/sci/ - Science & Math

>>10494757
Why do you guys keep posting this meme? It's just a self-congratulatory way to confuse children.

Threadly reminder to work with physicists.

>>10494775
Do you mind posting something that it's worthy?

Why do people even still use Leibniz notation? Lagrange notation is just so much clearer.

>>10494775
>it's unreasonable to expect this when advanced knowledge is at its peak availability
get fucked brainlet.

>>10494775
>It's just a self-congratulatory way to confuse children.
What do you mean?

>>10494757
Thanks for making the thread for me, but please don't attach bait to the OP. We all knew the first post would happen.
>>10494784
Because Leibniz is intuitive.
>clearer
That's not quite true when you're working with an autistic mess of subscrips and superscripts, such as in differential geometry. Euler's is consistently better then.

>>10494854
>Leibniz
>intuitive
Oh, you know these letters we've been using to represent numbers? Half of them actually represent functions instead! No, we're not going to tell you which are which, or when we're applying them, or what we're applying them to, why would we? It's all "intuitive".

Any good math books that are nice for someone starting a math ba soon? Maybe some philosophical or motivating ones?

>>10494888
Linear algebra.

>>10494888
Hopf algebras.

https://www.maths.nottingham.ac.uk/plp/pmzibf/files/May2020.html
>This workshop is one of the workshops of a special RIMS year "Expanding Horizons of Inter-universal Teichmüller Theory". The workshop will review fundamental developments in several branches of anabelian geometry, as well as report on recent developments.
>The list of speakers includes major contributors to anabelian geometry, birational anabelian geometry and binational anabelian geometry for varieties over C.

>>10494757
What is a good introduction to affine and projective geometries?

saved by the bell >>10495275

>>10494888
look at the cambridge list for reading before starting their math degree

>>10495179
eduardo casas alvero's projective geometry book is pretty detailed and has a shitload of problems, and there is a problem book in projective geometry whose name I can't remember that assumes knowledge from his book so that helps to reinforce. It assumes knowledge in affine geometry, although I can't say that the knowledge of affine it requires is really that much.

I've never needed to learn affine geometry from a book, but i heard Berger's trilogy is godly

>>10495351
*correction: not a trilogy, i was confusing it with another set of geometry books, it's a two-volume set but still Berger is what i meant

>>10494784
Lagrange is useful for single variable problems and maths.
Leibnitz is better for multiple variables and for Physics, in which you can just play with differentials without a care in the world.

>>10494777
I'm a math/physics goblino and while I wish it was true, sometimes the gap in methodology causes a lot of misunderstandings. There is also the issue of a vastly different approach to the fundamental philosophy of maths.

>>10494784
>>10494854
>one variable
Lagrange > Euler > Leibniz > Newton
>Multi variable
Euler > Leibniz > Lagrange >>>> Newton

>it's another highschoolers argue over notation episode

>>10495413
Notation can be very important in how you attack a problem. Physfags use Leibnitz all the time and treat df/dt as a fraction. Because they deal with well-beheaved objects, you can write df/dt = x implies df = xdt implies f = X(t) + constant most of the time, but the "df = xdt" step is not acceptable in most mathematical fields.

>it's another highschooler thinks he knows what he's talking about episode

>it's a I will greentext instead of actually talking episode

>>10495373
Newton's multivariable notation is pure kino.
>>10495413
>it's an undergrad complains about high schoolers episode

>>10495493>>10495497

look kids, maybe one day you'll grow up and think back on this moment and hate yourself for being so dumb.
but of course right now you think you're at the top of the world, you think you know everything, and better than me. i was also a kid once, i guess
but perhaps it would be helpful for you to know that stewart's calculus or whatever other meme you've been reading is not the be all end all, and certainly what's going on there is not representative of what happens in "most mathematical fields" or what "physfags" do

I found a neat proof that [math]|\mathbb R|\leq |S_{\mathbb N}|[/math], the latter being the set of permutations.

The alternating series [math]\sum_{n\in\mathbb N}\frac{(-1)^n}{n}[/math] is conditionally convergent, so by Riemann rearrangement theorem, there is a way to rearrange the sum to get any real number. The amount of rearrangements are in bijection with the elements of the set of permutations of naturals. Hence there is an injection from the reals to this latter set.

>>10495502
It's litteraly what is used to solve most differential equations in engineering and physics, at least where I'm from.
It is also completly unacceptable in any serious mathematical reasoning - again, from my uni. Because you just can't consider the thing has a fraction.

I know you need to feel superior about yourself but it's a known fact that there is some abuse of notations - which is assumed and accepted - in a few physics textbooks and courses, and that said methods are not used in maths textbooks
Of course it goes deeper than "hurr notations are better" ; Leibnitz notation just "leads" to some techniques that are unrigorous in a way.
Another example is how vectors and matrices are noted in the Dirac formalism ; different notations lead to different methods of reasoning.

>>10495516
"Most" might be an hyperbole here though, now that I think of it. "Some" would be a better choice of words.

How can I, a sub 100 IQ individual learn math?

>>10495516
> said methods are not used in maths textbooks
you have not read math textbooks then. it's obvious that the fundamental thm of calc implies that you can treat it as a fraction, as well as chain rule.
>completly unacceptable in any serious mathematical reasoning
it's acceptable past first/second year
>you just can't consider the thing has a fraction
you can, since it works, and everybody does
>Leibnitz notation just "leads" to some techniques that are unrigorous in a way
only to high schoolers and stupid undergrads, literally no one else
>Another example is how vectors and matrices are noted in the Dirac formalism
wow im impressed u know babby qm
>I know you need to feel superior about yourself
i don't feel superior about myself, i just think you're stupid

>>10495547

>>10495560
Lang is a meme.

>>10495560
Thanks friend

Making an AG from scratch guide, anything you think i should add?

i still prefer this chart

>>10496038
AG better not stand for Algebraic Geometry.

>>10496316
Why not?

>>10496108
>no Rudin
weak

>>10496038
>Making an AG from scratch guide
why
https://rigtriv.wordpress.com/ag-from-the-beginning/
already exists

>>10495562
Why so?

What is a good book on graph theory? I have zero knowledge on the topic, and close to zero knowledge of combinatorics.

>>10494757
What? Why are the numbers in OP's pic eastern Arabic? I didn't know there were ar-apes that browsed /sci/. Interesting.

>>10496468
OP is a subversive jew trying to distract the white man from the maths he truly belongs to, Topos theory and models.
Come home white man.

>>10495514
Very nice

>>10495514
Nice

can someone redpill me on "lifts" in algebraic topology? i'm reading hatcher at the section where he computes the fundamental group of the circle, and he uses lifts. what is the motivation for using lifts?

>>10496711
Two paths on the real line are homotpic (keeping end points fit) if and only if they start and end at the same point. That's because the line is contractible essentially. That's a really useful property, because checking homotopy groups is otherwise very hard.

There is a covering map that wraps the line around the circle, the fibres of which are a discrete set (i.e. the integers in R).
Given a path on the circle and a choice of starting point there is a unique path on the line that starts at that point and which gets mapped down to path on the circle, this is the lift. In particular, if your path is a closed loop on the circle its lift has to start and end in the same fibre above the starting=ending point.

