the coefficients on their own, to simplify things? Also, how are you going to represent things like, e.g. the Jacobian, or the Lie group SO(3), or the inertia of an object?

>>11604406
>Do you guys really not read math history?
No. And I never will.
>Do you literally just jump into the textbook
Yes.
>without any curiosity about the origin of the concepts?
No. I am curious about the concepts themselves, not their origin.

>>11604416
>I don't know why you're trying so hard to look smart,
I was just shitposting.
>I was only pretending to be retarded!
but unironically
>>11604428
idk what a Lie group is except that supposedly the universe is in the shape of a LIe group
the universe is E8
read the exceptionally simple theory of everything

>>11604590
Don't shit on things until you know the full picture. If something commonly used appears stupid to you in math, you're probably uneducated.

>>11604302
>incredibly simple compact representational method for a massive class of objects and maps between objects in vector/matrix notation
>even has sensible generalizations to infinite-dimensional cases
>somehow this is bad

>>11604590
>i don't know what a lie group is
second year undergrad fuck off
a group which is a manifold

>>11604406
>Do you guys really not read math history?
Yes. I just don't understand what you wrote
>I don't know who in their right mind thought trying to explain connectedness by talking about something most people conceptualize as continuous
What do most people conceptualize as continuous? How does continuity relate to connectedness period, unless you mean path-connectedness?

>>11603866
>List more things like this.
My personal favorite are weak derivatives, but there are a lot of other great constructions, e.g. the Lebesgue measure (and more generally the Hausdorf measure). What makes the Lebesgue so cool is that you can actually prove that it was the *right* one and really the only way to measure volume in a way compatible with our daily experience.
Weak derivatives are also pretty cool, since there are at least 3 different ways of how you could *obviously* define them and all these ways work out to the same concept.

>and know it would have the power to be the object you were looking for to finish a construction
Dedeking almost certainly didn't. The justification for it's usefulness was post hoc (like with pretty much any mathematical construction).
There were probably hundreds of failed attempts to construct the reals.

>Please stray away from equations, they are boring and pleb tier.
What is that even supposed to mean?

>>11604406
>Do you guys really not read math history?
I read a book on the history of Analysis.

>Do you literally just jump into the textbook without any curiosity about the origin of the concepts?
Far more important then the origin is the motivation.

Good morning, /mg/!

>>11603598
Try using this: https://en.wikipedia.org/wiki/Lebesgue%27s_number_lemma

>>11604063
>the start of the stuff I dont know
When you know you don't know something, you learn it! Post a problem you are struggling with and we can try together.
>>11604087
It's actually more like the "her" in "Herriot". The operation is a function [math]S \times S \to S[/math] defined by [math](a, b) \mapsto a*b[/math],

>>11604078
Well, yes, given your perspective it is a 'natural construction' because most mathematics has been nicely packaged for you to see it that way. Did the pioneers see it that way? Fuck no. They probably tried hundreds of different things that didn't work before finding the perfect thing that worked. If you can't look at math as "How the fuck did someone come up with this at all?" I don't think you're understanding it beyond face value, you're certainly not seeing the forest through the trees.

>>11604135
>>11604907
When I say equations are boring and pleb tier, most can be derived from a series of identities or first principles. I'm talking true mathematical creativity, something that seems completely out of nowhere and perfect and constructed to solve a deep problem. e^ipi + 1 = 0 is pretty cool and all but anyone would have come to that eventually. I'm so sick of people giving me that answer when I ask for a beautiful math concept. I'm talking something truly creative and something that defined an entire theory.

>>11604940
>I'm so sick of people giving me that answer when I ask for a beautiful math concept.
I gave you like two concepts, one of these leads to my favorite equations (which obviously requires some more context and requirements to comprehend in any way):
[eqn]\int_{\Omega} u \ \partial_{i} \varphi \ \mathrm{d}x = \int_{\Omega} \partial_{i} u \ \varphi \ \mathrm{d}x [/eqn]

>something that defined an entire theory.
The above equation defines the entire field of non-classical PDE theory.

The other examples were the Lebesgue and Hausdorf measure. They are pretty cool too...

>>11604918
>Good morning, /mg/!
>3AM
euro? or bad sleep habits?

>>11604960
8:35:05 ante meridiem. Although, I do have bad sleep habits, too. This was probably the second or third night this year that I got even 8 hours of sleep.

>>11604968
transatlantic sexpress tranny slut lmao

What do i have to do to be able to study 8 hours a day with laser-like concentration on the subject?

>>11605147
Be genuinely interested, sleep enough, make sure you get all the nutrients.

>>11603711
he didn't just hate jews. he was an enthusiastic and active nazi

>>11605253
how to be genuinely interested?

>>11603060
do you think he ...you know.. had the jungle fever?

>>11605147
meth

>>11605314
>he was an enthusiastic and active nazi
based!

>>11605339
fuck off, nazis ruined german math

>>11605429
>nazis ruined german math
What did they not ruin?

Marco (the waiter) Cameriere meets Pigritio (Piggy) Risorgiamento in the automat.
>“Say, Marco. You know that guy Watermaker?”
>“Watermaker the differential geometer?”
>“Yeah. Know something? He uses infinitesimals. Infinitesimals aren't rigorous.”
>“Aw, c'mon, Piggy. Everybody uses infinitesimals these days. They can be made rigorous. They are rigorous.”
>“They're not rigorous, Marco. Watermaker needs a lesson in rigour.”
>Idly chopping his cigar in bits with poultry shears, Pigritio emphasises the point. “What he needs is rigour.”
The following day, a shocking announcement is made at the conference on singularities at New Rochelle.
>“Dr Watermaker is in hospital, apparently with a shoulder wound caused by a sawn-off shotgun or some similar instrument. Fortunately, we have been able to fill in a replacement speaker."
>"Professor P. Risorgiamento will talk on "The treatment of obstructions".

>>11605439
Spaceflight
War technology
The Global Jewish Order (in the long run)
Germany (in the short run)

>>11590389
I looked a bit into the topic of this paper - actually computing zeta function zeros.
I was aware that it's a theorem that the Riemann hypothesis was [math] \Pi_0 [/math] in the arithmetic hierarchy, i.e. there's a natural number problem such that to find a counter example you can brute force it discretely. But I always found it dubious how one would verify an actual zero on the critical strip and I'm still a bit nebulous about Riemanns computation thereof in the 1850's. But it was through that paper brought to my attention that
[math] \pi^{-s / 2} \Gamma(s / 2) \zeta(s) [/math]
is real for [math] s= tfrac{1}{2} + t [/math].
(Here's a small task for u: prove this!)
It means that a sign change long t actually guarantees you got the bounds of a zero.
Sidenote, it looks like Turings last paper is on finding zeros and hell this is an ugly looking subject
https://en.wikipedia.org/wiki/Turing%27s_method

Now I tinkered around with my own approaches, in particular via the Eta function representation convergence for Re(s)>0 and

[math] \zeta(x+i\,y) = \dfrac{2^{i\,y}}{2^{i\,y}-2^{1-x}}\cdot\sum_{n=1}^\infty \dfrac{(-1)^{n-1}}{n^x} \left[ \cos(y\, \log(n))-i\sin(y\, \log(n))\right] [/math]

I do some introductory discussion here
https://youtu.be/Fl3XgPpvSNI

Is there a name for a weaker form of continuity where [math] x_n \rightarrow x [/math] implies that there exists a subsequence with [math] f(x_{n_k}) \rightarrow f(x)[/math]

>>11605627

Has there been any attempts to generalize the zeta function? I remember reading a paper where they generalized the collatz conjecture (which is admittedly much easier to do) although I don't remember if it led to any special insight.

