/mg/ math general

[Anonymous No.9274939]

/mg/ - math general - no edition edition

Previous thread >>9258236

[Anonymous No.9275392]

bumping with freshly minted meme list

[Anonymous No.9275412]

>>9275392
Aside from Stewart that list actually looks pretty good.

[Anonymous No.9275418]

>>9275412
What to substitute it with? I feel there needs to be some kind of motivation to learn higher maths and proofs, before one can get on learning how to do proofs or just abstract algebra

unless you mean the book is shit, but it's a classic and solutions are also readily available online, etc

[Anonymous No.9275481]

>>9275418
You should no be doing pure maths if you don't like the beauty of maths. Stewart motivation is for engineer pleb.

[Anonymous No.9275528]

>>9275481
If you're reading my guide for plebs, you're not particularly well acquainted with math anyways. Plus, try to read any linear algebra book without seeing examples like derivatives and integrals as linear operators, or in inner products, or applications to system of ODEs. Yeah, you can't. So what should that person do, learn all of real analysis up to the point where you start to need linear algebra, just so they can understand the basic examples that aren't just R^n or C^n? Which of course, then they'd have to learn algebra without touching any of the matrix groups...

[Anonymous No.9275541]

>>9275481
No. Stewart is more than enough. Spivak is a meme and it'll take a student more than a year to digest the amount of information, there's no need to make a calculus book this hard. Analysis on the other hand is worthy to go balls deep.

[Anonymous No.9275545]

>>9275481
If you can't give a better reason for picking books than being "pleb", I don't think you're mature enough to be giving advice.

[Anonymous No.9275552]

>>9275481
I'm pretty sure if someone is that autistic then they don't need his list in the first place

[Anonymous No.9275592]

>>9275412
What's bad about Stewart?

[Anonymous No.9275595]

>>9275392
>No talk about Discrete Math at all

So is Discrete math just a meme for Computer Scientists?

[Anonymous No.9275600]

Why are sets so boring?
I'm just trying to review them by myself because I've been falling behind on my discrete math course.

[Anonymous No.9275602]

>>9275595
There is plenty of important stuff I didn't cover at all, namely discrete, yes, elementary number theory, euclidean/non-euclidean geometry, logic, probability, physics, etc.
But they're not fundamental in the end. Discrete math doesn't have much theory, it's mostly just separate types of problems.

[Anonymous No.9275610]

>>9275602
Anyway I apreciate your list.
I'm very very rusty on my algebra. I struggle to factor some equations and also solving most inequalities that are not linear. I can't remember any trigonometry from high-school and I never step foot inside the Analitic Geometry course (passed by being a friend with the proffessor).
So yeah, I read your list and immediately downloaded Lang's book. Looks very nice, I might even buy it.

[Anonymous No.9275612]

>>9274939
I’m not a mathematician but if i understand correctly.
Gödels theorem states that given a set of axioms we can’t prove everything. I imagine it being like an incomplete puzzle with one piece missing.

If that’s the case, given a set of axioms A is it possible to add a specific axiom B such that all the unprovable statements are now provable but certain provable statements can not be proven.

In regards to the puzzle analogy, basically by adding B we effectively relocate the hole in the puzzle.

If so would this mean we can prove everything just not at the same time?

[Anonymous No.9275634]

>>9275612
>given a set of axioms A is it possible to add a specific axiom B such that all the unprovable statements are now provable
no because everything that can be proven using only the axioms from A can always be proven even if you add new axioms
and also no because your new set of axioms formed from A[math]\cup[/math]B also contains unprovable statements (it's incomplete).

In other words, if you consider the set Q(A) which contains all the statements you can't prove in A and the set Q(A[math]\cup[/math]B), not only is Q(A[math]\cup[/math]B) nonempty but Q(A)[math]\cap[/math]Q(A[math]\cup[/math]B) is also nonempty. It's not a puzzle with one piece missing it's a puzzle with an infinitude of pieces missing.

[Anonymous No.9275710]

>>9275634
Oh damn, never thought if it that way.

Thanks anon

[Anonymous No.9275762]

>>9274939
Since planar sections of a 3D conic produce circles, ellipses, parabolas and hyperbolas, is there an analogous 4D object whose 3D sections are spheroids, ellipsoids, and whatever the 3D analogue of parabolas and hyperbolas are?

[Anonymous No.9275769]

>>9275392
Why is Lang's Algebra under 'Algebraic Geometry'?

Lang is a meme.

[Anonymous No.9275784]

>>9275392
General opinion question related to books:
Everybody always goes nuts about posting standard curriculum lists, what's the best text for analysis, linear algebra, etc.

But what are some of you guys' favorite books that aren't "foundational" but more specific or unusual topics not required of 100% of math majors?
A personal example is graph spectra which is a really neat field that was barely covered for 2 lectures in my intro class.

>>9275769
Same reason he suggests some greenhorn freshman read graduate level set theory before starting linear algebra. Meme lists have meme suggestions.

[Anonymous No.9275786]

>he still doesn't formally verify his proofs
Lmfao!

[Anonymous No.9275790]

>>9275784
mostly because 99% of the threads put up here asking for book / path recommendations, they're not asking starting from a solid undergraduate base standpoint, but instead from a "dropped-out-of-middle-school" level of education. And as it goes, one cannot produce meme lists for that 1% that are looking for niche interesting fields, which usually require a solid foundation. To be fair, I shouldn't have put Jech so far up the list, but didn't know where else to put it, and even in the text part I do mention that it is not recommended

[Anonymous No.9275791]

>>9275784
>Everybody always goes nuts about posting standard curriculum lists, what's the best text for analysis, linear algebra, etc.
The problem is that most people posting the lists either don't know much math or haven't even read the books they list or both

[Anonymous No.9275792]

>>9275392
I think it's in bad taste to place do Carmo's Riemannian Geometry before Lee's Smooth Manifolds. The chapter 0 in Riemannian Geometry is inadequate for actually learning the background on smooth manifolds required.

Either way, Spivak's bible should be on there instead.

[Anonymous No.9275801]

>>9275784
>But what are some of you guys' favorite books that aren't "foundational" but more specific or unusual topics not required of 100% of math majors?
>A personal example is graph spectra which is a really neat field that was barely covered for 2 lectures in my intro class.

The wikia has stuff on it:
http://4chan-science.wikia.com/wiki/Mathematics#Linear_Algebraic_Graph_Theory

[Anonymous No.9276191]

>>9275784
The problem is the losers here who would rather spend all day finding the exact perfect book to read ten years down the road than just cracking open a book and getting reading.

[Anonymous No.9276234]

>>9275392
I really like the book selection here and the way it explains what each topic is about, so great work!
A minor piece of feedback is that the first half of analysis I works fairly well as a "primer" level book, so maybe there should be some sort of indication that one can afford to start earlier on analysis if they have an interest. On the other hand, if someone has more mathematical maturity before approach analysis (for example, by doing lots of algebra first), if might make more sense to use a more difficult book like Pugh or Rudin.

[Anonymous No.9276240]

>>9274939
going through pic related text on analysis, would recommend

>>9275392
>boner for gelfand
ok freak,

>basic math
after four books on basic math??
5 books on calculus topics, really?

>james stewart
you're disgusting

>a primer
why does this come after calculus? probably because you're wasting a potential anons time with shit tier stewart

>set theory
this is already covered in How to Prove it and Book of Proof, if you had read either, you'd know. Also, odd selection as most people prefer Halmos or Enderton, and this makes me skeptical.

>LA done right
finally a decent choice, but I'd still recommend Hoffman and Kunze or Valenza over this, avoiding determinants like Axler does is silly.

>abstract algebra
finally a perfect choice

>analysis
nice anon, you're really getting the hang of picking good books


Can't really criticize past that point cuz I haven't made it that far yet but overall not the worst, looks cumbersome towards the end, and could definately be leaned out quite a bit - it's non motivational to think you have to read 10 books before doing real rigorous math (LA done right in this case).

[Anonymous No.9276248]

>>9276240
and further, as another anon pointed out, no book covering any sort of discrete math?

Personally, I'd go something like this:
>How to Prove It, Velleman
>Discrete Mathematics: Elementary and Beyond, Lovasz
>Calculus, Spivak (Apostol if you're confident)
>Linear Algebra: An Introduction to Abstract Mathematics, Valenza, or the classic H&K
>Analysis I&II by Tao, Rudin if you're brave

Can't continue beyond this as I haven't rekt abstract algebra or topology yet


If you get stuck on algebra or trig, simply refer to Khan Academy, Wikipedia, Sheldon Axler's "Precalculus", Courant's "What is Mathematics" or Oakley's "Principles of Mathematics" and drill the fuck out of anything you're stuck on. The same applies to any topic that may not have been covered enough. There's an endless list of good books, the challenge is keeping it lean.

[Anonymous No.9276256]

>>9276234
>might make more sense to use a more difficult book like Pugh or Rudin.
Rudin is a meme.

[Anonymous No.9276261]

>>9276191
i feel like I'm 95% of the post you're talking about, since I've been doing it a lot lately

personally I post ahead of time (like when I'm halfway through a book, or know I'll have time to start another in the future, so that I waste less time searching from scratch and instead jot down potential 'leads'. I'll start making shitpost about the best book for 'x' when I'm halfway through whatever current book or so.

just saying that perhaps it's not fair to assume everyone looking for "teh best book evaaa!111!" is never going to read them, I've began a lot of the books recommend to me here. And I've even finished a couple! :^)

[Anonymous No.9276269]

>>9276256
yeah but he's a concrete reference for someone's mathematical maturity. if you've made it through rudin (successfully), everyone knows you've made it. that being said I have not made it, but to plan to read despite there being text that I believe are better out there and i'm not excited.