Now the joke is that two loops downstairs are homotopic if and only if the lifts (starting at the same point) are homotopic (with start/end points fixed).
But it's very easy to check when two paths are homotopic on R: they have to have the same start and end points.
By assumption the start points of the lifts are the same so it the paths are homotopic if and only if the end points, which are points in the discrete fibre, are the same. This implies that there is exactly one homotopy class for every point in the fibre. The fibre is exactly the integers, so the fundamental group is Z.

This idea works in general if you replace R by the universal covering space of whatever space you're looking at. The fibres will be a bit more complex though.

>>10496778
thanks anon, helped me get through the proof more intuitively.

>>10494757
?

>>10496778
You're sexy.

Are sets of measure zero necessarily disconnected?

>>10496778
What does it mean for a space to have a trivial fundamental group?

>>10497025
No. For example, we can give R the trivial measure where literally every set has measure zero.
Any two paths are homotopic.

>>10495562

>>10496711
>lifts
There's lectures all over youtube for this too
start watching at 41:15 https://youtu.be/h26w-MyHEW8

>>10496454
Look up some beginner lectures https://youtu.be/J21ZvWWrWwg
Then any Dover Books on Graph Theory will work

>>10497272
>computer science
uh.. ok.. I guess
Thanks anon

>>10497299
its cmu, they use this book like everybody else but dover is fine too http://diestel-graph-theory.com/

>>10495179
Geometry David A Brannan

>>10496454
http://www.zib.de/groetschel/teaching/WS1314/BondyMurtyGTWA.pdf

Probably a tard question but when should I learn how to read and write proofs for maths?

I am a turbobrainlet reviewing very basic geometry, furthest I got in school was trig. Understanding helps me remember and I figure proofs are a good way to do that. At what point in learning maths should someone start focusing on proofs?

And any good online resources anons would recommend specifically for this? Prefer online material to a book cause it’s just more convenient to have constant access, and I find videos/lectures easier to understand.

>>10497677
Start focusing on proofs as soon as you think that you actually care about pure mathematics as a hobby (or career) and not just as a tool you need to pass a test or study engineering.
Plane geometry happens to be where most high school students are introduced to proofs for the first time, so it's not a bad time to start if you want to. It's actually a much better idea (in my opinion) to learn proof techniques through basic geometry than through an "intro to proofs" book or course; those books are usually pretty stupid.
So if you want to do that, just find a rigorous geometry course. I can't recommend a particular one but it shouldn't be too hard to seek out.

>>10497707
Thanks for the advice.

I guess what I find interesting about maths is figuring out how to get from any x to any y in as few steps as possible, and representing logical problems as mathematic ones. I guess the practical goal is to get better at programming things using as few steps as possible, but it’s also just stuff I think is interesting.

I guess I should just be studying sorting algorithms, but I feel like being able to make proofs showing the logic of the algorithms would help me retain stuff.

>>10496331
Thanks for this, it's pretty good.

>>10497677
Proof writing is common sense. Literally whenever you assert something, wonder "wait, why is that again ?". Either this comes from a known theorem or it is something else you have to prove, in which case go back to step 1.
In most cases this process terminates in not too many steps.

>>10494874
...but it IS intuitive

I like math but I've yet to find a subject I really adore. I don't like analysis or differential equations very much, and anything too combinatorial makes my eyes bleed. I do like algebra, but when I did my REU it ended up being a lot of basically combinatorial arguments. (Which seems to be a common occurrence, eg, group theory is cool but then you study groups through rep theory which is meh). I've been planning on reading some intro alg. topology and alg. geometry to see if I enjoy either of those, what other subjects should I dip my feet into? I'm concerned with my grad school prospects if I can't muster real passion for what I'm studying.

>>10498175
Have you taken any differential geometry?

Also, algebra is extremely broad. You can study tons of algebra without getting too deep on combinatorics.

what's an example of a non-bijective isometry?

>>10498175
>is algebraic topology fun
Yes.
>is algeo fun
No.
>>10498205
The inclusion of the unit interval into the real line.

>>10498214
What is an explicit example of that map?

>>10498175
Well you have to understand that, deep down, there are only few things humans can "really" do and those are: basic algebra, combinatorics, finite-dimensional linear algebra, elementary arithmetic and calculus.
Everything else in math is an attempt to reduce very complicated problems to problems in these manageable areas (obviously, these reductions sometimes require a lot of imagination).
If you look hard enough at some papers (especially in algebraic fields), you will find a lot of clever formal mumbo-jumbo and a key argument that is actually a counting or linear algebra result.

>>10498058
Maybe for you.

what's the metric induced from arc length?

>>10498283
d(x,y) = infimum of the lengths of all arcs from x to y

>>10498228
f:I->R defined by f(x)=x.
Isometry, not a bijection.
>>10498283
We have a topological space X and a device that measures paths L: γ->R. We define d(a, b) as the infimum of all the measurements of paths from a to b.

>>10498288
>>10498290
where can I find this more rigorously presented?

>>10498336
Let X be a topological space and [math]a, /space b \in X[/math]. We denote [math]\Omega [/math] the space of continuous functions [math] \gamma : I \rightarrow X[/math] and denote [math]\Omega_a^b[/math] the subset of [math]\Omega[/math] such that [math]\gamma (0)=a[/math] and [math]\gamma (1) b[/math].
A measure function is a map [math]L: \Omega -> \mathbb{R}[/math] such that L is nonnegative, L(a*b)=L(a)+L(b), where * denotes the usual path concatenation, and L(a)=L(-a).
We set [math]d(a, b)= inf L( \gamma)[/math] where [math]\gamma \in \Omega_a^b[/math].

>>10498359
>measure function
My finger slipped, length function.

>>10498336
any book on riemannian geometry

Fetch me a good linear algebra textbook torrent

>>10497677
It is not a prerequisite whatsoever, your own capacity of abstract thinking is the only prerequisite, but I'd say that really elementary algebra (solving 2x + 3 = 5 for example), basic geometry knowledge and intuitions related to the common numerical systems are really helpful.

Obviously, you'll need some of this prior knowledge to apply what you learn about proofs.

Velleman's How to Prove it and A Transition to Advanced Mathematics (by Smith, Eggen and St. Andre) will suit you fine.

I feel kind of guilty using all this math that was discovered by others before me. It would be a shame if I didn't contribute something myself.

>>10498512
Most don't contribute anything, they just keep the lights on.

>>10498519
I know.

>>10497677
Go through this playlist starting at vid 1
https://www.youtube.com/playlist?list=PL5A714C94D40392AB

It's perfect for learning proofs because everything is done with basic arithmetic. After you can go through a book with proofs and have an idea what's going on.

>>10497025
>Are sets of measure zero necessarily disconnected?

In Lebesgue measure, the only connected, nonempty sets of measure zero are points.

Proof: if there is some point x in R that isn't in the set X but which has points both less than and greater than it in X, then we can write [math]X = [-\infty, x) \cup (x, \infty)[/math].