>>11605775
holy shit I am retarded this implies continity.
Good though, I was asking because my actual goal was proving continuity and I thought had to settle on weak shit.

>>11605627
Brainlet here, isn't every statement a [math] \Pi_0 [/math] statement because you can (discretely) enumerate all possible proofs and check for each one if it is a correct proof of the statement?

>>11605796
I'm kind of curious on something. What does your proof look like? I've scratched out the following:
We take an arbitrary open set [math]A[/math] around [math]f(x)[/math]. This lets us define the subsequence of the [math]x_n[/math] whose images are outside of [math]A[/math]. This subsequence "converges" to [math]x[/math], so it has to have the given subsequence. It can't, and thus it's actually a finite sequence, which implies that, for some [math]N[/math], [math]n > N[/math] implies [math]f(x_n) \in A[/math] .

is Hoffman & Kunze, Axler and Roman the holy trinity of linear algebra books?

>>11605971
>is Hoffman & Kunze, Axler and Roman the holy trinity of linear algebra books?
Axler is a meme.

>>11606000
"Axler is a meme" is a meme (unlike "Lang is a meme" which is 100% true), it's a very good proof-heavy book that focuses on the abstract parts of linear algebra. Perfect for a math major's first course on linear algebra. Its only sin is the omission of matrices, arbitrary dimension and arbitrary fields throughout most of the book.

>>11605971
>>11606000
>>11606013
Friedberg

Which of these are actually true?
>Rudin is a meme
>Axler is a meme
>Lang is a meme

>>11605803
What you seem to propose is to enumerate all possible proofs. E.g. enumerating all strings and, as a middle-step, checking if they are actually proofs and then checking of what they are proves.
Then every provable statement would eventually get proven and every disprovable statement would eventually get disproven. One issue is that via Gödel, there's statements for which there's neither a proof of it, nor of its negation.
Moreover, this way before proving "1+2=3", you may end up proving "1+2+5^2=3+25", "1+2+6^2=3+36", "1+2+7^2=3+49", and all that crap.
In any case, that's a bit more meta than what Pi_0 is, however, which is more tangible:
A statement S being Pi_0 means, roughly, that it is itself of the form, "forall n·· A(n··)," i.e. the statement S is the claim that a first-order arithmetic statement A holds about all the numbers.
So to disprove the statement, you may just find a counter-example, i.e. an m such that an _arithmetical claim_ A(m) fails. I.e. to disprove is possible in Peano arithmetic, which is nice.
The Riemann hypothesis being Pi_0 means that you can find a counter-example while enumerating over natural numbers like that - which is not apriori obvious (to me anyway) because it's usually formulated about a statement about some complex values function on subsets of C. I.e. it naively sounds like you'd have to find possibly transcendental numbers c satisfying f(c)=0 exactly.

Btw. what you want to get at can in some sense also be done in the logic, and is called Skolemization.
https://en.wikipedia.org/wiki/Skolem_normal_form

>>11605789
>Has there been any attempts to generalize the zeta function?
Of course, in many ways. After all, it's over 150 years old.
Here's a long list of zeta functions, many of which have Riemann's as special case.
https://en.wikipedia.org/wiki/List_of_zeta_functions

A pretty one is an algebraic approach to reason about "primes" in a broader sense, see
https://en.wikipedia.org/wiki/Dedekind_zeta_function

>>11605971
I’m unfamiliar with Roman but Valenza, Axler and Shilov form my personal holy trinity of LA, I guess

>>11605439
Ummm, the white race?

>>11606098
>Ummm, the white race?
How did Hitler not ruin the white race?

Been to Germany/France/Britain recently?

>>11603866
Was your undergrad not in math? I can’t see why analysis wouldn’t be covered otherwise

>>11606105
This. In the Eastern European uni I did my bachelors, the undergrad degree was essentially just analysis with a little bit of algebra and logic on the side.

>>11606082
>Rudin is a meme.
I can only speak about baby Rudin. Its presentation is great, but I'd say is not as good as everyone say it is. I'd say it's more of a middle ground between, say, Bartle & Sherbert (which is more like rigorous calculus) and Carothers (which is purely abstract, dealing with advanced topics such as metric spaces and the Lebesgue integral).
>Axler is a meme.
See: >>11606013
>Lang is a meme.
This one is true.

>>11606098
>Valenza
Beautifully organized, but not very interesting, it's a really basic book.
>Shilov
Never tried it. I'll take a look at it.

>>11605893
Dont have time to read but you can show it by contradiction. Assume that f(xn) os far from f(x) infinitely often. Then pick a subsequence where this is a case and you get a contradiction becausw fpr this subsequence it converges.

>>11606105
I did analysis but not Rudin. I took a second year analysis course using 'Analysis with Introduction to Proof' by Lay and a complex analysis course, and this satisfied my analysis requirements. I did rings and fields, groups, Galois theory, and algebraic geometry, and then more applied courses like ODEs, numerical analysis, and variational calculus.

Any mid-late 20s returning students made it in research? Asking for a friend

Need someone to screen share and give me the answers to my chem test this tuesday (1 hour). $30 reward in monero.
discord: _____#3678

is Grothendieck the greatest mathematician of the 20th century

would you agree with that?

>>11606082
Rudin is a meme, but not a bad book. It's a good book. It's just also a meme.
Axler is not a meme.
Lang is a meme and it's shit.

>>11606193
go away

>>11606193
this is the fucking math general
go fuck yourself you cheating rat piece of shit

>>11606193
Why do you post the face of your bully?

>>11606198
>gromov: *exists*

>>11606084
You mean Pi_1 statement. Pi_0 statements are computable.

infinities and irrational numbers seriously need to FUCK OFF from math

>>11606450
no, YOU fuck off, Wilderberg

>>11606450
Who even deals with infinite stuff?

>>11606264
yeah

>>11606499
boomers

>>11606450
For me, as long as it's [math]\Pi^0_1[/math], it's ok.

i wish i could study maths for 16 hours a day but my brain is rotten

>>11606578
It's all about fostering little habits anon. make the habit of studying half and hour a day, then a whole hour, then two, and so on. Restricting cellphone and internet usage could help, too. 16 hours seems a bit far fetched, but there's nothing stopping you from studying 6-8 hours daily, for example.

>>11606578
8 hrs sleep
2 hrs exercise
12 hrs math/being productive/taking care of necessities
2 hrs chillin, dispersed throughout the day

not a timetable, just a loose day meakeup

What book did you use for first time in abstract algebra? I'm thinking artin but it seems a bit hard, so lang?

>>11606168
Very nice, thanks.

>>11606605
do Fraleigh and maybe Herstein. complement them with Dummit & Foote. Don't use Lang

>>11606605
Stillwell, Elements of Algebra
Hands down the best algebra book for beginners. It actually motivates the algebra instead of the bourbacucks who just spam def prop proof without end.

>>11606605
my first algebra class was a mix of AA and number theory - pretty comfy and with a lot of useful! results. also the prof was one of the best educators I ever had so that's that.
it's fair to say we didn't use anything in particular but there were some obvious similarities with his book "elements of linear algebra" (in polish, great book if you can't into space).
overall I'd say lang is good (the undergrad lang especially) - used it for a bit. also read some of the "course in the group theory" which I can't recall the author of.

>>11606618
That's three books for one topic, do I really need that much AA? Why not lang?
>>11606621
Thanks for rec
>>11606624
Thanks for the recs, but group theory is covered in these AA books why should i pick up another book on this topic specifically?

>>11606605
In English: Herstein.