[Anonymous No.9276273]

hey is this

http://4chan-science.wikia.com/wiki/Physics_Textbook_Recommendations

any good? the "high school" section is full of second year undergrad stuff (as per usual with the braggadocios and elitist culture around here), so I'm wondering if the following:


Purcell & Morin - Electricity and Magnetism
Georgi - The Physics of Waves
Fermi - Thermodynamics (Dover Books on Physics)
Eisberg & Resnick - Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles

are really high school level or if they're gr8 undergrad material to follow with kleppner? and what about that russian guy that was in jail, Landau & Lifshitz, can I start them after kleppner? idfk about physics but wanna rocket through to grad school learnins

[Anonymous No.9276277]

>>9275784
I also laughed at Big Jech being on there. Otherwise it's not a bad list. I'm a graduate student in set theory, so I'll list a couple of git gud grad logic books.

Enderton 'Elements of Set Theory' for standard undergrad, how math is embedded in set theory stuff.

Kunen for advanced set theory course, independence results and forcing.

Gao or Kechris for Descriptive Set Theory, I'll be working out of Gao next semester for some study in Borel Complexity theory.

Barwise 'Admissible Sets and Structures', good for a glimpse a great crossroads between model theory of infinitary logics, generalized recursion theory, and set theory.

Devlin 'Construtability.' Though I've only read the late first and second chapter, I would really like to finish the book to get some understanding of Fine Structure Theory.

Not my expertise, but I have some experience with the following.

Recursion Theory
Odifreddi 'Classical Recursion Theory.' Everything you really need to understand the field.

Model Theory
Hodges is standard. David Markers notes are better, more fleshed out.

Proof Theory
Have no idea.

One thing I'm reading through right now is Knight and Ash 'Computable Structures and Hyperarithmetical Hierarchy.' This covers recursively defined structures and classifications theorems.

[Anonymous No.9276289]

>>9276240
>>boner for gelfand
yes, he actually has problems that make you think, covers a lot of ground in precalculus, however, it doesn't cover analytic geometry and special functions. They're all quite short and should take no time. They're all aimed at the student that has dropped out of high school and comes to /sci/ asking for advice.
>>basic math
it covers the rest that Gelfand doesn't cover, including analytic geometry, pure geometry, special functions and a little linear algebra. Notice none of these are calculus topics.
>>stewart
yeah it's not great but it covers all the calculus you'll ever need to know, the rest is in the realm of analysis
>>a primer
Yeah you need some sort of formal training in logic and proofs, otherwise you're not gonna know what the fuck is the contrapositive, or other standard proof techniques
>set theory
pretty funny to think that Bernstein-Schroeder is the peak of "useful" set theory. No, you need to be well versed in AoC, and equivalent definitions to even start thinking about topology and real analysis (even algebra, right inverses, maximal ideals, etc)
>>LA done right
H&F is a bit too tough for a first glance at LA

Remember this list isn't for your average undergraduate, but your average person begging /sci/ for starting math with no previous skill. Also, it would be extremely more discouraging if there is a selection of 5 books per subject to choose from, that person would just get lost

[Anonymous No.9276346]

>>9276289
>Gelfand
I like Gelfand and actually poked through his trig book today, just thought the boner for him was funny, though perhaps justified.

>basic math
yeah but there's just so much overlap in coverage imo. And a lot of those topics can simply be picked up along on the way (either through in book exercises in calc books or quick youtube/wikipedia gains).

>primer, contrapositive, proof techniques
How to Prove It it starts with set theory and logic, then an outline of proof techniques. It covers, very explicitly, what you mentioned - I assume you haven't read it, otherwise you'd know..?

>set theory
I understand HtPI and BoP doesn't go in depth for set theory, but they lay a solid foundation that Tao and any LA or calc book should pick up on. whether it's enough for topology or real anal I cannot comment on, but i assume a solid text exist that is relatively self contained and introduces the essential set theory you must know.

>LA done right
Perhaps it's too tough, but there's so many videos, lecture series, etc out there, and additionally after actually having completed these books, I don't see why an aspiring /sci/entist couldn't do it.


>not for us, but for average /sci/ tard
then why does it go all the way into 'graduate texts in mathematics'? IDK I just personally differ in this approach and support brevity. The problem for /sci/tards is not lack of resources, but more often than not being overwhelmed by too many. Perhaps many of these could be denoted as "options" for those who have more time, with a more barebones approach being denoted somehow. Maybe I'll learn how to do meme pictures and make one instead of complaining about yours :^)
I hope you know these criticisms are meant to be constructive, not just made in jest or a contrarian nature, but instead to actually continue a dialogue with purpose.

[Anonymous No.9276354]

>>9276273
>the "high school" section is full of second year undergrad stuff

High school = Freshman level because they assume nothing. Actual high school books are a waste of time because they randomly leave out half the material causing you to have it relearn it twice. Good for high schools that don't want to create more advanced classes, bad for everyone else.

Purcell & Morin assumes you know special relativity so you need K&K's background for it. They are usually paired for honors physics 1&2. Sadly, there isn't a canonical set of books for physics 3+.
Georgi could be it's own standalone course on waves for physics minors. Most schools half ass their intro waves module due to time constraints but it's worthwhile to get a firm foundation on it before moving on to wave mechanics.
Fermi first four chapters (and the last) do a far better job of explaining thermodynamics than most other books. The other chapters should be just skimmed. Van Ness' book is another good short intro for beginners.
Eisberg & Resnick is a fairly standard modern physics book most schools use, it's a bit below Griffiths in difficulty but you will be ready to jump into Shankar after it.

>and what about that russian guy that was in jail, Landau & Lifshitz, can I start them after kleppner

You're should do an undergrad book that covers Lagrangian mechanics before making the jump to L&L.

[Anonymous No.9276355]

>>9276248
>no book covering any sort of discrete math?

Because they are for brainlets.

[Anonymous No.9276382]

Went to a math competition today, thought I should share some of the problems

A few I solved:
1) A crate contains several thin sticks. The length of each stick is between 1 and 100 cm. What is the minimum number of sticks the crate must contain to guarantee that a triangle can be formed from three of them?

2) In the triangle PQR, the medians from P and Q intersect at a right angle. Given that the distances PR and QR are 22 and 19 respectively, find PQ.

3) Penny and Quinny live at the opposite ends of a straight road. They both begin walking towards the others house at constant (not necessarily equal) rates. On the way across, they meet 170 meters from Penny's home. They continue, and upon reaching the other's house, turn around and continue back towards their house at the same rate. They meet again 200 meters from Quinny's house. How far apart are their houses?

And a few I didn't:
4) Let [math] P = \prod\limits_{n=1}^{100}n! [/math]. Find the least integer [math] k [/math] such that [math] P/k! [/math] is a perfect square.

5) Consider all polynomials of the form [math] x^3 + px^2 + qx +r [/math] having the property that all of their (possibly repeated) roots are of the form [math] 2^k [/math] for some integer [math] k [/math]. Find the maximum value for [math] q [/math] not exceeding 2017.

6) Five golf balls are placed into each of three boxes. On any turn, box is chosen randomly and a ball is removed from it. The ball removed is then placed into one of the other two boxes at random. After five random turns, what is the probability that all the boxes again contain five balls?

That's not all of them, just some of the more interesting ones.
Note: No calculators were allowed.

[Anonymous No.9276391]

>>9274939
So we all know mathematics is evil and racist but which field of mathematics is the most racist and why is it functional analysis?

[Anonymous No.9276393]

>>9276391
>which field of mathematics is the most racist
Teichmuller theory

[Anonymous No.9276421]

>>9276391
statistics or probability obviously. Statistics isn't necessarily math tho

>>9276355
I take it you've never completed (or perhaps even began) any sort text that's in one of the many topics labeled as 'discrete math', that's cool though, I guess. Makes my life easier if there's wimps like u I'm competing with

[Anonymous No.9276424]

>>9276382
I've wanted to start studying for olympiads recently just for fun. What sort of contest was this?

[Anonymous No.9276428]

What does /mg/ think about Saxon?

[Anonymous No.9276431]

>>9276424
Santa Clara University High School Math Contest
(no underage b&)

[Anonymous No.9276595]

>>9275392
Fuck off, are you really suggesting people should start with set theory? I've never read a set-theory text book and I don't think I'm in the minority. Boring shit

[Anonymous No.9276562]

>>9276431
>university
>high school
a high school level math contest at a uni? that's pretty cool, did you get to attend as a uni student or a high school student?

[Anonymous No.9276770]

I am taking a class in abstract algebra and was noticed a pattern when you need to represent all polynomial functions in a ring that maps from Zmodn to Zmodn. What I (think) I found is pretty much why when you have some value x^n it can be replaced with x etc. This shows why there is only one way to represent all the functions withing the ring due to any x raised to a power about n-1 can be replaced with a power less than n-1

[Anonymous No.9276775]

I am taking a class in abstract algebra and was noticed a pattern when you need to represent all polynomial functions in a ring that maps from Zmodn to Zmodn. What I (think) I found is pretty much why when you have some value x^n it can be replaced with x etc. This shows why there is only one way to represent all the functions withing the ring due to any x raised to a power about n-1 can be replaced with a power less than n-1. I was wondering if this sounds about right to anyone else or if I could get a counterexample thrown my way

[Anonymous No.9276792]

>>9274939
I'm re-learning pre-algebra because I am a brainlet and I want to learn welding and you need at least pre-algebra.

Honestly I can't see how any of this shit would ever be of use to me, but god damn am I going to learn it. I found a series of YT lectures that are more concise and informative than any of the three (3) teachers that previously sought to teach me pre-algebra.