Otherwise, if [math]\sup X > \inf X[/math] we can find x < y in X. But then X contains the nonempty interval [x, y] and has nonzero measure.

>>10498538
er, every proof is done with basic arithmetic. You do enough of those then you have a sort of logical framework how to claim something is true. When he says 'you should prove the other cases' after showing one case, actually try to do it.

>>10497025
>Are sets of measure zero necessarily disconnected?
Not necessarily. A line in R^2 has measure zero

Does it matter if you state the projection like pic related or stated via the extended complex plane: that is [math]P: \mathbb{C}\cup \{ \infty \} \to S^{2}[/math], etc. ?

>>10498727
pic related

Can't wait to crack this baby open when the semester ends. Even if the answer is no I will still read it but does having a good grasp on set theory translate into having a better understanding of other fields? Does it make you a better "mathematician" is basically what I am asking.

also, what do they mean by "subset metric"?

>>10498731
Literally two functions with different domains and codomains.
>>10498740
No.
>>10498743
What it sounds like. Take the two elements in the subset, take their distance in the original space, done.

>>10498753
But the mapping I have still gives the Riemann sphere if I define [math]P(\infty) = (0,0,1)[/math], right?

>>10498740
Well any math you actually work through is going to make you a better mathematician, but set theory is not the best choice to have a better understanding of math as a whole.
It is a fairly isolated part of math and, aside from some fringe combinatorialists and functional analysts, few mathematicians will encounter any serious set theory in their work.

>>10498740
If you want to understand general structures of math, learn abstract algebra - intro to the study of groups, rings, and fields, etc - and topology.

>>10498753
>>10498782
Okay thanks.
>>10498792
I will be taking the intro abstract algebra course next semester. I just really want to get a better taste of set theory.

>>10498740
Reeaaally depends on other stuff. For me it works greatly. I can't really grasp another subject without getting its basics with set theory (set-theoretic definition of a graph, set-theoretic construction of the natural numbers, probability theory axiomatization, etc), but there are many areas that need more than set theory to get the intuitions. Number theory and abstract algebra are two examples.

Let's be real. No working mathematician needs any bit of set theory other than the naïve stuff and equivalences of the AoC. Prove me wrong.

>>10498740
It's good to know some set theory. Often, when something like Tychnoff's theorem or Zorn's lemma comes up, if you don't have any experience with set theory, you'll probably be confused why they're making such a big deal out of it.

>>10498821
>>10498844
Thanks as well.

>>10498873
I believe Enderton is great for developing the best set theory foundations possible. Jech's is more formal and complete, but that book is just for pure set theorists.

I'm taking a course linear algebra and another on intro differential geometry (called Calculus on Manifolds) and I must say
seeing the determinant and all its properties derived twice simply from the wedge product was a pretty great experience.
It really demystified what was otherwise an essentially arbitrary, magical object (it still is, in some ways).

Anyway, just wanted to share this feel.

>>10494757
Suppose [math]f(x)=(x-a)(x-b)(x-c)(x-d)[/math] is a polynomial in [math]\mathbb{Q}[x][/math] whose Galois group over [math]\mathbb{Q}[/math] is [math]S_4[/math]. I'm trying to find all intermediate field extensions of [math]\mathbb{Q}(a,b,c,d)[/math], which of course correspond to subgroups of [math]S_4[/math]. It's easy to get some of these just by identifying elements of [math]S_4[/math] with permutations of [math]a,b,c,d[/math]; so for instance, [math]\mathbb{Q}(d)[/math] is the fixed field of [math]\langle (12), (123) \rangle \cong S_3[/math], [math]\mathbb{Q}(ac+bd)[/math] is the fixed field of [math]\langle (24), (1234) \rangle \cong D_4[/math], etc.

What I'm struggling to find are the fixed fields corresponding to [math]A_4[/math] and [math]\langle (12)(34),(13)(24) \rangle [/math], the Klein four subgroup inside it. Are there any obvious algebraic expressions involving [math]a,b,c,d[/math] that are easily seen to be fixed by precisely these groups?

>>10498831
https://en.wikipedia.org/wiki/Whitehead_problem
https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem

>>10498919
It's not obvious, but I think you will find the case of the 4-group oddly satisfying:
https://en.m.wikipedia.org/wiki/Cross-ratio

Is it true that, if [math]x_n[/math] is a sequence such that [math]x_{n+1} - x_n \rightarrow 0[/math], then the set of its adherent points is [math][\liminf x_n, \limsup x_n[/math]?

For convergent sequences (say, x_n = 1/n), it is trivial, and for sequences that have no convergent sequences it is also true. I'm scratching my head over the nonconvergent sequences that have convergent subsequences.

I wrote that statement while studying real analysis last year but I didn't save any proof of it. I think I saw it on a book. Any help?

>>10498949
How would you go about doing that ?

>>10498949
you mean is contained in [liminf x_n, ...]?
if so, it's pretty trivial, since any convergent subsequence is bounded above and below by liminf and limsup, almost by definition

>>10498946
Damn, that is satisfying

>>10498963
no it is exactly equal to that interval (why would there be such a strong assumption to state a trivial fact ?)
The idea is very simple: the assumption u_{n+1} - u_n -> 0 means that the sequence varies "increasingly slowly".
Now, by definition, arbitrarily far in the future, the sequence is going to go near lim inf (u). But it is also going to go near lim sup(u). And since it goes "very slowly" from one to the other, at one point it is going to come close to every value in between.
Now the only challenge is to make this precise with quantifiers etc.

Algrebraic differential equations when?

>>10499016
So just differential equations?

>>10494757
>low-res pics of covers
>half of the posts don't have authors' names not included
>so many topics!!!111

Yeah, let me just match each of the cover images to thumbnail pics on Ebay. If you were trying to get a rise, well done

>>10499016
algebraic algebra when?

>>10495550
no, it's not, we laugh at physicists every day for treating dy/dx az a fraction, it's a formal symbol, nothing more

>>10499168
> we laugh at physicists
ok, first year undergrad

Does there exist any rigorous relation between a stochastic process and its realizations? For instance, when we find a numerical approximation [math]\hat X(t)[/math] to a strong solution [math]X(t)[/math] to some stochastic differential equation, we merely have
[eqn]\mathbb E[ \hat X(t) - X(t) ] \le \varepsilon[/eqn].

All right, it means, on average, we obtain a good approximation. But, literally, there may be concrete realizations of [math]\hat X(t)[/math] whose respective realizations of the error are arbitrary bad!

That's discouraging. It seems we have to merely believe that we (probably) solve an SDE well. And we you look even deeper, it turn out you can't even generate realizations -- all you have is just a pseudorandom generator which gives you "pseudorealizations".