>>11606181
can't give you personal advice, but there's a postdoc in his 40s working for the same prof as me
so it's at least technically possible

>>11606588
>12 hours

Bro I can manage 3 hours max, that's of actual concentration. How the fuck can you do 12? And everyday? I literally don't do anything for 12 hours straight every day. I would become absolutely sick of it by the third day unless you were paying me in gold.

>>11606605
>What book did you use for first time in abstract algebra
Schaum's Outline of Abstract Algebra.
>do you recommend it
No.

>>11606636
I felt the pain in this

>>11606629
>why should i pick up another book on this topic specifically?
diversify, get another point of view. also read a bit of the grad lang during that period.
either way you asked what we used so there's your answer. it worked well enough for me.

>>11606634
>How the fuck can you do 12?
the trick is to disperse the hours - read a bit here, think about it understand it do something with it and when sick of it do something else for a while. also remember it doesn't have to be straight uninterrupted 12hrs math, being productive with your time is enough.

>>11606605
>What book did you use for first time in abstract algebra?

>>11606659
I would put a butterfly on the cover instead of that.

>>11606629
Fraleigh is only good as an introduction, Herstein will challenge you more, and D&F is the best reference book for all your questions.

I'd say just pick Herstein and D&F as a complement, but if you're unfamiliar with proofs/algebra in general, and since you said you found Artin difficult, my recommendation would be to get into Fraleigh first.

btw, Artin is also a good book, it's just that there are better ones.

>>11603154
Functional analysis is Jewish math.
>>11603284
Set theory.
>>11603608
Memory-holed Jewish events.
>>11603866
>spending hours on a formal construction of a notion that is intuitively obvious
More Judaism.
>>11604302
Based.
>>11604369
T : V -> V
>>11604499
You WILL lose understanding if you don't look at historical developments from time to time.
>>11604639
... such that the group operations are smooth. If you're gonna be a snob, do it right.
>>11606198
Good, but overhyped and politicised.
>>11606629
>Why not lang?
Shitty writing without any motivation. Also some chapters are just senseless.

>>11606666
Nice quads. What's your opinion on Aluffi? It t wasn't there when I learned the subject but I've heard lots of enthusiasm on it.

>>11606651
>diversify, get another point of view
This makes sense, never thought about that
Why should I try grad algebra that shit looks extremely difficult
>>11606666
I see, thanks I will do that! Yeah I suck at proofs.
You never explained why not to choose lang though

Please tell me the advantage of endowing the Lubin-Tate space at infinite height with a perfectoid structure.

>>11606605
My intro class used this http://abstract.ups.edu/download/aata-20190710.pdf (actually free not a pdf upload http://abstract.ups.edu) and we skipped chapters 12-15 and stopped at chapter 18. I liked it

>>11606605
I self-taught algebra by reading MacLane's book. I really liked it; MacLane is an extraordinarily good writer. It's like Aluffi in that it uses categorical concepts freely, but unlike Aluffi it doesn't shoehorn the shallowest little bit of them everywhere as a cheap marketing gimmick to trick trannies into thinking it's deep. It just uses them where it's actually natural to do so.

>>11606629
Lang's undergrad algebra book is honestly probably fine. When people say "Lang is a meme", they're usually referring to the 900 page yellow brick intended as a graduate reference text. Or the "basic mathematics" book for high schoolers.
Artin's a really nice book, and it's not all that hard. Artin is very geometric compared to a lot of algebra books, which you personally may like or may not like. Herstein's a standard recommendation too, but I haven't actually looked through it so I can't say anything other than that.
You absolutely don't need 3 undergrad algebra books. You might need 3 or more in total, but one of them should be a graduate text and then some more if you're particularly interested in something (the guy recommending you do more group theory probably likes groups). The idea of getting a different point of view when you get stuck is good, but reading 3 different books is a dumb way to do this. You have stackexchange, or youtube, or even here, if you need an alternative explanation because the book isn't working for you.

>>11606676
>Why should I try grad algebra that shit looks extremely difficult
you answered your own question.
reminds me of that time i went first semester undergrad pure math + cs double degree.
since i wanted to go pure spec, I've taken a good at the classes I'd have to take and realized that since I've slacked in hs I don't really know combinatorics all that much (skipped lessons and never studied anything apart from informatics) - so I decided that I either fail while taking this class now or I'm worthy of pursuing this degree.
Of fucking course I was worthy aced that shit best fucking class of that semester 10/10 would take again this is how winning is done.

>>11606698
*taken a good (look) at the classes

>>11606689
I have no idea what the fuck is Lubin-Tate space, but pic related is my opinion on structures.

>>11606697
Thanks, this is the best answer I could've hoped for!
>but one of them should be a graduate text
oh wow
>>11606698
damn that's interesting af, good job! (your post motivates me to use Lang instead LOL)
I also slacked in highschool im trying to be very serious about math so I'm studying ahead

Brainlet here. If I have a non-increasing function on the reals, is it always true that [math]\sup_{x \geq 0} f(x) = \lim_{x \to 0^-} f(x)[/math]?

>>11606763
I meant [math]\lim_{x \to 0^+}[/math]

>>11606763
>is it true that all non-increasing functions are continuous at 0
no.

>>11606790

what's a counter-example

>>11606795
f(x) = 3.50 (x <= 0)
f(x) = 1-x (x > 0)

>>11606795
0 for x at most 0, -x-1 for x>0. The limit would be -1

Are there rational numbers that are not integers and whose squares are natural numbers?

>>11606864
>Are there rational numbers that are not integers and whose squares are natural numbers?
No.

>>11606864
hint: [math]\sqrt{p}[/math] is always irrational

>>11606864
If a and b don't have a common denominator then a*a and b*b don't either, since none contain any novel numbers.

>>11606864
If you had some situation like that, let p/q be the most reduced case such that its square is an integer, then p^2 would be an integer multiple of q^2. Two possibilities: the square root of this integer would be itself an integer, and so p=nq and thus p/q = n; or this square root is not an integer, but in that case it would be irrational, and so p would be irrational or q would be irrational, and this would contradict the assumption that p and q are integers.

>>11606864
>>11606872
for [math]p[/math] prime obv

Remember OATS at 3pm British time (GMT+0)! https://sites.google.com/view/nialltaggartmath/oats

>>11607141
>British

>>11606082
Depends on which Lang you mean. If you mean Lang's Algebra, it's unironically the best algebra book I know and I don't give a fuck what >>11606200 and >>11606153 think if they mean Lang's algebra, because you have to master that if you're serious about heavy duty shit.

>>11607229
I will read your explanation of why that is bad when I wake up.

Nighty-night, /mg/! And you keep up the good work >>11604063!

>>11607244
Anglos aren't white

>>11607244
>I will read your explanation of why that is bad when I wake up.
https://youtu.be/WJTDpiDH7b4?t=1093

>>11607244
>I will read your explanation of why that is bad when I wake up.
https://youtu.be/WJTDpiDH7b4?t=1093

>>11607243
lang's algebra was precisely the book i was talking about. there are so many better basic algebra books and lang writes like shit and has no idea what he's doing. exercises are garbage.

>>11607244
Hey, updated my current state on stuff, almost finished my sets problems (yesterdays section), and read through the mappings topic, going to reread all the definitions and do the exercises, im thinking of only doing like maybe 1-2 easy problems, all the ones that have a * (means it might be related to future material iirc) and try some hard ones too;
Also I saw your post saying to ask what i dont know, but only a bit before doing my exercises did I see the comment saying what * meant as a mapping lol.. Gn

>>11607482
The introduction of category theory is one of the best ways to teach abstract algebra at a higher level. I've found it easily one of the best textbooks I've ever read, personally.