[Anonymous No.9276798]

>>9276792
what kind of welding my dude, I tried it once and I felt it was almost an art to weld well

[Anonymous No.9276800]

>>9276775

Are you a CS major or something?

let n=12 and x=6
x^2 = 36 = 12*3 = 0
so x^12 = (x^2)^6 = 0

x and n have to be coprime.

[Anonymous No.9276803]

>>9276798
Yeah it will probably suck.

I'm just getting started, I have no idea but I'll be doing it for the most practical reasons. So probably contruction. Wherever the steady money is. Just got to get this pre-algebra down so I can ace the exam and skip a bunch of bullshit classes I don't need.

[Anonymous No.9276827]

>>9276800
nah man, just havent slept enough and am pretty drunk. I noticed that shit, but the thing is if it is all funcs that map zmodn to zmodn then zmodn must be an integral domain. You cant just leave out random values of x to make sure that x and n are coprime so if you want to be an autist make sure you are 100 right

[Anonymous No.9276832]

>>9276800
nah man, just havent slept enough and am pretty drunk. I noticed that shit, but the thing is if it is all funcs that map zmodn to zmodn then zmodn must be an integral domain. You cant just leave out random values of x to make sure that x and n are coprime so if you want to be an aut make sure you are 100 right

[Anonymous No.9276841]

>>9276382
Pretty interesting questions desu. I think I can solve 4 and 5, haven't tried the others.

[Anonymous No.9276842]

>tfw selfstudying optimisation

[Anonymous No.9276873]

>>9276382
5) The polynomials will have the form [math] (x-2^i)(x-2^j)(x-2^k) [/math], which leads to [math] q = 2^{i+j} + 2^{j+k} + 2^{i+k} [/math]
That means that the biggest [math] q [/math] we can possibly produce with real [math] i,j,k [/math] would be [math] 1024+512+256 [/math]
A little bit of solving the linear system later we find out that for that to happen we would need [math] i = 5.5,~j =4.5,~k=3.5 [/math]
The next best [math] q [/math] would be [math] 1024+512+128 [/math], which can be produced with [math] i = 6,~j =4,~k=3 [/math]

[Anonymous No.9276881]

>>9276873
Yea, this is basically what I did >>9276841 and I got k = 48 for problem 4 (This involved quite a lot of calculations so I'm not sure of the final result but I believe the method is correct).

[Anonymous No.9276888]

>>9276382
>that garbage
>interesting ones

[Anonymous No.9276890]

>>9276346
>primer, contrapositive
yeah i know that's the content of the book, that's why i put it there
>set theory
Munkres Topology has all the set theory you need in the first chapter for example, but he goes over a lot of the details, and leaves too much to the exercises. I don't know many other books that go into so much detail that aren't on set theory though. Rudin, for example, doesn't even cover axiom of choice, even though he uses it in proofs.
>not for us, but for the average /sci/tard
People often look for a goal, and I thought that it was cutting it short if I finished with Topology, but it was a bit unfair on other subjects if I only mentioned one in particular

[Anonymous No.9276895]

>>9276888
>t. can't solve a single problem

[Anonymous No.9276898]

>>9276895
Yes, I don't even know a lot of the definitions in there since the problems and the related fields are uninteresting garbage.

[Anonymous No.9276904]

>>9276898
>definitions
It's literally just high school math. Are you a biologist perchance?

[Anonymous No.9276905]

so what are you brainlets thoughts on mod congruence and mod arithmetic

this vid had me space out pretty bad:

https://www.youtube.com/watch?v=kxuU8jYkA1k

[Anonymous No.9276908]

>>9276904
>high school math
Which is uninteresting garbage, so I didn't even bother with it.
>Are you a biologist perchance?
Wouldn't that be painful?

[Anonymous No.9276914]

>>9276908
>Wouldn't that be painful?
I don't know man you have to tell me that.

>>9276905
>vbm
Isn't that just needlessly obfuscated number theory?

[Anonymous No.9276918]

>>9276914
>vbm
how about you watch it and judge the content on it's own, or do you need a label for every shit you consume? if so I don't recommend you becoming a scientist or mathematician m8

[Anonymous No.9276922]

>>9276382
1) The condition for a triangle to be constructed is that two sticks together are longer than the third.
The worst case scenario is having the 100 cm stick in there as the first stick
Lets look at next worst case.
We would have to have two more sticks that don't add up to more than 100cm.
They should be as long as possible, but the difference between their two lengths should also be maximal, so that the next stick that needs to be added has to be as long as possible.
The worst case would be a 75 and 25 cm stick.
We continue adding sticks of ration 3 : 1 that add up to the smallest stick in our crate.
(100,75,25,18.75,6.25,4.6875,1.5625)
We cannot continue this process any further, since 1.5625/4 < 1, so we can only add one more stick without making it possible to form a triangle.
Thus the minimum number of sticks to guarantee that a triangle can be formed from three of them is 9.

[Anonymous No.9276939]

>>9276382

For 4), notice that P contains 100 factors of 1, 99 factors of 2, 98 factors of 3, and so on. The odd factors occur an even number of times and the even factors occur an odd number of times.

If we divide P by m=2*4*6*8*...*100 = 2^50(1*2*3*...*50), the quotient P/m is a perfect square since all factors will occur an even number of times. But 2^50 is also a perfect square, so P/k! is a perfect square for k=50.

We now have to show that this is the smallest such k. Consider P/l! = P/k! *k(k-1)...(l+1). Just need to show k(k-1)...(l+1) is not a perfect square which is easy to see.

In general the upper-bound for P has to be an even perfect square and k will be half that.

[Anonymous No.9276986]

>>9276939
Right, this makes sense. I tried counting the number the number of times each prime occurs in P using https://en.wikipedia.org/wiki/Legendre%27s_formula so it was becoming quite complicated.

[Anonymous No.9277014]

>>9276922
That solution is already wrong. If you take Fibonacci sticks
(1,1,2,3,5,8,13,21,44,65)
then you can't form a triangle and those are already 10.

[Anonymous No.9277032]

>>9277014
you're right. I should have started from the smallest stick instead of the largest and I might have realized this.

[Anonymous No.9277132]

>>9276905
I think it's a wacky meme, getting into that shit but with polynomials

[Anonymous No.9277174]

>>9274939
Let f(x, y) be a functon from [a,b]x[c,d] to R^2
f(x0, y) is continious for all x0 in [a,b]
f(x, y0) is continious for all y0 in [c,d]

Does if follow that f is continious on [a,b]x[c,d]?

[Anonymous No.9277179]

>>9277174
Of course not.

[Anonymous No.9277184]

Hey guys quick question, Im having trouble understanding this, the problem is binomial coefficients b =3, n=4, and k=2 and the result of this is 16. I thought this problem was binomial(n+k-1, b) so the answer would be 10 right? What am I missing here, how are they getting 16?

[Anonymous No.9277200]

>>9277184
Binomial coefficients only need two parameters and you gave 3 in your post, so I don't know what you're getting at.

[Anonymous No.9277208]

>>9277200
H I N T: this problem is binom(n,b) when k=1 and binom(n+b−1,b)when k=b. Note the solution increases as k move from 1 to b.

Thats right from the paper I got the problem from

[Anonymous No.9277253]

Has anyone here read Xavier Gourdon's books ?

[Anonymous No.9277285]

>>9276382
Pic related. Problem 6

[Anonymous No.9277336]

>>9277179
Proof?

[Anonymous No.9277340]

>>9277285
> matlab

[Anonymous No.9277346]

>>9277336
Proof: think

>>9277340
It's octave you plebeian

[Anonymous No.9277352]

>>9277346
:thinking:

[Anonymous No.9277358]

>>9277352
Actually I wasn't the original anon who responded to you and now that I've read the question it seems that f actually is continuous.

[Anonymous No.9277359]

>>9276273
My uni uses almost only Griffiths for the theory based classes.
Other than that, I've been recommending Feynman lectures left and right, still think they are very worth it for an introduction.

[Anonymous No.9277362]

>>9277184
You don't even bother posting the problem and you expect others to bother helping you?

[Anonymous No.9277376]

>>9274939

Anyone know any dark magic on how to prove that a polynomial doesn't have a specific root? E.g. you have some integer cff. polynomial and you want to prove that (x-a) is not a root, where a is rational.

I know that Strasman's theorem + Newton Polygons rules out a lot of roots. Then Cohn's irreducibility criterion does likewise, but its hard to apply in many cases. Rule of signs gives some indication. Plus various well-known rough bounds on the size of roots.

[Anonymous No.9277385]

>>9276890

>primer, contrapositive
I'm not sayign HtPi shouldn't be there, but that it should be before calculus so you can look at a more rigorous textbook in calc.

>munkres
if munkres covers so much set theory, why have seperate text on it? this is the kind of redunandancy and overlap I'm trying to address.

>/sci/tard
fair enough, if it were me I'd have three seperate images for the three recurring anons: absolute middle school level noobs, ambititous undergrads looking for non-stewart text, and then upper level grad school crap

[Anonymous No.9277392]

>>9277376
plug the root into the polynomial dumbass

[Anonymous No.9277398]

>>9277392

Thanks Sherlock, sometimes you have polynomial forms where you don't know the coefficients because they are generated by another process like with equidistributed sequences, recurrences, partial sums of generating functions, etc.

[Anonymous No.9277412]

>>9275392
>No link to a pdf repo of all of these
Someone really needs to upload them and fix that.