>>10498540
Typo, that should say [math]X = ((-\infty, x) \cap X) \cup ((x, \infty) \cap X)[/math]

What should I do if I literally can't into math? High School was no problem but uni is really fucking me and I still have another semester of mandatory math.
I'm pursuing a degree in astronomy, but if I keep getting fucked like this I might just jump ship and get a degree in artificial intelligence instead.
Do you guys have any advice for a mathlet pursuing a physics/astronomy degree?
Sorry about the blogpost

Is there an efficient algorithm to generate an incidence matrices of a certain projective plane of order n?
We want the matrix to be of size (n^2+n+1) and to follow the 4 properties :
1) All rows have n+1 "1" (the rest of the coefficients being zeros)
2) All columns have n+1 "1"
3) Two rows have exactly one " 1" on the same column
4) Two columns have exactly one "1" on the same row.
The algorithm I've found (it's pretty bad, but that's what I've found so far) is as follows
-The first row consists of n+1 "1" and then the rest of the coefficients are zeros
-Each new row is generated like this: take the first n+1 columns that don't have n+1 "1" and label them i1, i2,...,i(n+1), then move the labels to other columns (with what would be analogous to a "for loop") that don't already have n+1 "1" until the sum of all the labeled columns is [1,1,...,1], then add a new row at the bottom of the matrix with "1" on columns i1, i2,...,i(n+1) and zeros everywhere else. (I know when I explain it like this it may sound very unclear but I can explain it in detail if you want)
-Repeat n^2+n times until you have your matrix.
Pic related is what you have when n=2.
I haven't proven it yet, but I'm pretty confident that if there is a plane of order n, the algorithm will generate an incidence matrix. This algorithm has complexity O(n^n), but I doubt it is possible to have an algorithm in polynomial complexity.
Is there a better way to generate an incidence matrix? Also if you have any idea on how to turn it into code, I would appreciate it.

I think I'm missing something here. Part (a) is easy enough, but I can't show (b):

Let [math]\sigma:U\to\bar{\mathcal F}[/math] be a section, and let [math]x\in U[/math]. Denote the elements of the disjoint union to be [math](y,s_y)[/math] with [math]y\in U[/math] and [math]s_y\in\mathcal F_y[/math].

Now suppose [math]\sigma(x)\in\mathcal V(V,t)[/math]. We can write [math]\sigma(x)=(y,t_y)[/math], and the condition [math]\pi\circ\sigma = 1[/math] implies that [math]x=y[/math], so that [math]\sigma(x)=(x,t_x)[/math]. However, this doesn't tell me anything about a "global" section [math]t[/math], just one constrained to any such element. I could imagine there is some glueing process but I'm not seeing it.

>>10499645
Eyes on the topology.

>>10498421
>torrent
Why not Library Genesis? Get whichever of these seems to fit your needs: https://4chan-science.fandom.com/wiki/Mathematics#Linear_Algebra
>>10499337
A mathlet into physics or astronomy is not exactly a good combination. Your major might only show one year of mandatory math but most physics courses will just teach you the necessary math along the way.

>>10499655
i still dont get it

the preimage of a basic open set is just [math]\sigma^{-1}\mathcal(V,t)=V[/math], and the section t is irrelevant

I may need you to spell it out please

>>10499681
Focus anon. If it swaps midway from a section to another, it breaks continuity. The essence is proving that.

>>10499645
It follows from the assumption of continuity that you have [math]\sigma(y) = (y, t_y)[/math] for [math]y[/math] in some neighborhood [math]W \subset V[/math] of [math]x[/math].
Doing this around each element of [math]U[/math], you will find such a section in a neighborhood of each element of [math]U[/math]. They will coincide on intersections (use the definition of stalk and the sheaf property) and therefore glue to a section on U (again using the sheaf property).

>>10496108
Smith logic is unreadable

>>10499337
You need math for AI too. It's more stats / linear algebra stuff though.

>>10499111
>universal algebra blocks your path

here's the daily putnam problem >>10499837

>>10499337
hire a tutor, pay some grad student $50 to spend an hour walking you through shit then figure rest out. Go to office hours, maybe try doing your homework

Calc 2 babby here, does anyone know of a good resource for conceptual understanding of everything involved in series? We’re only taught the calculations and the textbook is not an efficient use of time.

>>10499676
>Library Genesis
Danke, I had forgotten aboot that.
Which one of the books in the wiki would you recommend?

>>10499952
The textbook.

>>10499965
XD

>>10499952
Series proofs are 75% "are you actually retarded lim a+b=lim a+lim b is fucking obvious", 15% nice trick, 10% Bolzano-Weierstrass and similar results. Evaluations are mostly cancer.
It's not worth it.
>>10499984
He's serious.

>>10499952
Absolutely this >>10499988 if you plan to study for evaluations. If you want a broader understanding, search different sources online for more stuff. Just dig on different places. Like this weird test that nobody talks about https://en.wikipedia.org/wiki/Cauchy_condensation_test

You can survive knowing the ratio test, the root test (you only have to use it in REALLY obvious cases), the test for divergence, and knowing how to "build" series. This is, if you want to prove that the series S = A+B is convergent, proving that A and B are convergent suffices, stuff like that.

>>10499965
>>10499988
>>10500000
I’ll do this, then. Thank you.

Anyone ? >>10499262

What area of math would the definition/study of properties of the gamma function falls into ?
I vaguely remember proving that it is equivalent to the factorial for real numbers with Stirling formula in my engineer baby-tier complex analysis course, but it looks number theory to me.
Is it just because the two are interwinded ? How is gamma defined exactly, where can I find a textbook about it ?
Thanks

>>10499676
Sorry about the late answer, I've literally been sleeping and having an exam in said math (lol)
Its 3 semesters of mandatory math. I'd be done with them already but I had to take a year off for health/personal reasons.
The math I learn in my physics courses come much easier than the pure math courses. All the math in programming courses are also no problem.
Thats why I'm getting so frustrated.
>>10499868
Fug. Linear Algebra really fucking sucks. Guess AI is a no go then.

How do you get into research? I have taken all of the anal courses at my uni and starting grad courses this fall.

>>10500627
I guess probably the part of number theory that involves a lot of complex analysis. Or the part of complex analysis dealing with stuff that comes up in number theory. Something like that.

I am working on the book that our class is based upon: discrete mathematics 2nd edition by norman l. biggs.

I have some arbitrary problems on some problems and get stuck far too long.
What's the idea here on how to learn this?

>>10499655
>>10499697
>>10499710
oh my it was just under my nose but i didnt see it, thanks anons
no more math past my bedtime haha

>>10499954
Strang or Axler. Both for maximum gains. I'd recommend only Axler if it wasn't for his determinant autism

>>10500000
>>10500134
dont forget alternating series test

>>10500627
any book on complex analysis will cover it. so look around for CA books. Perhaps one that has a dedicated chapter to it

>>10500830
Use distributivity and assumption that k is natural. It's not written very well

/sqt/ is dead, can someone help explain what this step is doing?

>>10498914
good feel
8/10 would like to feel

>>10500865
>converges but not absolutely
This makes me so mad

Jesus, PDE classes make me want to kill myself.
How did you get through it? Is there any way at all to make these classes a tiny bit less depressing?

>>10500930
>>10495514

Do all regions in [math]\mathbb{R}^{3}[/math] admit a parameterization?