>>11607716
dilate

>>11607734
Git gud

Does anybody have that list of books to study made by a professor? I remember seeing it, but didn't save it, nor could I read the title of the books due to the low resolution at points.

>>11607901
>Does anybody have that list of books to study made by a professor? I remember seeing it, but didn't save it, nor could I read the title of the books due to the low resolution at points.
see >>11604555
the titles are included within the text

>>11607931
I assume the list of topics are what are covered by the authors, which are in parenthesis?

>>11607934
>I assume the list of topics are what are covered by the authors, which are in parenthesis?
Yes

Who's the Max Stirner of maths?

Hi guys newfag here with a lot of free time looking to learn math.
>>11604555
Is this list a meme or is it something I can actually follow to get a holistic education in math

>>11608057
i'd consider it a meme. it's ridiculously ambitious, especially for a self-learner

>>11603060
>Go into class (log into zoom)
>"okay class, today we'll be covering the proof of [insert graph theory result] using [insert algorithm]. A lot of people claim to make new results in this direction, but all they do is make minor improvements to the algorithm and publish the result.
Do any of the proofs you've had just casually shit talk other people in their field? It has happened on multiple occasions with this guy I'm taking a class with. Some other examples
"Yeah, people just prove analogues of certain results for the tutte polynomial and that somehow qualifies for high level journals."
"people just rehash the burning algorithm and apply it to uninteresting circumstances all the time."

>>11608057
It's a meme. The freshman and sophomore stuff is what I'd expect for incoming PhD candidates at a top 10 school. The junior stuff only if they're very strong. Senior is where I expect them to be a year or two after their qualifier exams. Specialist-level stuff is your dissertation topic. There's no mathematician alive who could look at that specialist reading list and say, "Yeah, I know all these fields."

Good morning, /mg/!

>>11607246
That's a "nothing", then.

>>11607457
Dark Vikernes talking about burning churches.

>>11607652
Gotcha! I saw your update and two comments:
(1) those *-problems will be important when you get to the point where the author considers the groups of automorphisms.
(2) The her thing was about how you pronounce Herstein's name. I was maybe a bit unclear about that.
The offer still stands, so feel free to ask!

Have a productive day.

>>11608097
lmao what a fucking chad

I have a kibda vague question about Turing completeness and computation.
I am studying a family of transformations on sets, so [math] T_i: \mathcal{\mathbb{R}^d \longrightarrow \mathcal{\mathbb{R}^d [math] with an indey set I.
Then I asked the question if it is possible to make any set convex. So for which sets does there exist finitely many such transformations where the result of applyung them all is convex.
Since the process of performing such a transformation is quite complex I felt like, on an intuitive level, it might be possible to do computation with this.
The idea is that the set is like a state and I can make a list assignig each state a corresponding transformation. Then I choose an input set and let the program run until it terminates. The output is the resulting set.
Since the transformations smooth out the set, I could say the terminating condition is the set being convex! What if it was impossible that every set can be made complex because then every turing machine would halt or something like that!

To be specific: my transformations are the Steiner symmetrizations, where you make a set symmetric and convex in a direction v while keeping the measure on each line in direction v the same (you can find a definition on wikipedia).
If you apply Steiner Symmetrization on polygons, and count a vertex with >180° angle as a nonconvex vertex and the others as convex vertices, then you get complex transformations of basically a string of 0s and 1s.

>>11608433
latex fix: [math] T_i: \mathcal{L}(mathbb{R})^d \longrightarrow \mathcal{L}(\mathbb{R})^d [/math]

>>11608035
That's tough, since he's quite a unique one. Let me try and find some commonalities and go with

https://en.m.wikipedia.org/wiki/Frank_P._Ramsey

>still no yukariposts
What a disappointment.

>>11608662
AAAAAAAAAAA YUKARI-SAMA IF YOU CUM INSIDE ME LIKE THAT I'M GONNA COOOOOOOOOOM

Is there a name for the [math]\langle Tx, y \rangle \leq ||T|| ~ ||x|| ~ ||y|| [/math] inequality?
>isn't it just Cauchy-Schwarz followed by the definition of the norm
Yes, but it's also one of the most basic tricks you have in applied maths, so I wouldn't be surprised if someone gave it a funny name a lá [math]\tau[/math].

Let [math]A[/math] be some set and [math]H[/math] a closed hyperplane. Is the closure of their intersection given by [math]\overline{A \cap H} = \overline{A} \cap H[/math]?

OATS in 15 minutes. Get ready! https://sites.google.com/view/nialltaggartmath/oats

>>11608982
No.

>>11608982
A = complement of H
what happens ?

>>11608035
Unironically me.

uhmmm is this the stupid question thread? How do you get the idea for a basic proof?
Prove that if x, y and z are three real numbers such that x^2 + y^2 + z^2 < xy + xz + yz, then x + y + z > 0

Since (x − y)^2 + (x − z)^2 + (y − z)^2 ≥ 0, it follows that 2x^2 + 2y^2 + 2z^2 − 2xy − 2xz − 2yz ≥ 0 and
so x2 + y2 + z2 ≥ xy + xz + yz.

What makes one think that (x − y)^2 + (x − z)^2 + (y − z)^2 ≥ 0 is a good starting point?

>>11609302
practice

>>11609302
>What makes one think that (x − y)^2 + (x − z)^2 + (y − z)^2 ≥ 0 is a good starting point?
throwing shit at the wall
realizing that those produce the squares and products you need from having dealt with polynomials
pure intuition
"genius"
god's inspiration
looking up the question
asking /mg/
and various other methods

>>11609302
>What makes one think that (x − y)^2 + (x − z)^2 + (y − z)^2 ≥ 0 is a good starting point?
It gives both the squares of x, y and z and the terms xy, xz and yz.

>>11603747
Since you take the sup over all neighborhood N of x you can restrict yourself to a subset of the neighborood whose lower bound is close to x. Since any neighborhood of x contains x, that bound is necessarily smaller than x.

For instance any interval of the form ]x-epsilon, x+epsilon[ is a neighborhood of x. For any neighborhood N of x there exist epsilon > 0 such that x - epsilon > lower bound(N). You can thus choose a increasing sequence of epsilon that converge to 0 such that the limit realizes the upper bou,d over all neighborhood.

So in R you're considering lim f(y) for y = x - epsilon and epsilon -> 0, in other words lim f(x - epsilon) for epsilon -> 0. All that works for R but not necessarily for another space.

>>11609348
>>11609334
>>11609319
I mean What idea prompts you to set up this starting point in that example? Would you have used the same starting point? If yes what was the thought process like.

>>11609397
I would because >>11609348

>>11609397
experience man, just play with these sort of expressions and it will seem more natural

If you have time to waste, try to read Equations and Inequalities by Herman, Kucera and Simsa. It is full of these sorts of problems and very well explained.
I read it before my 1st year of undergrad and it did wonders for my computational skills

This is a physics question but it's the math I'm struggling with. In classical mechanics W = F.ds (an integral) and when mass is conatant, through substituting F = m.dv/dt and ds = v.dt we get
W = m.(dv/dt).v.dt
then: W (1/2).m.(d/dt).v^2.dt

My question is where the fuck does the 1/2 come from? I don't understand this step

>>11609387
Pretty sure it works for any metric space, it's just a pain to prove.
Consider [math]B(x) = \{ B_{\epsilon}(x) : \epsilon > 0 \}[/math].
The inequality [math]\sup_{N \in \mathcal{N}(x)} \inf_{y \in N} f(y) \geq \sup_{N \in B(x)} \inf_{y \in N} f(y) [/math] is par the course.
The reverse inequality is obtained by considering that if [math]N \in \mathcal{N}(x)[/math], there's an [math]N' \in B(x)[/math] such that [math]\inf_{y \in N} f(y) \leq \inf_{y \in N'} f(y)[/math].
Showing that [math]\sup_{N \in B(x)} \inf_{y \in N} f(y) = \lim _{\epsilon \rightarrow 0} \inf _{y \in B_{ \epsilon} (x)} f(y)[/math] is just a formality.