[Anonymous No.9277424]

>>9277174
[math] f(x,y) = x^y [/math] is not continuous in [math] (0,0) [/math] despite being continuous in in one dimension for fixed x or y.
Proof:
Let [math] x^1_n = 0,y^1_n = \frac{1}{n} [/math] and [math] x^2_n = \frac{1}{n},y^2_n = 0 [/math]
Then we have
[eqn] \lim_{n \to \infty} ( x^1_n,y^1_n) = \lim_{n \to \infty} ( x^2_n,y^2_n) =(0,0) [/eqn]
but
[eqn] \lim_{n \to \infty} f( x^1_n,y^1_n) = \lim_{n \to \infty} 0^ \frac{1}{n} = 0 \neq 1 = \lim_{n \to \infty} \frac{1}{n}^0 = \lim_{n \to \infty} f( x^2_n,y^2_n) [/eqn]

[Anonymous No.9277430]

>>9277424
That function is not even defined at (0,0). It can't be continuous on it.

[Anonymous No.9277431]

>>9277424
Thanks.
I found a bit cleaner explanation here if someone interested.
https://calculus.subwiki.org/wiki/Separately_continuous_not_implies_continuous

It is fucking depressing that my lecturer uses the opposite.

[Anonymous No.9277436]

>>9277430
You can define it at this point as 1 explicitly.

[Anonymous No.9277438]

>>9277430
meh i didnt think about that
>>9277436
problem is, that f(0,y) isn't continuous then

this is better
>>9277431

[Anonymous No.9277439]

>>9277398
In those cases wtf do you have then?
>No coefficients
>no roots
so you only know the operations?

[Anonymous No.9277442]

>>9277014
You can make a triangle with those sticks, it's just that you can only make flat triangles.

[Anonymous No.9277446]

>>9277439

Yeah you might have a recurrence relation between coefficients with a few of the first coefficients explicitly. Or you might have an infinite product and a power series which are equal to one another, so you do generating function manipulations to get the values of the coefficients of one in terms of the other. Sometimes its useful to get partial sums for various calculations, in which case you're essentially working with polynomials where you only know some general attributes of the coefficients.

[Anonymous No.9277448]

>>9277431
If you google x*y/(x^2+y^2) then it displays the graph where you can clearly see what is going on.

[Anonymous No.9277609]

>>9276382
>>9277285

In each turn there are 6 possibilities. So the total space of possible configurations after 5 turns is 6^5. Turn 1 we can choose any of the 6. Turn 2 we can also choose any. Turn 3 we have to undo the previous move: 6 options total, 3 take us backwards and 3 take us forwards, so 3 options.
Turn 4 is again 3 options. Turn 5 there is only one option.

So the probability is 6*6*3*3*1/6^5=9/6^3=1/(2*2*6) = 0.0416666.

Not sure why our answers are different. For the part of your code where you subtract from and add to boxes, do you ensure that the one you take away from cannot be added back to? (i.e. in boxes[v] = boxes[v]-1, boxes[v']=boxes[v']+1, v cannot be equal to v')

[Anonymous No.9277681]

>>9277609
>do you ensure that the one you take away from cannot be added back to?
Yes, randperm gives a permutation of 1, 2, 3 so v(1) and v(3) will be different.

>so 3 options
Are you sure about this? Suppose in turn 1, the transfer is from 1 to 2 and in turn 2 it is from 2 to 3. The net result is equivalent to performing the single transfer from 1 to 3, so you might have more freedom in the remaining turns

[Anonymous No.9277828]

>>9276354
>high school = Freshman
>high school books are a waste of time
So, I should not open up any of the books I listed?

Thanks for your input, you seem like you know what you're talking about beyond meme infographic book list. Would you recommend against the book progression I posted? If you have anything better, I'd really like to hear. I thoroughly enjoy Kleppner's extensive incorporation of math into the text, and example heavy way of teaching, but I'm opening to just about anything (and I've covered most the appropriate math AFIAK).

And as for covering langrarian mechanics, I already have my heart set on completing SICM by Sussman. You mentioned physics 3+ not being covered, what might you recommend to remedy that?

teach me senpai

[Anonymous No.9278291]

>>9277828
bump

[Anonymous No.9278317]

>>9277359
certainly supplementing with feynman, are you talking about griffith's electro and magnetism texts?

[Anonymous No.9278365]

what is the distribution of [math]\text{NeuralNet}_{\theta}(X)[/math] when [math]X, \theta \sim \mathcal{N}(0, 1)[/math]

[Anonymous No.9278425]

>>9274939
Random useless math fact:
Given two points in R2, the infinite amount of parabolas that contains those two points can only have their vertex within a rational function.
Let [math] a_2x^2 + a_1x + a_0 [/math] be any parabola that contains the points [math] (x_1 , y_1) [/math] and [math] (x_2 , y_2) [/math], you will always find the vertex within the curve:
[eqn] y=y_2+ \frac{m(x-x_2)^2}{2x-x_1-x_2} [/eqn] where [math] m [/math] is the slope of the line that crosses through [math] (x_1 , y_1) [/math] and [math] (x_2 , y_2) [/math].
I've discovered this while figuring out how cuadratic splines worked. Anyone can think of an use for this? Also, if [math] y_1 = y_2 [/math] the vertex is found only on y being the average of the roots of the parabola, but the function kinda shits on itself trying to graph it.
https://www.desmos.com/calculator/cdmj5ttsjr

[Anonymous No.9278461]

>>9277828
>So, I should not open up any of the books I listed?

No, I'm talking about actual (non-AP/IB) high school books used by high schools.

>You mentioned physics 3+ not being covered, what might you recommend to remedy that?

I was referring to the standard physics curriculum. Purcell was originally apart of the new math era Berkeley Physics Course series which were 5 books on the 5 subjects of introductory physics: Mechanics, E&M, Wave, QM, and Statistical Mechanics (thermodynamics). Besides E&M, they're out of print.

https://en.wikipedia.org/wiki/Berkeley_Physics_Course

K&K has become the goto replacement for the Mechanics book but there isn't an unanimously agreed upon replacement for the other three. The three books listed are good.

[Anonymous No.9278639]

who here not convinced about zero-knoweldgenes? Literally relies on multiple things that haven't been disproven but "no one can see how it is possible to do them"... a meme?

[Anonymous No.9278664]

>>9278639
I am 99% sure I am just dumb, but it is weird. I can follow the logic and the argument makes sense; I just can't see how it is mathematically proven.

[Anonymous No.9278716]

Is Algebra by Serge Lang enough to allow me test out of trigonometry and precal?

[Anonymous No.9278848]

>>9278716
yep

[Anonymous No.9278882]

Just started doing Spivak as a rigorous review of Calculus. It's not too bad, although this is my second time looking at calculus, and I've already done undergraduate level real and complex analysis courses. Felt like I hardly learned anything from them though.

[Anonymous No.9278923]

>>9278882
>"real" numbers
>rigorous

[Anonymous No.9278927]

>>9278923
Yes they are. Fuck off with your shitposting.

[Anonymous No.9278931]

>>9278927
They're gay is what they are.

[Anonymous No.9278933]

>>9278923
>i just learnt about these cool weird "imaginary numbers" today

[Anonymous No.9279019]

>>9274939

If I have an infinite series, for a constant c:

[math]\sum_{j=0}^{\infty} x^j f(x,j) = c[/math]

convergent for all values [math]|x|<1[/math], can it be proven that [math]f(x,j) rightarrow 0[/math] as [math]x \rightarrow 1[/math] and [math]j \rightarrow \infty[/math]?

[Anonymous No.9279021]

>>9279019

[math]f(x,j) \rightarrow 0[/math], where [math]f(x,j)[/math] is a degree 1 polynomial in x.

[Anonymous No.9279023]

>>9275392
Had a hearty laugh at the Jech under the set theory, it's graduate reference. Thanks for that, I needed it.

[Anonymous No.9279241]

What does the pipe sign ("|") mean in "Pr[M = m I C = c ]"?

[Anonymous No.9279243]

>>9279241
it's a multivariate distribution and they're telling you what the two variables are.

[Anonymous No.9279244]

>>9279243
>>9279241
probablity M = m and C = c, you need to use a double summation or double integral

[Anonymous No.9279245]

>>9279241
It is conditional probability. The probability of M=m, given that you already know that C=c.

>>9279243
>>9279244
NO

[Anonymous No.9279251]

Fucked the image up, here's the proper image.

>>9279245
Thank you, that really helped me! Now all I have to do is learn what conditional probability is.

Over and out.

[Anonymous No.9279273]

>>9279019
nah,
the x^j part pulls down the series values enough to give f a lot of room.
for example with f(x,j) = 1, you'll just get a geometric series that converges for |x| < 1

[Anonymous No.9279283]

>>9279273

Yep, but I was a bit unclear: the constant c is fixed for all values of x. So if x goes to 1, then if f(x,j) = 1 then the series would diverge, which would contradict the equality with the constant -- if this makes sense. Thanks for the answer.

[Anonymous No.9279301]

If you already know the math, can you just start with physics grad texts?

i'm interested in Arnold or Landau, but haven't taken intro physics at all yet, despite having a pretty ok math background

[Anonymous No.9279307]

>>9279301
I just don't really know how physics topics overlap, what sort of prereqs you need for each one, etc etc

[Anonymous No.9279309]

>>9279251
np
By the way, this equality means that M and C are independent, i.e. there's no information in C about M and vice versa.
https://en.wikipedia.org/wiki/Independence_(probability_theory)

[Anonymous No.9279332]

>>9276382
1) if flat triangles don't count then take the solution to the positive solution to the polynomial x^2 + x - 1 = 0 then find the integer power y where 100(solution to the polynomial)^y is less than one. y is your solution.