>>10498914
>taking linear algebra and smooth manifolds at the same time
What.
>>10501056
IIRC every connected open set in R^3 was homeomorphic to R^3, but I don't know what you mean with parametrization.

>>10501065
Wait no, every star shaped open set was, and maybe every n-connected for all n.

>>10501056
What do you mean by parametrization? It is well known that elliptic curves are not rational, ie there does not exist a (locally rational) polynomial map from R^n or P^n

>>10501022
Pretty neat but not enough quell my distaste

>>10500627
>How is gamma defined exactly, where can I find a textbook about it ?
read the wikipedia page for it, but it does get really involved
https://en.wikipedia.org/wiki/Gamma_function
wiki may suck ass, but at least they can't fuck up math pages with their politics shit

>>10500930
[math] 1 - 1 + 1 - 1 + ... = \frac{1}{2} [/math]
[math] 1 + 2 + 4 + 8 + 16 +... = -1 [/math]
boo!

Why don't papers include information about how the authors arrived at their conclusions / chosen proof techniques instead of just showing you some cryptic proof? Is it because they don't want to reveal their secrets and threaten their job security?

>>10501263
Probably because experts in the subject would understand how they got there in the first place, because it would make the paper longer and more tedious to write, and because the whole point of giving a lecture on your paper is for people to ask these questions. Your best bet is to send an email to the authors asking how they came about with the results and 8/10 they will be happy to answer

>>10501263
Scratch work with no results might as well include the entirety of human knowledge. Papers are there to show how to do something, not how not to do something.

>start calculus, really enjoy it
>move to differential equations
>it's the most boring shit i've ever taken
I hope this is just a low point

>>10501374
No, it IS the worst thing you'll lay your hand on.

>>10501374
nobody enjoys DEs, seriously. Just learn what you need to learn and move out quickly

Boys, I need to get my maths knowledge up for an upcoming masters I'm doing. Haven't touched maths in 4 years. Is Khanacademy any good? Think I want to redo the high school maths and get my level up to at least undergraduate.

>>10501566
Handbook of mathematics.
http://booksdescr.org/item/index.php?md5=5AE296C34050FAB50136D08E779DA44C

>>10501602

Thanks man I appreciate it

what should I know before taking differential geometry?

>>10502178
Linear algebra, topology, multivariable calculus.

>>10502181
How heavy is the topology? I've taken analysis and have the basic gist of things.

>>10494757
Lol

>>10502182
You should take general topology first.

>>10502186
Alright. Any recommendations for a good differential geometry textbook?

>>10501374
ODE is the basis for the theory of flows, which is a basic and incredibly powerful tool in differential geometry. Also, if you don't think the qualitative theory of ODEs is based, I don't know what to say, you just have no taste.

I'm a junior undergrad math major now and I've been getting C's in all my math classes for years. I feel like I haven't absorbed anything. aaaa

>>10502188
Introduction to Smooth Manifolds, John Lee.
You can be a retard and just read the appendix, but I wouldn't particularly recommend doing so.

>>10502470
At least you aren't me. I'm not even passing anymore.

>>10502470
>>10502604
m-me too

why can't i retract a disk into a circle? why do i need to take out a point out from the disk for it to be able to retract?
pic related, let's say i make the points on the disk, except the center, move to the circle with linear speed, and the center point moves to the left with linear speed, wouldn't this be a retraction?

>>10502614
oops here's the pic.

>>10502614
No.

>>10502623
i know i won't from everything i've read but i just can't figure out why.

>>10502641
Man, the no retract theorem is just about the single most intuitive result in algebraic topology. You might as well completely give up on the subject.

>>10502650
thanks mr perelman

>>10495560
Lang is a meme.

>>10502614
>>10502618
It's discontinuous. You're tearing any interval that goes through the center into two pieces.

Daily Putnam Problem >>10502958

>>10502614
An important thing to note is that, if you need to prove that a transformation is impossible, you should try to find a property of the starting state that should be preserved and is not shared by the end state.
Do you notice anything here ?

>>10502614
the middlepoint starts being in a neighbourhood of all the points around it, yet ends up not being so. So you're breaking the continuity.

The easiest way to argue this is by a sequence argument, ie: using the sequence definition of continuity. A function is continuous at a point x iff every sequence {x_n} that converges to x is such that lim f(x_n) = f(x).

Now if you decide to move the centrepoint to any side in the circle, pick a sequence that converges to the centrepoint from the opposite side.

>>10503579
sorry, that first line should read
**any neighbourhood of the middlepoint has points around it in any direction, yet ends up not being so.

How do I prove that there are only two isometries from [math]\mathbb{R}[/math] to itself under the standard metric (reflection and translation) ?

>>10503612
Take a real number x.
Take another real number y>x.
Any isometry f can only take y to f(x) + (y-x) or f(x) - (y-x).
This choice of f(y) completely determines the map (why?).
The former is a translation only, whereas the latter is both.

Is there a field where none of the elements look like normal numeric values? Say, one that doesn't contain an isomorphic copies of the (possibly modular) integers?

>>10503669
No. Every field has a prime subfield, which depends on the characteristic. That is, if the field is of characteristic 0, then it is an extension of the rationals. If it is of characteristic p, then it is an extension of Z/pZ

Sometimes when I look at resources I keep bumping into Chicago REU summer papers... What the fuck? Do they even do anything? It's 8 weeks and most of them literally cover at most chapter 1 of a book. I saw one that covered the first chapter of Reid's UGAG, which is literally just intro stuff. Then they always say oh yeah this is beautiful mathematics

>>10501374
Do you know linear algebra? DE stuff is just gonna feel like a bunch of arbitrary shit if you don't know linear algebra.

>>10503739
>Do they even do anything?
Almost invariably no, literally nothing. REUs are CV padders for undergrads, and they barely even function as that anymore since everybody knows they're just bullshit CV padders and the only "research" that goes on is in problem sets.
Although I did meet a professor once (graph theorist) who managed to wrangle his REU students into doing some kind of group research project that was legitimate enough to be published at the end after some cleaning up. I don't expect it was anything super impressive but it's pretty cool anyway.

>>10499819
i went through the entire book
can confirm. it isn't a textbook

Are multiplication operator s densly defined on finite measure spaces?

>>10503218
Remimi!
>>10503612
>isometries from R to itself
Can't you determine those from two points?
>>10503669
Take the smallest subfield that contains the identity. It's isomorphic to either Q or Z_p for some p.
It also works for random rings with identity. Take the smallest subring that contains the multiplicative identity. Either isomorphic to Z or Z_n.

>>10502188
Kristopher Tapp - Differential Geometry of Curves
Check it on libgen.io, very nice intro

I can't read wikipedia "latex" ie those images with math text. Does anyone know what could be doing it? I tried turning my adblocker off already

>>10504484
Part of it is loading correctly, so it's probably a page problem.

>>10504487
it's been like this for at least a week, on every page I try. I've also just tried turning all my add ons/extensions off and still nothing. It'd be weird that something like this could happen and no one says anything. However, if I 'drag' the images where the math text is, I can kind of see the text

>>10494757
Anyone here use to suck at math and somehow manage to turn it around?

when did /mg/ go to shit?