>>11609439
*an [math]N'[/math] such that [math]N' \subseteq N[/math], and then the inequality follows trivially.

What is the mathematical intuition behind wave equation?

Laplace operator measures how much is the function averaged out. The heat equation therefore says that the function becomes more averaged with time. This I understand. But I have no idea how can one see that the wave equation should really give some kind of wavey motion.

>>11608144
> The operation is a function S×SS defined by (a,b)a∗b,
Alright so Im going to assume that Im not missing anything so all i know is that it takes 2 elements and outputs another element, BUT we don't have information on how * does it, or we don't know what a * b returns; like 2h after starting to type this (got interrupted) think i got a proof but im not sure if it looks good, will post later;
Im curious on a solution without drawings for ex 12.a) which is prove the De Morgans Rules for interssection;
Pic related is ex.14 to prove the number of elements of m(A union B) is pic related, I could only solve it via writing, how can i get a more formal solution?

>>11609544
I'm assuming you can prove it in the case where [math]A[/math] and [math]B[/math] are disjoint. Induct on B's cardinality or something.
Then, since [math]A \cap B[/math] and [math]A - (A \cap B)[/math] are disjoint, we can use [math]m(A) = m (A - (A \cap B)) + m (A \cap B)[/math]. Also notice that [math]A \cup B = [ A - (A \cap B) ] \cup B[/math] is also disjoint for [math]m(A \cup B) = m(A - (A \cap B))+ m(B) = m(A)-m(A \cap B) + m(B)[/math]
Also, ask this sort of stuff in /sqt/.

>>11608689
S-sauce?

https://www-app.uni-regensburg.de/Fakultaeten/MAT/sfb-higher-invariants/index.php/Regensburg_low-dimensional_geometry_and_topology_seminar
Potentionally neat!

>>11609544
Okay, so the way I would do this problem number 14 would be like this:
(1) Given any [math]x\in A\cup B[/math], we know that [math]x\in A[/math] or [math]x\in B[/math] (not exclusive). If we go through first all the elements of [math]A[/math] and then all the elements of [math]B[/math], then we have gone through all the elements of the union, but it is possible that we count the same element twice, and this happens precisely for those elements in the intersection. Therefore, we fix the thing by subtracting the number of elements in the intersection to get the number of elements in the union.
Alternatively:
(2) Replace [math]B[/math] with [math]B' = B\setminus A[/math]. Then [math]A \cup B = A \cup B'[/math] and the sets are disjoint. Then [math]n( A\cup B) = n(A \cup B') = n(A) + n(B')[/math]. Moreover, [math]n(B') = n(B) - n(A \cap B)[/math], so the result follows.

>we don't have information on how * does it, or we don't know what a * b returns
That is true. However, when you get to the point where groups are introduced, then there will be additional information like for example [math]a*(b*c) = (a*b)*c[/math] that can be used to prove things on a general level.

For the de Morgan, if some element is in the complement of [math]A\cap B[/math], then it is not in [math]A[/math] or not in [math]B[/math] (or it is in the complement of the set it is not in). This leads to the fact that it is then in the union of the complements (we need it to be in only one of the complements for this to be true). That gives you [math](A\cap B)^c \subseteq A^c \cup B^c[/math]. If an element is in the union of the complements, then it is in at least one of the complements, and so it can't be in both of the sets, and therefore not in the intersection. This gives you [math]A^c \cup B^c \subseteq (A\cap B)^c[/math].

>>11608983
Kinda doubt anything not know to Mahowald since the dark ages is gonna be in the talk.

>>11609673
I have no idea what you are trying to tell us.

>>11609690
Maybe I misunderstood the purpose fo the seminar, it's not supposed to be original work is it?

>>11609704
I guess it's more like sharing ideas and giving new perspectives etc. At least I hope so, since I am pretty much retarded but still able to follow.

>>11609723
Well, let me know if anything fun shows up.

>>11609744
The last talk could at least be something nice, as Markus Szymik is a pretty good speaker. Who knows? Not me.

>>11609750
I was referring to the current talk, can't go because of scheduling.

* | a | 2a | 3a | 4a | 5a
a a 2a 3a 4a 5a
2a 2a 3a 4a 5a a
3a 3a 4a 5a a 2a
4a 4a 5a a 2a 3a
5a 5a a 2a 3a 4a

>>11609760
Ah, that was a few hours ago. The slides for that: http://www.maths.gla.ac.uk/~ajb/dvi-ps/Talks/OATS%202020.pdf

>>11609767
>609767▶
>* | a | 2a | 3a | 4a | 5a
>a a 2a 3a 4a 5a
>2a 2a 3a 4a 5a a
>3a 3a 4a 5a a 2a
>4a 4a 5a a 2a 3a
>5a 5a a 2a 3a 4a

>>11609767
a 2a 3a 4a 5a

2a 3a 4a 5a a

3a 4a 5a a 2a

4a 5a a 2a 3a

5a a 2a 3a 4a

>>11609787
There's already a mention of this within The Abstract algebra
Just Grouping
Want a whole entire chapter on groups go to
http://mandal.faculty.ku.edu/math791/spFteen791/PIGroups.pdf

>>11609770
I really dislike how he uses kO, it's possibly the worst of all the choices he could have made for notation. At the end he gets to a fun toy problem where he asks about self-dual cyclic A-modules. Did he say anything more about this in the talk than in the slides?

Rate this typo.

Why are physics majors so much less popular than math majors?

>>11609616
stop advertising shit whore

why does this man trigger this general so hard?

>>11609614
https://asmhentai.com/gallery/240594/9/
You're the same guy who asked last time, aren't you?

>>11609871
Pointing out the fact that this man is a liar and a cheater doesn't imply being triggered though.

>>11609871
he's delusional

>>11609894
>he's delusional
or maybe you're just biased?

>>11609809
To be honest, I don't remember if he did. I am not personally that interested in spectra, so my mind was half there and half somewhere else at times when listening to him.

>>11609839
Rude.

>>11609918
Anon is biased, Wildberger is based. That's a fact.

>>11609839
she is a beautiful girl which deserves to be loved.

>>11609918
He definitely triggered me with his video on mathematical statements which are true, but unprovable, which turned out to be nothing but schizo math. The statement was something like

The first 10^20 digits of sqrt(2) don't contain 10^10 8's in a row.

First of all, this is totally provable.

Second of all, he never gave better arguments than "just look at this, obviously this is almost certainly true" and "just look at this, how in the world would you prove that?"

>>11609961
>schizo math
>implying all the math after 1400 isn't schizo

>>11609961
Yes, I think his view on mathematics is quite reductionist, arguing that objects only exist if they are "computable" in some way seems like a very easy way to rob a lot usefulness from mathematics.
And I also fail to see a good reason why considering objects which aren't "computable" is such a bad thing.

how reasonable is it to transistion into physics for
grad school?

>>11609999
nice get

>>11603060
Does any of you know if there is some publicly available large dataset of various mathematical and science related constants? With an option of downloading some subset of data in some form suited for exploration with R, Python even VBA, ideally csv?

>>11609952
Everybody does! Did you do anything interesting today?

>>11610110
what is large? how many do you expect?