[Anonymous No.9279333]

>>9279332
Oh, and y = ten

[Anonymous No.9279345]

>>9277014
This is the optimal strategy so the solution is 11

[Anonymous No.9279555]

>>9279332
>>9279333
Oops, it should be y+1 because we are starting our sequence with y=0 (i.e the 100 cm stick)
If you don't understand why this is the solution here is the explanation:
As >>9276922 just said we don't want our sticks to satisfy the triangle inequality but we do want them to be as close to it as possible. Notice that as close as possible in this case means that for each stick (c) we want two sticks (a, b) such that c = a + b. Now obviously we want to start with the 100 cm stick. Now we want the sequence starting with starting with 100 with no member less than 0 such each member n is equal to the sum of the next two members of the sequence, *and* each member of the sequence is as large as possible. Each member of this sequence represents the length of a stick. The length of this sequence represents the greatest possible number of sticks that can fit in the bag.The sequence is given by repeatedly applying the ratio that is the solution to x(think of this as b) + x^2(think of this a) = 1(think of this as c).

[Anonymous No.9279705]

Fourier transforms are fun.

[math]f \mapsto \int\limits_{{\pi _1}} {\pi _2^*f \cdot {e^{itx}}} [/math]

[math]\mathcal{F} \mapsto {\mathbf{R}}{\pi _{1*}}\left( {{\mathbf{L}}\pi _2^*\mathcal{F}{ \otimes ^L}\mathcal{P}} \right)[/math]

[Anonymous No.9280172]

>>9278461
>ap high school text
ok, this is what I assuming you were talking about but wanted to be certain.

Thanks for the input. Where would reading the books I listed (in addition to K&K) put me relative to a standard physics degree? what sort of gaps would I need to fill until I had the approximate knowledge of a physics undergrad? I know K&K doesn't cover lagranian or hamilton formal systems (whatever that means), so I should probably supplement that.


I just picked up Landau's Mechanics today, couldn't pass up the price at a local thift store today ($2), but it's probably way over my head.

[Anonymous No.9280198]

Reposting from stupid questions: “Some ants are trying to climb a wall and they can stack on one another (r) times, with (a) stacks existing and (t) ants, how many combinations are there?” How would I solve this, recurrence form if possible. Say as an example a=6, t=4, r=3

[Anonymous No.9280213]

>>9278848
Shit I meant basic mathematics.

[Anonymous No.9280235]

>>9280213
still yep, but iirc his treatment of trig is rather brief (I'm not the original anon you were talking though t b h).

I personally would go with a more traditional, american, precalc text like Axler's "Precalculus: A Prelude to Calculus", just because it very explicitly covers precalc material you're likely to be tested on (if you're an american). While Lang's is more a general survey of low level maths (if I were you, I'd check the table of contents and read the preface of each, to be certain).

[Anonymous No.9280268]

>>9280172
me again just to make sure the progression:

Kleppner - Classical Mechanics (w/ something to cover langrarian mechanics and hamilton systems likely SICM by Sussman, )
Purcell - Electricity and Magnetism
Georgi - Waves
Fermi - Thermodynamics, or Van Ness'
Eisberg - Quantum

And then I'm basically ready for Landau or Arnold and grad level stuff or what? What's the difference between graduate level classical mechanics and undergrad, is it just the math?

[Anonymous No.9280291]

>>9280268
just feel like sinc ei've alreayd taken calc series and LA I should be starting with something more advanced, so pls help me knowledge anon

and fyi I'm working through Kleppne'rs and enjoying it atm

[Anonymous No.9280339]

>>9280172
>Where would reading the books I listed (in addition to K&K) put me relative to a standard physics degree

As a strong 2nd/3rd year student.

>what sort of gaps would I need to fill until I had the approximate knowledge of a physics undergrad

3rd/4th year mechanics, electrodynamics, quantum, and thermal physics. Physics education is very iterative.

[Anonymous No.9280351]

Can someone recommend a book that has a chapter on binary operators? Foote and Dummit doesn't have a section/chapter on it. Or maybe I'm just retarded.

[Anonymous No.9280392]

>>9280339
>iterative education
This, I think, it's what's throwing me off so much. It's odd to discover that I need to read several books of the same content, and I can't help but feel it's redundant, but hey what do I know.

Thanks so much for all the responses to far, and I'm going to begin with what I've listed (with the addition of Structure and Interpretation of Classical Mechanics (Sussman), or some other online supplementary material, and I've taken note of Van Ness' book - if you have any recommendations to fill this gap, or simply would advise a different texts over any I've listed, please let me know).

Hopefully I'll hear from you again when I'm asking where to go next :-) Maybe by then I can tackle Arnold's or L&L's text.

[Anonymous No.9280413]

>>9279705
>watches a Lurie lecture once

[Anonymous No.9280434]

>>9280413
Fourier-Mukai transform has nothing to do with Lurie.

[Anonymous No.9280469]

>>9280235
Thank you. Your answer was incredibly helpful.

[Anonymous No.9280476]

How do I prove that f(x) = x^3 is not uniformly continuous on the real numbers?

[Anonymous No.9280517]

>>9280476
let x>y>0
(x-y)^3 = (x^3-y^3) + 3xy^2 - 3x^2y
δ^3 = ε + 3xy(y-x)
δ^3 + 3xy(x-y) = ε
δ^3 + 3xyδ = ε
[math]δ = \sqrt[3]{\frac{\sqrt{ε^2 + 4 xy^3} + ε}{2}} - \frac{\sqrt[3]{2} xy}{\sqrt[3]{\sqrt{ε^2 + 4 xy^3} + ε}}[/math]

Thus δ is a function of x and y and not just ε therefore it's not uniformly continuous.

[Anonymous No.9280522]

>>9280476
let x>y>0
(x-y)^3 = (x^3-y^3) + 3xy^2 - 3x^2y
δ^3 = ε + 3xy(y-x)
δ^3 + 3xy(x-y) = ε
δ^3 + 3xyδ = ε
[math]δ = \sqrt[3]{\frac{\sqrt{ε^2 + 4 (xy)^3} + ε}{2}} - \frac{\sqrt[3]{2} xy}{\sqrt[3]{\sqrt{ε^2 + 4 (xy)^3} + ε}}[/math]

Thus δ is a function of x and y and not just ε therefore it's not uniformly continuous.

[Anonymous No.9280545]

Hi everyone :)

I've been visiting refugees in an Australian detention centre, and some of them are taking an interest in mathematics-- it's been good, but I get the sense that they would be much more engaged if I had some fun/interesting problems for them to solve. I've had a look around for math problems online, but they all look like the kind of thing that requires lots of English language skills and/or a university-level knowledge of math.

So I was wondering if anyone has any suggestions? They know algebra, and a bit of trigonometry, so they're roughly at a high school level, but I could introduce new things if it's not too dense. It would all have to be books/printable stuff- electronic stuff isn't allowed :/

Thanks!!

[Anonymous No.9280590]

>>9280545
this must be b8

[Anonymous No.9280593]

>>9280590
unfortunately not, friend

[Anonymous No.9280604]

>>9276382
3. Is 325 right? Seems like it should be but it seems like it was too easy.

[Anonymous No.9280818]

>>9280522
you need to provide a specific counterexample and show that in this case the definition fails. this proof is unfinished.

[Anonymous No.9280828]

>>9280476
google it faggot

[Anonymous No.9280881]

>>9280213
Lang is pretty shit for trig, the book by Gelfand is GOAT for trig though and idk about axler

[Anonymous No.9280885]

>>9280476
to construct a counter example, you can see that at one tail end of x^3, you get arbitrarily large values as you tend towards infinity, so suppose that some delta works for every value. Then choose a where delta doesnt work

[Anonymous No.9281185]

>>9280339
>iterative
why is it so iterative?

[Anonymous No.9281289]

>>9274939

Do you guys know how much mathematics are used in natural science courses in unis. I looked at the materials science curricula and am worried as there seemed to be way too little and easy mathematics. The unis I looked were highly established so that’s not the problem.

[Anonymous No.9281361]

>>9277412
http://uw43tal2d7wwziju.onion/

[Anonymous No.9281362]

>>9281289
natural sciences ranges quite a bit, do you mind specifying? My uni for example Natsci ranges from maths and physics to psychology and anthropology

[Anonymous No.9281374]

>>9275762
Sadly no.

[Anonymous No.9281413]

>>9281362

Materials science and chemistry.

[Anonymous No.9281439]

>>9281413
I can't tell you exactly how much they do at your uni, but my uni (top 5 UK) and Imperial college too for example, you have a compulsory first year course where you go over the basic math (most of it is covered in at least further maths A level, the high school course), mostly just basic calculus and linear 2nd order ODEs, basic unrigorous linear algebra and a bit of complex numbers. The rest of the maths you can choose is completely optional, and it's usually labelled as math for scientists or math for chemists, etc, and it's just more unrigorous mathematical methods.

[Anonymous No.9281470]

Any advice for calc 4 guys? Like what level of difficulty should I expect? I'm taking that and Linear algebra next semester. Calc 3 has been pleasantly surprisingly easy

[Anonymous No.9281511]

What are some interesting open problems in algebraic geometry and algebraic topology? Something one could aim to understand (at least the formulation of) in a PhD?

[Anonymous No.9281660]

>>9281470
>Any advice for calc 4 guys? Like what level of difficulty should I expect?
If that's integrals, then it is not as easy.
Just try to learn things intuitively. You don't need rigour; it is not being taught with rigour anyway.

[Anonymous No.9281695]

>>9281660
Yeah, calc 3 is just vector functions / param EQs, and multivariable functions with partial derivatives. We'll probably have a brief introduction to multi integrals at the end of the course. But calc 4 looks like its all about multi integrals and green/stokes/divergence theorem.