>>10504953
wait i meant to post this one

Anyone know what book and or site I can go to get a head start if im taking algebra 2 currently as a Junior(HS)?

I've found my love for mathematics too late and love to hear discussions about it if only I understood it so. I'm unsure whether to read my Algebra 2 textbook or just start reading the pre-calculus text books here, maybe both? Or focus on alg 2 before going into precal?

Thanks

>>10504953
>>10504964
What a mess.

>tfw no qt gf to help me understand schemes

>>10504976
try one of these from the /sci/ wiki though I have no idea if either of them are good

>>10503130
yeah, i thought about it again today and realized it's discontinuous... small changes from the center will make the image ''jump''.

>>10504484
> I can't read wikipedia "latex" ie those images with math text.
They're just images in SVG format (vector graphics, using XML). Try using "view source" to get the URL then copying it to the address bar. The URLs have the form:
https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad9b0047f8437f7b012041d7b2fcd190a5a9ec2

>>10504976

khanacademy is a good resource

>>10505460
yes i know that but you can understand i dont want to do this to look at the letter "M" in math tex

How would you parameterize the region of this surface ([math]x^{2}+y^{2}=4[/math]) between [math]z=x[/math] and [math]z=2[/math]?

>>10506108
x=2cost
y=2sint
t goes from 0 to 2pi

>>10506120
I mean the missing part of the cylinder.

>>10506123
write y=2sint, x=2cost, and note -2<z<2.
compute intersection z=x and x^2+y^2=4
get t = arcsin( plusminus sqrt(1 - z^2/4) )
parametrize as -2<z<2, and t going from the two above points in x and y

>>10506123
Do it like >>10506119 told you.

>>10506339
>>10506245 even.

>>10495560
How to Prove it is a meme

Here is a very anal stylistic question.

How to abbreviate the plurals: Theorems, Propositions, Lemmas, etc.

Should it be: Thms. Props. Lems.

Or: Thm.s Prop.s Lem.s

The second was recommended by some online source, since the thing being abbreviated is the first term, but honestly I have no idea.

friendly friday reminder that axiom of powerset is wrong

>>10506476
I've seen thm's, prop's, lemma's, or perhaps with the s as a superscript. It shouldnt be a problem anyways, i think most people would understand thms

>>10506529
Which set doesn't have a power set?

>last semester of undergrad
>employers don't give a shit about my BS in math
>bankers want accounting students
>software companies want comp sci students
>high schools want math **education** students
>wageslave jobs would rather hire high school grads
It's all so tiresome. The really shitty part is that I can't focus on my last few courses with all of this career bullshit in the air.

>>10506576
having both choice and powerset is responsible for the vast majority of pathological nonsense that crops up in, eg, measure theory. Getting rid of choice causes nonsense itself, but powerset can be removed without too much breaking.

>>10506615
yeah I read that shit on reddit /r/math too

>>10506605
yo-you did learn to code, right anon?

>>10506529
just like Regularity

>>10495562
Show that:

"...""Lang is a meme" is a meme" ...is a meme"

with n instances of "is a meme" implies the same for n+1 instances. Use this to prove that the memeness of Lang fails to converge.

>>10497677
Proof writing comes from reading textbooks with full proofs written and explained, and assimilating the style and techniques.
Some fields have the same methods applied everywhere.

>>10506649
>textbooks with full proofs written and explained
This doesn't exist.

How long do you guys take to work through a book on average?

>>10506650
The series on which I got myself some basic algebra and how the real are constructed had it, and it was pretty neat
Unfortunately I don't believe it's been translated into English

>>10506653
I try to give 2 hours every day to self-study, but even with that, it takes time.

>>10506623
Except you need regularity if you have the axiom schema of specification.
>>10506650
I've seen a few for Calc.
>>10506653
A 400 page book? 3 weeks.
30% absorption, but the forgetting and revising model works nicely.

>>10506653
Took me about a month to read first two chapters of shafarevich

>>10506664
>400 page in 3 weeks
you're not impressing anyone by skimming through a calculus book

>>10506605
The only reason you should ever pursue a BS in math is to eventually get a PhD and work in academia. That's assuming you're a good student and eventually a good researcher with a decent network. Over here, people who fail to do this (i.e. average-ish or underachieving students) study a year of education and become math teachers for the HS or college freshman level — dunno if this is an option in the US or Europe.
>>10506653
Depends a lot on the level of the textbook. I worked through Spivak's Calculus in Manifolds in three weeks, but I'm a babby undergrad and also lazy as fuck

>>10506653
people die if they are killed

I hate to fill this thread with trash but whats /mg/s thought on this thread >>10504209 ? It's actually kinda interesting albeit I don't know shit about mathematical logic,foundations and all that, I kinda just turned a blind eye to it all.

Thoughts?

>>10506785
I meant to post both woops
>>10502045

>>10506616
>you wouldn't sleep with sleep_with_crazy

welp, i guess someone doesnt like schemes

>>10497707
>those books are usually pretty stupid.
Don't say that, I like my Book Of Proof.

>>10506821
That's a message the postdoc shitter before left for Pierre and you paused at the moment Pierre notices it. The postdoc was heard crying at the female restroom later that day and he transfered to applied math department.

>>10506832
Learn to code.

>>10506707
You're not impressing anyone by skimming through a post.

Please recommend something on functional analysis. Preferably not a book but video lectures or something visual.
I understand the theorems and other stuff but can't memorise, it's too much overabundance of information. Something that would help me structure it nicely in my head.

>>10506615
>doesn't answer the fucking question

>>10498963
>>10499003
sorry guys, I messed up writing the problem

Let's say [math]x_n[/math] is a sequence such that [math]x_{n+1} - x_n \rightarrow 0[/math]. How do I prove that the set of adherent values/partial limits of the sequence is [math][\liminf x_n, \limsup x_n][/math] (or empty, as it is in the case of the harmonic series)?

I've been trying to use what >>10499003 said but I don't know exactly how.

Also, is it an if and only if situation? I mean, if [math][\liminf x_n, \limsup x_n][/math] is the set of adherent values, is it true that [math]x_{n+1} - x_n \rightarrow 0[/math]?

>>10506961
Wow, you're literally hitler.