The standard /sci/ library has a submodule
>scipy.constants

E.g. the physics ones are
>scipy.constants.physical_constants
https://docs.scipy.org/doc/scipy/reference/constants.html

>>11610187
Such that Bradfords law would be evident. I have some shitty presentation asignment, we can choose whatever topic we like, this one shows up i such a wide range of different kinds of data that it would keep people intrestedto choose seemingly unrelated examples.

>>11610169
I was doing my functional analysis homework.
I got a bit stuck on the following subproblem:
Consider the set
[math]\{ \frac{|a|^2 - |b|^2 + 2i Im(a\bar{b}r)}{|a|^2 + |b|^2 + 2 Re (a\bar{b}r)} : a, b \in \mathbb{C} \} [/math], where -1 < r < 1 is a real parameter. Show that this is an ellipse with foci -1 and 1, and compute lengths of both axis.
i'll try again tomorrow :)

>>11607748
kill yourself

>>11610269
so my math level is basic derivatives and integrals (had basic calc in my economics major). Can someone explain how is this kind of thing even approached? i don't know set theory just to let you know.

>>11610269
You may have already tried this, but I would probably try the following: Given any point on the ellipse, the sum of the distances between this point and the foci is constant. I'd try checking if that happens if one chooses -1 and 1 as the foci. Once that has been proved, the lengths of the axes are quite easy to find.

Is there any interesting math /physics where these are used?
[eqn]e^{\frac{d}{dx}}[/eqn]
[eqn]e^{-\grad^2}[/eqn]
What are the eigenfunctions?

>>11610236
i meant Bendford law. shit

>>11610317
Actually, Benford.
fuck

>>11610317
>>11610320
go back you disgusting adhd newfag retard

>>11610331
no u

>>11610317
Yeah I mean try it out with the scipy package, it won't get much more comfortable than that

>>11610310
>Is there any interesting math /physics where these are used?
They are central in math, so yes they have some applications also in physics.

Note that
[math] f(x+d) = {\mathrm e}^{ t \frac{d}{dx} } f(x) [/math]
is just another way of writing the Taylor series.

And, always for the right Banach spaces,
[math] g(t) := {\mathrm e}^{ t A } f [/math]
solves the differential+ equation
[math] \frac{d}{dt} g(t) := Ag(t) [/math]
with [math] g(0)=f [/math]

>What are the eigenfunctions?
If h(x) has a series expansion an Av=a·v, then you can well hope for h(A)v to be h(a)·v.
As for your example, the Laplace will be better behaved than it's square root.

For something more formal, here's some inputs, follow the links
https://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups

d = t

Recommend me a book on functional analysis /sci/

Background: linear algebra, fourier theory, calculus

>>11610269
Are you sure this is all info there is on a and b? I tried, but got stuck, and now that I think about it, we can divide by 0 and also get both foci on the ellipse, neither of which are OK. For example, let a=1 and b=0. Then the real and imaginary parts of the product are 0 and we get 1/1 = 1, or just blow everything up by choosing a=0=b. Without imposing any restrictions, that seems pretty hopeless, or maybe I am the hopeless one. Let's say that there is relative hopelessness between that problem and me.

>>11610339
Kinda worked,

>>11609426
Chain rule

>>11610339
>Note that
>f(x+d)=etddxf(x)f(x+d)=etddxf(x)
>is just another way of writing the Taylor series.
very nice, butters. very nice.

>>11610392
Sorry, I worded my previous post poorly.
I should have said "a, b are not both zero", apart from this - no restrictions.
I also meant "ellipse with filled-in insides", just like "disc" is a "circle with filled-in insides".

The original problem is more or less:
A linear map [math]A : \mathbb{C}^2 \rightarrow \mathbb{C}^2[/math] has eigenvalues [math]-1, 1[/math] and corresponding eigenvectors [math]x, y[/math], we assume that these vectors are normalized i.e. [math]||x|| = ||y|| = 1[/math]. Let [math]r = |\langle x, y \rangle| [/math].
Prove that the set
[math]\{ \langle Av, v \rangle : v \in \mathbb{C}^2, ||v||=1 \} [/math]
is an ellipse with loci -1, 1 and the lengths of axes can be expressed in terms of r.

>>11606689
It ensures the existence of a sheaf of analytic E-infinity rings, allowing for direct interplay between homotopy theory and local number stuff.

>>11609593
>since A∩BA∩B and A−(A∩B)A−(A∩B) are disjoint, we can use
Alright thanks, I was just missing this part as I wrote this on text instead of mathematical logic

>>11609616
Yeah that was more in-depth than im thinking when solving these problems, I understood the solution thanks.
For De Morgan I also got it and was basically turning drawing to text, for some reason i thought there would be some other steps I was missing

>>11603284
Category theory 100%. Quote it's results when you need it and even know some of its basic theory so you can study algebraic geometry ect but absolutely do not study it for it's own sake. Every mathematician has a horror story of someone who did.

>>11610545
Okaydokay, now it makes a lot more sense! Now you should try to show that if you take any point on the boundary of that set, the sum of the distances from that point to 1 and -1 is constant.

>>11610568
A good way to get started with things like de Morgan or any other case where you are supposed to show that two sets are the same is to simply check that they contain one another. If we can show that [math]x \in A[/math] implies [math]x\in B[/math], then [math]A \subseteq B[/math]. If we also have [math]x \in B \implies x\in A[/math], then [math]B \subset A[/math], and so [math]A = B[/math]. If you have energy and time, you can try [math](A \cup B)^c = A^c \cap B^c[/math] like that. Did you make progress today?

>>11610545
>deforms the problem into illegibility before posting it
What the fuck is wrong with you.

>>11610616
For today I have redone ex1 (* exercise), changed text to mathematical logic in the m(a union b) exercise, and now going to quit trying to write the formula for calculating m(A1 union A2 ... union An), I know how what it is I just can't write the sommatory of the second part, the one that in m(a union b union c) would give the - (A interssect B) - (A interssect C) - (B interssect C);
So uh kind of did progress in that sense, now going to try some more of these exercises then reread all the definitions / lemmas in the mappings section and attempt to do all the exercises with * on them because theyre useful in the future, but I think even just those are too much

Has anyone made a blog about math? What did you post?

>>11610669
Protip: you can think of the general finite union as pairs. More precisely, [math](\cdots ((A_1\cup A_2) \cup A_3) \cup \cdots) \cup A_n[/math]. You know what happens to a pair, so you should try applying that here. Good luck!

>>11610699
I tried, made a few homotopy posts and deleted the whole thing because depression hit and I started hating my own text.

Good night, /mg/.

first for algebraic geometry and first for alg toplogy? looking for an intro text at the graduate level

>>11610709
There's a lot to be said about this.
Basically, you must accept that nothing you put out will be flawless but you get better the more it becomes a routine. Writing, or everything, really.
>>11610699
I've been shitposting my stuff here whenever it made sense. I think formulating things out for a broader audience always helps.

Why do I always, ALWAYS, forget how to do logarithmic equations, no matter how many fucking times I go over them?

What am I doing wrong? Engineering Math 1 nigga here

>>11610768
when you forget how to do them, do you go look it up or do you figure it out?

>>11610771
Usually, look up how to do it again.

>>11610768
You just gotta memorize those logs man, there's no other way.

>>11610777
try the other way next time
if your first reaction to not remembering something is "I'll just go look it up" your brain gets comfortable with the idea of "why should I hold onto this? I'll just look up the recipe when I need it"
if the next few times you forget you take 5 or 10 minutes and actually talk yourself through what's going on you'll understand it and won't even need to remember anymore.

whats the deal with math? how do i understand it better? this is a serious question, im just being vague on purpose

>>11610814
>im just being vague on purpose
why would you do such a thing

>>11610819
to get a widen the possibilities of an answer; not influence the answerer

Where can I learn more about infinite dimensional vector spaces?