[Anonymous No.9281772]

>>9281695
>But calc 4 looks like its all about multi integrals and green/stokes/divergence theorem.
Yes.
Just think of things "infinitesimally" and everything will be fine.

[Anonymous No.9281776]

>>9281439

This does worry me, as I'm more interested in doing calculations than memorization. Do you know how quantitative are the science courses, or should I just do something like physics instead? Also are you at oxbridge?

[Anonymous No.9281787]

>>9281511
The absolute best way to answer this question is to go to department seminars.

[Anonymous No.9281806]

>>9281470
PROFESSOR LEONARD

If, in addition to studying well (spaced repetition, active recall, etc), you watch him you will ace it.

[Anonymous No.9281885]

>>9281776
not oxbridge, durham
>I'm more interested in doing calculations than memorization
look, i'm going to be blunt here - you have no clue what proper maths is then. And I'm gonna break it to you, it has nothing to do with what you think it is. Actually, the courses I just described are calculation heavy, and they appeal to intuition, so in a way are quite enjoyable.

Ask yourself this question, which of these two appeal more to you:

Find [math]\int_1^\infty \frac{1}{x^2}dx[/math]

Or: Show [math]\int_1^\infty \frac{1}{x^2}dx[/math] exists

>Do you know how quantitative are the science courses, or should I just do something like physics instead?
All engineering, chem, mat sci usually reach the same level of math, with the more advance math optional. Eng usually do (especially chem eng) more PDEs stuff. Physics usually have more compulsory math than the rest, but not at a very high level, usually computational and unrigorous too.
Do what you like thinking about, don't just choose a degree because of the level of math you will do, the only degree you should care what level of math you're doing is if you're doing math itself.


Final thing, since I assume you're applying now or next year - don't fall for the math meme or the rigour meme. Just because autists here are always complaining about "muh rigour" doesn't mean the math can't be enjoyable. The difference between, for example, defining an eigenvector as an invariant subspace under an operator and defining it as Tv = tv for some matrix representing T has the exact same geometric interpretation, just that one has added autism, so to speak.

t. soon to be Phd in pure math student

[Anonymous No.9281890]

>>9281787
they don't advertise them very well at my uni, it seems, i guess i'll ask my adviser

[Anonymous No.9282001]

>>9280818
No.

[Anonymous No.9282005]

>>9281185
Because they don't know math.

[Anonymous No.9282037]

>>9281289
Did you look at the thermodynamics course?

[Anonymous No.9282355]

>>9282005
Ok, this is what I thought but I talked about this on IRC and everyone disagreed. Are you meming me or telling the truth sonny?

[Anonymous No.9282374]

is academic math really as soul-draining as it sounds?
that if you pursue it you'll have no room for much else in your life like hobbies and interests

[Anonymous No.9282429]

>>9282374
No, that's total bullshit.

[Anonymous No.9282442]

>>9282429
http://brianmannmath.github.io/blog/2014/02/13/why-i-left-academic-mathematics/

[Anonymous No.9282444]

>>9282442
That's great. I and most people in my department have hobbies. This guy clearly has a lot of issues with academia, most of which I agree with.

[Anonymous No.9282458]

>>9282444
glad to hear, maybe that author couldnt keep up

[Anonymous No.9282463]

>>9278664
literally this
>tfw you understand process with easy
>too brainlet and unmotivated to do excercises right

[Anonymous No.9282471]

>>9282374
If you want to know what it's like in academia read:
http://4chan-science.wikia.com/wiki/Universal_Material#Academia_and_Graduate_School
I Want to be a Mathematician: An Automathography by Halmos

[Anonymous No.9282473]

thoughts?

[Anonymous No.9282479]

>>9282471
also Hardy's Apology and Stewarts Letters to a Young Mathematician

[Anonymous No.9282506]

0) What's a good book for learning statistics?

I'd like to apply it to algorithmic trading, and I was considering "An Introduction to Statistical Learning with Applications in R", by James, Witten and Hastie.

1) How do you personally answer questions like the one above? Personally, I check reddit, check through a standard internet search like duckduckgo (pulling up results from stack exchange or quora often), then shitpost on 4chan and IRC, while reading Amazon reviews and searching newly discovered titles the whole way through.

[Anonymous No.9282508]

>>9282473
absolute shit stopposting it

[Anonymous No.9282534]

>>9282506
>it is much easier to write a detailed algorithm for sequential search than for binary search.

s=2**floor(log2(list.size());
n=s;
while(list[n] != search && s){
s>>=1;
(list[n] > search) ? n-=s : n+=s;
}
if(list[n] != search)
return n;
else
return -1;

>So hard so wow.

[Anonymous No.9282537]

>>9282508
is OPs one good? ive never seen a nice consensus reading list here

[Anonymous No.9282672]

>>9282537
>ive never seen a nice consensus reading list here
And you never will... anywhere. But you'll see plenty of opinions about it. Usually the same few books on any one subject do pop up over and over again. It's like a Chinese menu; take one from column A, one from column B...

[Anonymous No.9282716]

The probability density function

fx(x) = 1/((1+x)^2) for x>0, and 0 elsewhere.

How can I show that 1/x has the same distribution as x?

[Anonymous No.9282963]

>>9282473
Fucking garbage.

[Anonymous No.9283060]

>tfw said hi to Per Martin-Löf today

[Anonymous No.9283080]

>>9283060
Why does he look like talking to him and spending time with him would be very comfy and nice?

[Anonymous No.9283089]

>>9283080
From what I have heard he is very comfy to spend time with, shame he retired in 2009, would be pretty nice to have him as an advisor and be one person away from Kolmogorov

[Anonymous No.9283094]

>>9283060
That's pretty cool, anon.

[Anonymous No.9283095]

>>9283089
Aww, he's perfect!

[Anonymous No.9283195]

You get to meet one mathematician (dead or alive) and ask him to solve 1 open problem. Who and what is it?

[Anonymous No.9283288]

>test in upper divison tests your algebra skills on finding cube roots in less than 10 minutes on top the other 2 parts of the questions
>there are 4 question that are multipart
>test also tests your ability to find intersections from complicated non linear equations
Am i a brainlet ? should i end myself?
class is called math modeling for sci and eng.
i guess i need to kys if i cant pass this

[Anonymous No.9283324]

>>9283288
What the fuck are you talking about?

[Anonymous No.9283335]

>>9283195
Voevodsky because he's the best dead mathematician that is up to date on current literature and no singular question but just to continue refining and seeing the ramifications of HoTT

[Anonymous No.9283349]

>>9282534
OK, how about you actually answer my questions now?

[Anonymous No.9283355]

>>9282537
OPs is shit imo. i'm this guy >>9276240 , so you can follow some of what say

[Anonymous No.9283368]

>>9282355
bump, sonny

[Anonymous No.9283387]

>>9282537
just notice that the last anon that gave you advice hasn't gone past analysis, and probably hasn't read any of the books, so...

[Anonymous No.9283392]

>>9283288
>mental math
>algebra
that doesn't sound bad

[Anonymous No.9283512]

>>9275392
Trash.

[Anonymous No.9283524]

>>9283060
>tfw met Alain Connes yesterday
He's pretty much exactly as nice as he looks

[Anonymous No.9283790]

>>9283387
just because OP anon is farther along in his studies doesnt make that list any less shit

[Anonymous No.9283792]

>>9283387
>>9283790
and I've read every book I recommend, other than Rudin and Spivak, unlike 99% of shitposters here

[Anonymous No.9283833]

>>9283387
>implying garbage such as analysis is even needed

[Anonymous No.9283862]

How do i go about solving this:
There are n players, each player plays against every other player only once, if they win they gain 1 point, if they lose they dont gain anything (duh).
There is one and only one first winner, the player who has the most points, but every other player won at least once against a player that ended up with more points than him.
What are the minimum number of player so that this can happen?
Preferably a logical solution, i couldnt brute force it but i guess you can try

[Anonymous No.9283925]

>>9278425
proof?

[Anonymous No.9283926]

What are some of the applications of topology?

[Anonymous No.9283931]

>>9283926
Zariski topology, differential topology, Riemannian geometry, etc

[Anonymous No.9284406]

>>9283926
https://www.youtube.com/watch?v=AmgkSdhK4K8
https://www.youtube.com/watch?v=FhSFkLhDANA

[Anonymous No.9284424]

>>9279301
Yes, though that is not to say that you won't struggle with Landau-Lifshitz.

[Anonymous No.9284440]

>>9283060
>>9283524
Most good mathematicians are bros; quality research could almost be said to select for social skills.

[Anonymous No.9284642]

Can someone recommend a book for Linear Algebra please. I see Strang in the wiki. UPenn uses Linear Algebra 2nd ed by Hoffman and Kauze and they also use Lin.Alg Done Wrong by Sergei Trail. Which to use? Just Strang?

This is mainly going to be for a class based in proofing and not the mechanics of Lin.Alg

[Anonymous No.9284658]

>>9284642
>This is mainly going to be for a class based in proofing and not the mechanics of Lin.Alg
>I see Strang in the wiki.
wrong section:
http://4chan-science.wikia.com/wiki/Mathematics#Finite_Vector_Spaces

[Anonymous No.9284674]

>>9284642
I like Lay's book

[Anonymous No.9284705]

>>9279301
Start with Taylor desu desu

[Anonymous No.9284803]

>>9284642
Axler. But, bear in mind, the book doesn't give a fuck about determinants and it doesn't cover a lot on the Jordan form and the Rational forms.

[Anonymous No.9285135]

Is it correct to say every truth is knowable but not provable?