>>10508033
>or empty, as it is in the case of the harmonic series
It is not empty, you just have [math]\liminf x_n = \limsup x_n = +\infty[/math].
So let's spell out what I said the other day. I will work under the assumption that [math](x_n)[/math] is bounded and not convergent (so that [math]-\infty < \liminf x_n < \limsup x_n < +\infty[/math]).
It works in the other cases but I don't want to type.
Now what do ? We start with some [math]l \in ]\liminf x_n, \limsup x_n[[/math] and we want to prove that it is a limit point of the sequence.
This means that, for each [math]\varepsilon > 0[/math] and [math]N \in \mathbb N[/math], there is an [math]n \ge N[/math] such that [math]|x_n - l| \le \varepsilon[/math] (arbitrarily far into the future, the sequence will come [math]\varepsilon-[/math]close to [math]l[/math]).
So we start with some [math]\varepsilon > 0[/math] and [math]N \in \mathbb N[/math]. We may choose [math]\varepsilon[/math] small, say [math]\varepsilon < \min(\limsup x_n - l, l - \liminf x_n)[/math].
This weird looking assumption means that [math]\varepsilon[/math] is so small as to satisfy [math]\liminf x_n + \varepsilon < l < \limsup x_n - \varepsilon[/math].
By our assumption on the sequence [math](x_n)[/math], there is an [math]N' \ge N[/math] such that, for each [math]n \ge N'[/math], [math]|x_{n+1} - x_n| \le \varepsilon[/math] (the sequence "moves slowly" beyond that point).
Now, since [math]\liminf x_n[/math] is a limit point, there is an [math]N_1 \ge N'[/math] such that [math]x_{N_1} \le \liminf x_n + \varepsilon[/math].
We also know that, because [math]\limsup x_n[/math] is a limit point, there is also a [math]N_2 > N_1[/math] such that [math]x_{N_2} \ge \limsup x_n - \varepsilon[/math].
In other words, the sequence has crossed from one side of [math]l[/math] to the other. But we know that it travels "slowly", so we know that it has come close to [math]l[/math] at one point. We only have to put this in symbols.
(cont.)

>>10508585
Let [math]n = \min\{k \in [\![N_1+1, N_2]\!], x_k > l\}[/math]. In other words, we look at the first time in the interval when the sequence crosses to the other side of [math]l[/math].
Then, by definition, [math]x_n > l[/math] but [math]x_{n-1} \le l[/math]. Moreover, since [math]n > N_1 +1 \ge N+1[/math], we have [math]|x_n - x_{n-1}| \le \varepsilon[/math].
But [math]x_n > l \ge x_{n-1}[/math], hence [math]|x_n - l| \le |x_n - x_{n-1}| \le \varepsilon[/math].
This completes the proof: we have found [math]n > N_1 \ge N[/math] such that [math]|x_n - l| \le \varepsilon[/math].

So this was a bit long-winded (longer than I thought it would be), but if you actually make a little drawing of an interval and put all the relevant quantities (liminf, limsup, epsilon, the x_i's), you should have no trouble understanding the argument.

>>10508593
Let [math]n = \min\{k \in [\![N_1+1, N_2]\!], x_k > l\}[/math]. Then, by definition, [math]x_n > l[/math] but [math]x_{n-1} \le l[/math]. Moreover, since [math]n > N_1 +1 \ge N'+1[/math], we have [math]|x_n - x_{n-1}| \le \varepsilon[/math].
But [math]x_n > l \ge x_{n-1}[/math], hence [math]|x_n - l| \le |x_n - x_{n-1}| \le \varepsilon[/math].
This completes the proof: we have found [math]n > N_1 \ge N[/math] such that [math]|x_n - l| \le \varepsilon[/math].

So this was a bit long-winded (longer than I thought it would be), but if you actually make a little drawing of an interval and put all the relevant quantities (liminf, limsup, epsilon, the x_i's), you should have no trouble understanding the argument.

>>10507276
>something visual.
>functional analysis
good luck with that

>>10508585
Let [math]n = \min\{k \in [\![N_1+1, N_2]\!], x_k > l\}[/math]. Then, by definition, [math]x_n > l[/math] but [math]x_{n-1} \le l[/math]. Moreover, since [math]n > N_1 +1 \ge N+1[/math], we have [math]|x_n - x_{n-1}| \le \varepsilon[/math].
But [math]x_n > l \ge x_{n-1}[/math], hence [math]|x_n - l| \le |x_n - x_{n-1}| \le \varepsilon[/math].
This completes the proof: we have found [math]n > N_1 \ge N[/math] such that [math]|x_n - l| \le \varepsilon[/math].

So this was a bit long-winded (longer than I thought it would be), but if you actually make a little drawing of an interval and put all the relevant quantities (liminf, limsup, epsilon, the x_i's), you should have no trouble understanding the argument.

>>10508585
Let [math]n = \min\{k \in [\![N_1+1, N_2]\!], x_k > l\}[/math]. In other words, we look at the first time when the sequence crosses to the other side of [math]l[/math].
Then, by definition, [math]x_n > l[/math] but [math]x_{n-1} \le l[/math]. Moreover, since [math]n > N_1 +1 \ge N'+1[/math], we have [math]|x_n - x_{n-1}| \le \varepsilon[/math].
But [math]x_n > l \ge x_{n-1}[/math], hence [math]|x_n - l| \le |x_n - x_{n-1}| \le \varepsilon[/math] (draw it).
This completes the proof: we have found [math]n > N_1 \ge N[/math] such that [math]|x_n - l| \le \varepsilon[/math].

So this was a bit long-winded (longer than I thought it would be), but if you actually make a little drawing of an interval and put all the relevant quantities (liminf, limsup, epsilon, l, the x_i's), you should have no trouble understanding the argument.

>>10507582
infinite sets

>>10506460
What's the best "Intro to Proof" book then?

>>10508668
Rudin, Hatcher, Dummit & Foote

this guy always reminds me of the J U S T guy, especially now with his new haircut

>>10508668
Bourbaki Elements of Set Theory

Recommended books for introductory algebraic topology? Right now I'm working with Massey's "a basic course in algebraic topology", any other recommendations?

>>10508668
Chartrand et al

What's the intuition behind Baye's Theorem?

What are some good books to read if I have taken up to multivariable Calculus and Linear Algebra, and just switched to a math major from Computer engineering? Because of computer engineering I know a decent amount of boolean algebra and set theory, as well as graph theory, inductive proofs, etc. Basic discrete mathematics, essentially. I am a complete babby. I don't even know what real analysis is. I always imagined it as just calc 4 because smooth brain.

>>10508668
How to Prove It

>>10509156
Baby Rudin and Artin's Algebra will suit you fine. Burkill's Second Course on Analysis if Baby Rudin is too basic for you. Apostol's Analytical Number Theory if you're interested in number theory.

Depending on what you consider "Linear Algebra", it may be necessary for you to check out Hoffman & Kunze.

Munkres' Topology if you're feeling brave enough.

>have a question for /mg/
>write a long post explaining all the context
>about to type the question when I finally realize what the answer is

>>10509156
Conjecture and Proof by Laczkovich
Combinatorics and Graph Theory by Harris, Hirst, and Mossinghoff
The Joy of Sets: Fundamentals of Contemporary Set Theory by Devlin
Linear Algebra by Shilov

>>10508668
Book of Proof by Hammack (http://www.people.vcu.edu/~rhammack/BookOfProof/)) if your background is weak
A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre if you're average

>>10509393
been there done that x100

>>10509418
It's the internet equivalent of asking a professor a question and talking yourself through the answer before the prof can even get a word in

Ax+by=1
Cx+dy=2

solve for x and y

How do this?

>>10509483
kek

>>10509505
from the first equation, x = 1/A(1 - by) (*)
therefore (C/A)(1-by) + dy = 2
from here you can solve for y and then replace it in (*) to solve for x

>>10509531
i figured it out.. Thanks dude.