>>11609889
haha that would be strange thanks though haha

>>11610709
Yeah that was also said on the previous exercise but didnt think of applying it to this problem, video uploaded I finished the exercises (that seemed interesting) about sets, reread the mappings stuff and did most of the easy problems, tomorrow Im thinking of doing more mediums and maybe hards and perhaps read some of the next topic A(s)
Gn

cant seem to even get started on this proof, assuming only Rudin chapter 1.

for x>0, p,q rationals, if p > q then x^p > x^q.

>>11611266
homework goes in /sqt/ >>11601289

>>11610711
this is what i used for algebraic geometry in a uni course:

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by Cox

>>11611266
hint: if [math]a, b, c, d[/math] are integers (with [math]b, d \neq 0[/math], then [math]\sqrt[b]{x^a} < \sqrt[d]{x^c}[/math]

>>11611278
Ah, damnit! that's exactly what I started out with and then got stuck somewhere.

Just made this.
Any opinions?

>>11611328
Version which says "Question".

>>11611335
You'd murder the thread if you actually tried to gatekeep that hard. Most of this thread is undergrads and the grad students are all specialized in their own shit that one other poster at max cares about.
Keeping out boneheaded sophomore homework problems is enough.

>>11611335
>>11611328
What about people self-studying math?

>>11611442
they just make book lists and spam anime pictures

>>11611278
goddamnit, theres no solutions anywhere to Rudin second edition. How do I prove [math]\sqrt{n}[x] \sqrt{n}[y] = \sqrt{x}[xy][/math]? I assume I prove the inequality both ways, but is there a good hint to get me started?

>>11611469
Why are you reading the second edition of Rudin?

>>11605627
Very nice script man. Really enjoyed that

>>11611481
It's the only one I have.

>>11611486
3ed is literally free. You don't even have to pirate it
https://notendur.hi.is/vae11/%C3%9Eekking/principles_of_mathematical_analysis_walter_rudin.pdf

>>11611489
I want a physical copy. 2e has completely different exercises than 3e though, much harder.

>>11611266
did you mean x>1? 0.9^1 is less than 0.9^0.1
>x^p/x^q = x^(p-q) = x^n where n>0
>if x>1, x^n > 1 so x^p > x^q
>we know x^n > 1 because, assuming rationality, the integer divisor d is the root power, meaning exactly d copies of some real must be multiplied into x^numerator. if x>1, x^numerator where numerator is integer is obviously > 1. in order to multiply into a real >=1, you must have reals >=1, or else multiplying it will effectively divide it further and further because decimals are fractions are division

dunno if the axioms required that last bit for your book but i thought it was worth it

>>11611495
harder exercises don't seem to be doing particularly well for you considering you have to keep asking for solutions to get through the first problem set

is every possible sequence of digits found in irrationals, or are there some sequences that could be barred. say, might we never find 108408109820495 in pi?

>>11611469
you havent told us what n is supposed to be? how long are you taking on these problems? if youre stuck you need to stick with them for as long as it takes, even if that means 3 hours.

>>11611506
This is an open problem for pi, and pretty much every other irrational.
It's currently basically impossible to decide whether a number has that property or not unless you explicitly build a very artificial number where it's obvious whether it does (like 0.1234567891011121314151617...)

Good morning [math]/ \mathfrak{mg} /[/math]!

>>11610730
That is true, but sometimes I just want to wipe the world clean all traces of my existence. My supervisor thinks I write well, my parents say my English is beautiful, yet in my eyes that whole blog looked like it was written by a blindfolded and drunk baboon with microcephaly using its toes.

>>11610768
Prove the identities like how products go to sums, how the base change works etc. That way I learned them.

>>11610814
>whats the deal with math?
It's a deal with the devil. Interesting and fun stuff, yet it makes you obsess on it and can really crush your self esteem.
>how do i understand it better?
By doing it and trying to think about what is actually going on. Do things here and there, notice similarities, try to find out why the similarities occur, and so on.

>>11606732
Nice meme anon

>>11611147
Oops I missed this post. I checked your video, and your ideas are correct. Keep up the good work! You have a nice voice, btw.

>>11609836
this is an interesting question and I would like to know also

Any math chads that can help. I’m trying to find theta2 and theta3 obviously there is 2 solutions. How do you approach a problem like this? It’s for a program I am writing.

>>11611729
well 2 1 or 0 solutions depending on x and y

>>11603060
>we can put the positive integers into one-to-one correspondence with the perfect numbers
>we conclude that the set of positive integers is not larger than the set of perfect numbers
Wtf? Doesnt this seem to contradict intuition? It is difficult to imagine there being the same amount of perfect numbers as there are positive integers. Do we say that any two sets that are countable infinite are the same size iff there is a bijection between them? Or am I interpreting this wrong?

>>11611871
Sorry. I meant perfect squares, not perfect numbers.

>>11611871
Countable infinity means that there is a bijection between the given set and the set of natural numbers. Hence there can be only one countable infinite.

>>11603060
Is it possible to explain Stirling numbers of the first kind in layman's terms? (Stirling numbers of the second kind are 'Santa Claus is putting n gifts to k bags...')

>>11611055
A book on functional analysis.

>>11611611
>yet it makes you obsess on it and can really crush your self esteem.
You're projecting. Math doesn't automatically lead to insecurity.
Don't swalling with glee in your psychological issues.

>>11612093
Maybe I am. I really don't know. Maybe I have normalised such small problems of mine so thoroughly that I can't see them as anything but the default state. I just know that many people, for example the guy I share my office with, thinks so too, and he's a happy chap. He seems so stable that you actually have [math]\Sigma(\text{him}) \simeq \text{him}[/math]. For me it's the thrill of being able to do or learn stuff and then free fall into the depths of despair after a long enough struggle with no signs of progress. That happens to everyone, but I feel it as if somebody kicked me in the face with a metal tipped boot (don't wear those in the winter, btw!). Now that I wrote that, I think I see how much I am actually projecting. I guess I should actually thank you for pointing out that, so thanks anon!

>>11612142
You're welcome.
What is this "progress" you talk about anyway? One can't learn the bulk of existing math anyway, so learning a bit of it is the same as learning two times a bit of it. Just enjoy the thing you do learn.
Who do you think you're competing with? Nobody's going to be an Ruler or Riemann at this point - do you think being a Terrence Tao will make you happy? Let me talk you, learning a handstand or how to cook a wok will make you more happy than hustling in academia. That's my approach, anyway. I achieved a 1 second Hamdstand after two weeks doing ten a day. Shit's cash.

>>11611343
That really is just how gatekeeping be, anon. You gotta tell the seniors to fuck off too if you want to keep the sophomores away.
To be entirely honest I'm more interested in shitposting about gatekeeping than anything.
>>11611442
In my magnificence, I have decided to elaborate a more complex system for use by undergrads and non grad students. BEHOLD.

>>11612229
>What is this "progress" you talk about anyway? One can't learn the bulk of existing math anyway, so learning a bit of it is the same as learning two times a bit of it. Just enjoy the thing you do learn.
I was struggling with a lot of very basic stuff for a few weeks, and that really hit me hard. I felt like a total pseud. I made very stupid mistakes everywhere and it just kept getting worse as I started losing sleep etc. Instead of feeling like I was learning anything, I felt like somebody was scooping parts of my brain out with a spoon. Now that I managed to do what I had to do, it feels good to know I learned a lot from that struggle, but it really felt like I had taken someone else's funding at that point.
>Who do you think you're competing with?
My ideal self, I guess. I know I'm actually pretty good, but I just sometimes forget that.
>I achieved a 1 second Hamdstand after two weeks doing ten a day. Shit's cash.
Nice! Now keep practicing until you get 5 seconds!

what would you rather have: an extra 30 years of your prime form or extra 30 IQ points?