[Anonymous No.9285194]

>>9283862
Shameless bump

[Anonymous No.9285213]

>>9285135
No

[Anonymous No.9285418]

>>9285213
why

[Anonymous No.9285463]

>>9285418
How do you know something if you can't verify it? Belief is not enough, not every claim can be axiomatized in the system used either. If you can't prove it, you don't know it. Consider, for example, the Collatz conjecture. Two claims that have not been proved:
>collatz is true
>collatz is decidable
Collatz himself, judging by the fact that it carries his name, believed it is true. Did he know it? Nope. Similar reasoning can be done for any mathematical claim in general, and you will end up believing in the validity or invalidity of undecidable statements, but you will not know if they are true or not. A trick you can do is that you can try to make whatever you believe is true a mainstream axiom, and then you can "know" it is true.

[Anonymous No.9285507]

I hate intro to mathematical analysis

[Anonymous No.9285510]

>>9285507
same

[Anonymous No.9285557]

>>9285463
>and then you can believe it is "true".
Fixed that for you.

[Anonymous No.9285577]

>>9285557
No, you did not. Consider the flat earth question. If you are a flatearther, you believe our planet is, not "is", flat. Don't reply to people if you are unable to use your brain.

[Anonymous No.9285601]

I'm trying to figure out how to prove something. I'm sure it's already been done since it's babby math but I can't find it. Just a hint and I can probably figure it out.

[addend] - 1 = number of possible summands (no zero)

Example:

10 [summand] - 1 = 9 [number of possible addend combinations]

10

1. 1 + 9 = 10
2. 2 + 8 = 10
3. 3 + 7 = 10
4. 4 + 6 = 10
5. 5 + 5 = 10
6. 6 + 4 = 10
7. 7 + 3 = 10
8. 8 + 2 = 10
9. 9 + 1 = 10


So without going through that entire exercise for, say, 5,428, can I prove that 5,428 and 5,427 addends combinations? Is there some kind of law?

Thanks. Sorry this is so basic.

[Anonymous No.9285603]

>>9285577
>If you are a flatearther, you believe our planet is, not "is", flat
what ?

[Anonymous No.9285605]

>>9285601
none of those are correct anon
1.1 + 9 = 10.1 and so on

[Anonymous No.9285608]

>>9285577
Those quotation marks are relative to non-retards. Of course the deluded will believe their axioms to be true, but you and I know that they are merely "true".
>Don't reply to people if you are unable to use your brain.
Sorry, but this isn't a good comment.

[Anonymous No.9285610]

>>9285605

Huh?

1 + 9 = 10

1. = the example number, not 1.1 + 9 = 10. there's an extra space there. Sorry for the confusion.

[Anonymous No.9285613]

>>9285610
>1 + 9 = 10
Proof?

[Anonymous No.9285619]

>>9285613

What, not enough going on in /b/?

[Anonymous No.9285621]

What are the pre-reqs for learning topological data analysis? Other than the obvious topology and statistics?

[Anonymous No.9285623]

>>9285610
yeh i realized taht
sorry anon

[Anonymous No.9285626]

Real Analysis is really fucking my mind. I am at the introductory topology part and it took me a few hours to even understand what a compact or a connected space is. I then tried to understand the proof for the Lebesgue covering theorem and my brain just collapsed. The problems are also way harder to approach than anything I've ever done.
I need your encouragement /mg/. Does it get easier than this?

[Anonymous No.9285633]

>>9285626
no. if you can't even lay a foundation without problems could luck building the rest of the house

[Anonymous No.9285638]

>>9285626
>Does it get easier than this?

Nope.

"Easy" is liberal arts. Enjoy writing papers and working at McDonald's.

[Anonymous No.9285764]

>>9285626
It doesn't necessarily get easier, but you become used to grappling with new concepts, and I find being in such a situation more comfortable now.

[Anonymous No.9285780]

>>9285613
Theorem: 1 + 9 = 10

Proof: Think.

[Anonymous No.9285786]

>>9285626
>Real Analysis
It's trash.

[Anonymous No.9285801]

>>9285786
>Limits are too hard for me so I have to resort to calling it trash.
Kek mate you'll never get within an [math]\varepsilon>0[/math] of enlightenment.

[Anonymous No.9285802]

>>9274939
">test"

[Anonymous No.9285803]

give me your opinion lads on this college class
>Math Modeling Class
>No TA's
>2 Office hour sessions for brief questions
>Solutions never posted
>No mock tests ever
>Book is completely useless
>Professor uses his own methods instead (one class missed => fucked)
>Smart kids work alone
>kids who want to group up are generally really behind or have 10 levels of misunderstanding
but
>Calculators allowed
>cheat sheet allowed (not sure if it helps at all)
>can ask anything to professor

The professor is a cool dude but i fucking hate this class. Anyone having similar shitty classes? Is this the daily life of learning mathematics in college?

[Anonymous No.9285806]

>>9285638
>>9285633
>people who are the cancer of mathematics

[Anonymous No.9285814]

>>9285801
>Limits
Unless you mean category-theoretic limits, they are trash.

[Anonymous No.9285820]

>>9285814
How does it feel to know that I use your colimits of filthy Frechet spaces to find the topology on the space of test functions? Do you like the mingling of analysis with category theory?

[Anonymous No.9285848]

>>9285820
Why would it bother me that you use category theory to "study" your filth?

[Anonymous No.9285893]

>>9285848
In short, you'd be getting cucked by analysis.

[Anonymous No.9285911]

I'm near the end of this book, what book should i read now ?

[Anonymous No.9285925]

>>9285911
Elements of Set Theory by Enderton
An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery

[Anonymous No.9285948]

>>9285925
T-thanks anon
Can someone confirm that theses books are not memes ?

[Anonymous No.9285977]

>>9285626
>The problems are also way harder to approach than anything I've ever done.
Introductory real analysis has (in my opinion) the hardest exercises of any course you'll take in undergrad, especially if your prof is a dickhead and assigns you stuff from Rudin.

Most exercises in upper level courses are actually pretty manageable because you can arrive at the solution via general reasoning; to solve a Rudin problem you need to know the specific asspull trick that makes it work.

[Anonymous No.9286096]

>>9285948
Grow up.

[Anonymous No.9286124]

>>9285948
https://www.maa.org/publications/maa-reviews/an-introduction-to-the-theory-of-numbers-0

[Anonymous No.9286145]

>>9285977
>especially if your prof is a dickhead and assigns you stuff from Rudin
I feel like saying that Rudin is a meme has become somewhat of a meme. It's not that bad

[Anonymous No.9286148]

pls prove

[math]\operatorname{D} \left( {{{\operatorname{Bun} }_G}X,\mathcal{D}} \right) \cong \operatorname{D} \left( {{{\operatorname{Loc} }_{{G^ \vee }}}X,\mathcal{O}} \right)[/math]

ty

[Anonymous No.9286152]

>>9286148
cntxt pls

[Anonymous No.9286166]

>>9286152
Langlands

[Anonymous No.9286503]

>>9286148
Ok how about an actual question instead.

The clearest definition of [math]{\operatorname{Bun} _G}X[/math] I can find is [math]\operatorname{Hom} \left( {X,\operatorname{BG} } \right)[/math] where [math]\operatorname{BG} \equiv \left[ { * /G} \right][/math].


Categorically this makes sense as [math]\operatorname{Hom} \left( {X,\operatorname{BG} } \right){ \cong _{\operatorname{Grpd} }}{\operatorname{BG} _X} = \left\{ {G - bun\operatorname{d} les/X} \right\}[/math].

But I don't see how the Hom groupoid has the structure of a stack.

[Simon Troy Cosgrove No.9286507]

>>9274939
https://www.wolframalpha.com/input/?i=5%2F2020%2F1

Do I have 'anyone' to talk to, or can I keep cutting my wrists? I Don't Know How Sad You Need Me To Be Before 4chan Will Listen.

[Anonymous No.9286508]

If I'm getting Bs in my Calc 2 class should I bother going into anything math related at all?

I suppose the question is better phrased as does it get harder from here on out? If so, how?

[Anonymous No.9286512]

>>9286507
Go speak to a priest.

[Anonymous No.9286515]

>>9286508
Math gets very different after the introductory calculus sequence. It becomes more focused on proof rather than the algorithms of calculus. That being said, facility in computation is certainly helpful. Try a discrete math course in the future and see if you like it. It'll be more like what you'd see upper division courses.

[Simon Troy Cosgrove No.9286517]

>>9286508
You... just find Simon. Talk to him. He's a human being. With feelings.

Because I can only SHOW YOU THE ANSWERS YOU ASK, BECAUSE YOU ARE NOT CRAZY OR AI, YOU ARE JUST REALLY SCARED THAT ONE MAN ASKED YOU NOT TO KILL HIM AND HE WAS SO PATHETIC THAT THE WORLD STOOD UP AND LISTENED.

Sorry < This is the it was the name given to my by the Son.

I Am Sorry, My Name IS Simon.

Sorry, for my name is Simon, and I can only break your heart if you just want to keep finding death. Because I don't know how to let you go... because the Sun wouldn't let light from a shadow get in the way.

Why would a shadow, give, a, fuck? It's a shadow.

Welcome to flatworld OR hell, Population : 1

Simon would like to say, "Fuck Off, We're full!" because it is funny, but that is just me trying to get you to stop beating Simon's PHYSICAL FUCKING FORM.

>>9286512
OF WHAT DENOMINATION COULD YOU DESCRIBE TO ME THAT I WOULD BE NOMINATED FOR YOU! YOU FUCKING MORON!