>>10509483
>wanted to ask the professor a question because he was specifying something and I was wondering whether it worked in general
>he said it worked in general
>as I was lowering my hand, he noticed and insisted I ask the question
Beyond embarassing.

>>10509114
No idea if it's good or not but I see Hatcher mentioned a lot

>>10509114
Fuchs-Fomenko if you're autistic, Strom if you're hyper autistic.

>>10509114
See the official /mg/ recommendations in the OP:
>>10494757

>>10494784
in physics it makes things a lot easier
the proofs and derivations are more intuitive when you treat the notation as actual fractions

Daily Putnam Problem >>10510005

>>10494784
>Lagrange notation
good for basic calculus
>Leibniz
good for vector calculus and physics
>Newton
good for mechanics

you guys ever get sleep deprived and let the fatigue build up until you feel absolutely fucked? i had the best sleep yesterday i've had in weeks and it's like all of a sudden i understand what i'm reading.

>>10509114
i'm reading both hatcher and w.s. massey's basic course right now.

>>10510014
A very tasteful man.
Want to borrow my Remi folder?

In Polya's Problems and Theorems in Analysis I, the solution to this problem begins stating "It is sufficient to consider the case where the lower bound is finite". Why is this so? Is the sequence [math]\frac{a_n}{n}[/math] decreasing? if so, why the fuck?

>>10510191
Assume that it's bounded below, but it's also below zero.
Then it diverges downwards.
In particular, consider [math]a_{m+m} \le a_m + a_m[/math]

>>10510102
I've had insomnia so bad that I started hearing voices

>>10510143
Yes!

>>10510220
Okay, fantastic, so I got that [math]\frac{ax_{2m}}{2m} \leq \frac{a_m}{m}[/math] and therefore the subsequence [math]a_{2^m}[/math] is decreasing.

That doesn't tell me anything about the whole sequence.

>>10510303
https://mega.nz/#F!Vp0Dha4C!kzmpUPHrMS4hmg3mAIs3xQ
It should be entirely sfw, but I might have saved one or two by accident.
>>10510344
Right, I'll give another tip about the sequence.
[math]a_n \le na_1[/math]

>>10509154
the intuition is as follows:
if you're not a brainlet, you'll get it

>>10509777
>>10510102

Yeah I originally tried Hatcher but it didn't really click for me, seems more of a book for somebody that already knows what they're talking about and wants to relearn the basics

Daily Putnam Problem >>10510739

So, I'm learning Abstract Algebra, It's a wonderful subject, but I gotta make a complaint to all of you who recommend Herstein. Why the hell do you do that? Seriously, Herstein is a very old book, with an outdated notation, some exercises are plain stupid and it doesn't cover some important topics like canonical projection. I started using it because people say it's a classic book and the most recommended but so far I found it quite bad. Paolo Aluffi's Algebra: Chapter 0, on the other hand, is an excellent book, it covers a lot of important topics in its introduction and it covers a hell of a lot more topics than Herstein, besides it uses a very modern notation. A much better intro to Abstract Algebra in my opinion so far.

>>10511568
Every criticism I've seen of Herstein comes down to >muh notation. Also his exercises are fantastic, what are you talking about? His prose is entertaining to read and often shows you the beauty or importance of the math.
>it doesn't cover some important topics like canonical projection
If you're just learning abstract algebra for the first time, what qualifies you to say which topics are important? The basics of a first semester undergrad abstract algebra course are covered in Herstein. For other topics not in that book, I recommend Dummit and Foote or Fraleigh.

>>10511772
>Also his exercises are fantastic, what are you talking about?
Well, I've only read his introduction and a little of the chapter on Groups. The exercises about equivalence relations/classes and sets were quite weak.
>what qualifies you to say which topics are important?
I'm not qualified to say anything, but my professor is and the notation he uses and the things he explained before Group Theory were not the ones used in Herstein. But anyway, since I haven't read the whole book and you were kind enough to provide an answer defending your thoughts, I'll keep using it together with Aluffi.

>>10511568
See the official /mg/ recommendations in the OP:
>>10494757 (You) (OP)

>>10501065
These are graduate courses, so it's not a first course in linear, if that's what you're thinking.

If you were just saying that for another reason, well, I don't have very much freedom with my class schedule (I'm a Junior in a BA/MA program and to graduate in 4 years my last few semesters here are very loaded)

>>10511834
Ah, I skipped the intro chapter because I'm already familiar with that stuff. You've hardly scratched the surface and the best is yet to come! It's true that Herstein's notation is outdated, but if you're following along with the book and your professor's lectures, you should be able to understand what he's saying. That's why I don't consider that a good criticism of the book. My classmates would complain about his notation but many weren't keeping up with the class and didn't understand basic concepts -- if you don't know what a permutation is, of course you won't be able to recognize it being denoted in two different ways. I hope your class goes well and you learn a lot.

Our professor gave us a practice question sheet for our test tomorrow and one of the questions is "prove any unbounded entire function is a polynomial" How do I do this proof?

I figured expand it as a power series centered at the origin but I'm not 'seeing' this proof. Maybe something to do with a residue/singularity at infinity, but I'm struggling with this one.

>>10512058
That statement is not true without extra assumptions. How nice do you assume your functions are?

>>10495514
Nice.

>>10512076
Yeah I omitted something by mistake sorry, obviously that would make no sense since the identity is holomorphic and trivially unbounded.

f -> infinity as z -> infinity (not the modulus, just f itself)

>>10512100
The identity is a polynomial, anon.

>>10512058
>>10512100
Since it's an entire function, then it has a power series expansion everywhere. It is enough to show that it has only finitely many terms.
Consider the reciprocal 1/f . It tends to 0 as z tends to infinity. What else can you tell?

>>10509154
>What's the intuition behind Baye's Theorem?
All reputable statisticians reject Bayes "theorem".

>>10509114
>Recommended books for introductory algebraic topology? Right now I'm working with Massey's "a basic course in algebraic topology", any other recommendations?

>>10512265
What did he mean by this?

>>10512293
>What did he mean by this?
I'm not a "he".

>>10512265
This post made by frequentist gang

>>10512301
Then your words are irrelevant.

>>10509114
Tanre

>>10511568
>shitting on Herstein
>muh notation
>muh canonical projection

Herstein's Topics in Algebra is flawless. His proof of the generalized First Sylow Theorem can be a little bit weird, but besides that it's a great book. Algebra: Chapter 0 is an unnecessary encyclopedia. We already have Dummit & Foote for that, or graduated level books which are going to be better than a fucking bible. Why would an undergraduate need to learn normal series, for example.

>>10512302
pls no

>>10512476
You are absolutely wrong. It covers all the algebra you would do in an algebra-oriented undergrad. It is not at all encyclopedic, but has an "encyclopedic completion" by doing all the exercises.
>why normal series
perhaps if you've ever gone past babby intro to abstract algebra, you would have noticed theyre used in Galois theory

>>10497707
Planet geometria in second year of High School is literally what made my interested in math and probably the single most important factor that made me do maths at university