>>11604933
This. It is so nauseating to witness pretentious undergrads talk about analysis or topology. Everything is so intuitive and "natural".
What is it about math based on continuity that encourages such odious posturing?
>>11606621
>Stillwell
Based. All his books are wonderful.

>>11612401

>What is it about math based on continuity that encourages such odious posturing?

kids just do that, hopefully they will grow up

>>11609812
7/10 dangerous and misleading

>>11612049
The usual "real-world" example for the Stirling numbers of the first kind (god I hate how unwiedly that name is) is
>You have N rainbow beads in all different colours, and k pieces of string. How many bracelets can you make? (an empty piece of string is not a bracelet you cheap bastard)

>>11612661
>How many bracelets can you make?
*how many ways can you make your bracelets, if that wasn't obvious.

>>11612356
30 years of my prime
I'm no gigabrain, but I'm pretty sure I'm already smart enough to make it, and getting old really fucking sucks. Go ask any old person how much they'd like to be 20 again.

>>11612401
>What is it about math based on continuity that encourages such odious posturing?
It has nothing to do with continuity in particular. Look how bad undergrads get when they find out about categories too early.
It's just the occasional dipshit who doesn't grasp that reading a book is not the same thing as writing one.

tfw you get into a good grad school, but their strength is completely different from what you did in undergrad.

Looks like I'm switching from combinatorics to non linear analysis. I always enjoyed Analysis, but I have never taken a course in ODE and PDE. My background has mostly been Discrete Mathematics (TCS, Graph Theory, discrete optimization, and enumerative combinatorics).

What books should I read in order to gain a background in non linear analysis and optimization theory?

>>11612714
never taken a course in (ODE or PDE)*

>>11612714
why would you even apply to a department with a totally different focus than what you're interested in?

>>11612753
They had a few people who do combinatorics, but then I realized later that it wasn't their strength. I want to try something new, anyway. Also, the big nibba combinatorics schools rejected me so I guess its God telling me to try something else.

>>11612714
How does grad school work? Im thinking of switching from engi bsc to maths masters in my school but lost on how would changing schools for phd work, or why would I (assuming I get in)

>>11612762
what are the big nibba comb schools?

>>11612229
stop giving "her" attention

>>11612843
1.) Rutgers
2.) UCSD
3.) Emory
4.) Ga-Tech

>>11612857
>Emory
oh shid my comb prof is a visiting prof there.
no wonder I liked his class so much.

So I've done: Rings, Fields, Groups, Galois theory, and Algebraic Geometry and Commutative algebra. This fall I'm taking Topology and possibly Topics in Topology after that (typically algebraic topology). What are the intersections of what I've done and what I'm going to do, particularly with Galois theory? I'd like to go to grad school for algebra.

>>11613002
If you do algebraic topology, you will end up learning the basics of some (co)homology theory, most likely singular or cellular. Once you have seen how that stuff works out, there is for example https://en.wikipedia.org/wiki/Galois_cohomology if you liked what you saw. Commutative algebra in general can be linked with homological algebra where you approach different rings via their modules using the tricks the kind of which you will learn in your AT class. If you liked groups, then there is group cohomology, too. Since that can be formulated using classifying spaces of groups, it will add a bit of homotopy theory in the mix.

>>11603284
Elliptic curves

>>11612661
Is bracelet ABCD the same as bracelet DCBA?

>>11608135
That's not accurate at all. Students at top schools are surprisingly varied, but say we exclude applied math students and combinatoricists (who won't know almost anything on this chart). Then, most incoming grad students know all the things listed as sophomore and everything listed and more in their own subfield. (The bar has gotten a lot higher as a generation of students who took charts like this seriously in high school move through the system.) Strong students often come in with a working knowledge of at least half of the stuff listed in fifth year. And if you restrict to the top 1 student per year you could honestly add a sixth year to that chart.
>no mathematician alive
Can you really tell me with a straight face that Kontsevich doesn't know every single thing on that list?

IMHO if you find yourself saying "nobody knows that much" you're wrong, but it shouldn't have mattered in the first place. If you do math for any reason other than your own enjoyment you'll burn out.

More to the point though, I think that this list is actually a fairly reasonable linear order of mathematical subjects to learn.

who's the most acomplished self-learner you know?

>>11613051
Thanks for the reply!

>>11613159
go away misha

>>11613543
Who's misha?

Maths is not real.

>>11613521
You're welcome. You are going to do group cohomology and be one of my coauthors one day, are you not? I have already Kaguya-anon on my list. Join my list and be one of the best!

>>11613552
Verbitsky, the book list man.

>>11613159
How would I go about learning that? You suggest following the meme guide?

What branch of mathematics is dynamical systems?

>>11613583
yeah, pretty much follow it.
I think it's subject selection is reasonable (less so for the books it picks) though personally I'd defer things like set theory until later and on 'as needed' basis. For example I would recommend Artin's algebra as a place to start learning algebra. The reason I recommend Artin is that he tends to restrict to the case of finite groups much less and his exercises push students to use things which don't come from dry definitions.

If you have a more specific question about where to read about things I can offer my own experiences.

>>11613574
Gotcha. Over the years I've defended the reasonability of that list several times, and I've probably typed out and then deleted my main misgiving a couple times which is that it's simple too Russian in it's approach. It's funny to learn this is actually correct.

>>11613611
>I think it's subject selection is reasonable
I'm assuming that you're some sort of Kahler geometer.
Isn't the analysis on the curriculum pretty weak, or am I just autistic?

>>11613598
Very roughly analysis, more precisely dynamical systems.

>>11613634
Yeah, it's definitely not aimed at analysts. I do geometry (though not often the non-algebraic Kahler part). That's fine though because as I understand there's less need for analysts to have read a huge amount. People will expect you to know large chunks of EGA/SGA but I think knowing Hormander cover to cover is much less common. The list is also weak on arithmetic topics and rep theory.
...at this point I might as well just write my own list with a disclaimer that it's intended for people who want to be "algebraic geometers broadly defined"

>>11613611
>Gotcha. Over the years I've defended the reasonability of that list several times, and I've probably typed out and then deleted my main misgiving a couple times which is that it's simple too Russian in it's approach. It's funny to learn this is actually correct.
I can't find the book list now, sorry, but it is his.

Reminder to everyone that Kähler is not Kayler but the "äh" is like a long version of the first e in "electricity".

>>11613657
No, I mean, weak compared to what you'd expect from a Kahler/symplectic geometer. More elliptic operators, more PDEs, more harmonic functions, heat kernels and the like.
Heat kernels might actually be in there somewhere and I haven't seen them.

>>11613688
Also holomorphic function spaces and more complex analysis on several variables.

>>11613568
neither is sex and yet here we are

PLEASE stop posting this book list, it makes me feel I've did nothing for the past 5 years.

>>11613886
Don't worry, anon. I've seen your info barrages many times. You're pretty good!

>>11613913
I wish I knew half the stuff you do, [math]\mathfrak{desu}[/math].

>>11613611
Uh sorry no questions, i dont have a maths background and my masters will have topology/algebra geometry analysis and something more, even if phd I would still get a set of classes from each of those 'disciplines', i was interested in geometry tho

>>11611699
Had a lot of course work due tomorrow, think Ill just do some medium+hard problems on mappings and call it a day, going to start after chilling for a bit, and thanks but my voice still feels weird on recordings