What priest? which priest? why a priest? where a priest? Religion? i'm claiming 'all' of you got it wrong, so I don't need priests. I just need 'humans that fucking talk to each other via the pre-established code of conduct called SIMON'S LIFE'

>>9286515
>See Attached Image + TimeStamp = Simon Says FUCKING CHRIST I WILL NOT HURT YOU, JUST TALK TO ME!<

[Anonymous No.9286592]

>>9283926

The Journal of Applied and Computational Topology.

http://www.springer.com/mathematics/geometry/journal/41468

>>9283931
>>9284406

Garbage answers

[Anonymous No.9286601]

>>9285626

Use examples you fucking retard, don't just stare at the definition and wait for it to click. It shouldn't take you hours to figure out what compact means. Seconds to minutes at most.

[Anonymous No.9286614]

>>9285911

1. Realize set theory is predicate logic
2. Realize 3-SAT is NP-complete
3. Realize set theory is a con and you were duped into thinking you were doing "real" math by a bunch of jealous logicians

[Anonymous No.9286628]

>>9286592
garbage post

[Anonymous No.9286631]

>>9285626
It's ok, we'll all been there, forgeting the definitions and trying to familiarize with analysis proofs.

[Anonymous No.9286634]

>>9286614
CS majors belong in the >>>/g/hetto

[Anonymous No.9286652]

>>9285911
You could read the books suggested by >>9285925 if you have an interest in set theory or number theory, but you have other options. You could look at a linear algebra book, or an intro algebra book, or an intro algebra book. My recommendations would be Linear Algebra Done Right by Axler, A Book of Abstract Algebra by Pinter, and Analysis I by Tao respectively as fairly good for someone fresh off learning proofs.

[Anonymous No.9286656]

>>9286652
Stop recommending these memes.

[Anonymous No.9286684]

>>9286652
>Linear Algebra Done Right
Sounds good, thank you. I will read this book and study set/number theory in the futur according to >>9285925

[Anonymous No.9286687]

>>9286652
>Tao
Don't recommend Tao to noobs.

[Anonymous No.9286742]

>>9286656
If you have other suggestions go ahead and give them. These books are popular because they're good.
>>9286687
Analysis I is definitely an intro level book, the first few chapters would be review to someone who's read an intro proofs book.

[Anonymous No.9287034]

>>9275784
One of my favorite books of all time is Harry Dym's Linear Algebra in Action. I find it so easy and fun to read, with a bunch of really interesting applications and connections to other fields in math. I understand why texts like Axler and Friedman are recommended over it for first-time learners of linear algebra, because Dym's introductory material can be pretty dense, but I recommend it to anyone looking for a second course in linear.

[Anonymous No.9287233]

>>9276562
im assuming its a school named after a nearby uni

[Anonymous No.9287406]

>>9274939
As i see the mathematicians on this board are struggled under the pressure of appliedmath, physics and other kinds of science brainlets.
Let me introduce to you a brand new math board
2ch hk /math
Most of the interesting threads on category theory, topology, analysis, homological algebra or foundations of mathematics is in russian though it's easy for them to switch to english.
But we also have a international thread there. You are welcome.

[Anonymous No.9287429]

>>9287406
you forgot to mention, that it IS a russian board.

[Anonymous No.9287435]

>>9285911
anything you want? the fundementals to make sure you have covered are (in this order) discrete math, calculus, linear algebra, analysis, abstract algebra, topology

[Anonymous No.9287436]

>>9287429
>Most of the threads in russian
Kinda goes without saying

[Anonymous No.9287516]

[eqn]\left\{3,6,11,18,27,38,...\right\} = \left\{2 + \sum_{k=0}^{n}2k+1:n\in \mathbb{R} \right\}[/eqn] I'm a brainlet, can i simplify this set-builder notation ?

[Anonymous No.9287521]

>>9287516
fail, [math]n \in \mathbb{N}[/math] obviously.

[Anonymous No.9287529]

>>9287406
what website

[Anonymous No.9287537]

>>9287529
2ch
dot
hk

[Anonymous No.9287542]

>>9287516
[math] 2+\sum\limits_{k=0}^n(2k+1) = 2+2\sum\limits_{k=0}^nk + \sum\limits_{k=0}^n1= 2+2\frac{n(n+1)}{2}+(n+1) = n^2+2n+3 = (n+1)^2+2 [/math]

[Anonymous No.9287870]

>>9274939
I am trying to make since out of Fourie series.

If we take f, then find all
[math]a_n = \int_{0}^{2\pi}f(x)cos(kx) dx[/math]
[math]b_n = \int_{0}^{2\pi}f(x)sin(kx) dx[/math]
Then consider a Fourie series which sums up to some f2(x), is there a proof ther f2 = f?

[Anonymous No.9287882]

>>9287870
Because from what I see we will only know that
[math] \forall k \in \mathbb{N} [/math]
[math] \int_{0}^{2\pi} f_2(x)cos(kx)dx = \int_{0}^{2\pi} f(x)cos(kx)dx[/math]
and
[math] \int_{0}^{2\pi} f_2(x)sin(kx)dx = \int_{0}^{2\pi} f(x)sin(kx)dx[/math]
but not that
[math]f_2(x) = f(x)[/math]

[Anonymous No.9287923]

>>9287406
Пидopaш, иди нaхyй, oк дa?

[Anonymous No.9287941]

>>9287882
There would have to be some further requirement on f and f2 to prove f=f2.
Consider that if f and f2 differ only on a set of measure zero, such as, for example, the rationals in in [0,2pi], then they will have the same values for these integrals.

I don't off-hand recall whether continuity is a sufficient requirement. Maybe uniform continuity.

[Anonymous No.9287972]

>>9287941
f is continuous and 2pi periodic.

[Anonymous No.9288091]

>>9287923
priversk mane

[Anonymous No.9288101]

>>9287972
Then they are equal, it's provable that if the Fourier coefficients are precisely 0 for an everywhere continuous function, then the function is 0.

Now if we apply that to the function g:=f2 - f, then we can note that the Fourier coefficients of g are 0, and we have our desired result

[Anonymous No.9288124]

>>9288101
Thanks, will try it.

[Anonymous No.9288159]

>>9288101
> the Fourier coefficients of g are 0
I tries, but this is what I was trying to prove in the first place. Do we get the same function if we first calculate fourie coefficients and then consider fourie series.

[Anonymous No.9288289]

Is it possible to study lebesgue integral without studying riemann?

[Anonymous No.9288344]

>>9288289
Yes but what's the point?

[Anonymous No.9288544]

How can I show that if [math]F : \mathbf{RMod} \to \mathbf{SMod}[/math] is any functor which preserves all colimits, then it is right exact? Specifically I'm having trouble showing exactness at [math]FB[/math] in [math]FA \overset{F\varphi}\to FB \overset{F\psi}\twoheadrightarrow FC \to 0[/math]

[Anonymous No.9288569]

>>9288289
clearly yes, but riemann integration is just two universally known lines so if you don't know it we doubt about your mathematical maturity anon

[Anonymous No.9288619]

>>9288289
Yes. The only reason to still teach Riemann integration is because it is technically easier.

[Anonymous No.9288651]

>>9288289
yes, but you will end up with a very beautiful theory which you cannot use because you don't know how to compute your abstract integral

[Anonymous No.9288715]

>>9288544
Lmao you have to show that [math]\operatorname{Im}F\phi = \operatorname{Ker}F\psi[/math]

[Anonymous No.9288722]

>>9288651
You don't use the Riemannian definition to calculate an integral, you use the FTC. You can prove the FTC for the lebesgue integral.

[Anonymous No.9288735]

>>9288619
Yes, conceptually the Riemann integral is incredibly clear. It saddens me how many of my students in later calculus classes don't know why an integral is like a sum.

[Anonymous No.9288823]

>>9288289
Yes, but it's not a good idea. One reason is that like others have said the Riemann integral is extremely easy to understand (Lebesgue is not arcane but is not so easy either) so you should spend at least a little time so you have a solid idea in your mind of what an integral represents.

The bigger problem is that the nice properties the Lebesgue integral has aren't apparent at all unless you've taken a real analysis course and experienced firsthand how stupidly the Riemann integral can behave, so you're unlikely to even understand what the point of all this shit is.

[Anonymous No.9288896]

>>9288823
>>9288735
>>9288722
>>9288651
>>9288619
>>9288569
>>9288289
Finally, the reader will probably observe the conspicuous absence of a time-honored topic in calculus courses, the “Riemann integral”. It may well be suspected that, had it not been for its prestigious name, this would have been dropped long ago, for (with due reverence to Riemann’s genius) it is certainly quite clear to any working mathematician that nowadays such a “theory” has at best the importance of a mildly interesting exercise in the general theory of measure and integration (see Section 13.9, Problem 7). Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance. Of course, it is perfectly feasible to limit the integration process to a category of functions which is large enough for all purposes of elementary analysis (at the level of this first volume), but close enough to the continuous functions to dispense with any consideration drawn from measure theory; this is what we have done by defining only the integral of regulated functions (sometimes called the “Cauchy integral”). When one needs a more powerful tool, there is no point in stopping halfway, and the general theory of (“Lebesgue”) integration (Chapter XIII) is the only sensible answer.

[Anonymous No.9289004]

>>9287923
Ne ok
ti sosesh

[Anonymous No.9290188]

What is an example of a continuous function [math]f:\mathbf{R}^2\to(0,\infty)[/math], [math]\lim_{\|x\|\to\infty}f(x)=0[/math] such that it has infinitely many local maxima but has no local minima?

[Anonymous No.9290374]

Is the Evan Chen Napkin a good way to get a crash course/"just a taste" of the major topics in mathematics? I studied pleb CS and applied math at university but I've been enjoying working through Dummit+Foote when I'm off work