/sci/ - Science & Math

What's math?

What’s your favourite math conundrum bros?

>>10960175
BABY DON'T HURT ME

>>10960172
Someone redill me on Alex Eskin and the Magic Wand Theorem. Is it just a meme?

Threadly reminder to work with physicists.

>>10960177
>What’s your favourite math conundrum bros?
Monty Hall

Why is this wrong?

>>10960662
Try putting amplitude 4 in there and see if you get it right. Don't know what else could be wrong.

>>10960662
Oh, you've used t as the variable. Try x.

>>10960678
Wow, thanks! I really should've used the variable from the graph but I never thought of that.

>>10960686
Also, confused anime girls should always have question marks nearby. So you get -1 point for that too.

>>10960697
I'll do better next time.

How many possible ways one can reach yes and one can reach no?

>>10960203
https://arxiv.org/abs/1105.0805

How do I dedicate myself to Mathematics?

>>10961000
>How do I dedicate myself to Mathematics?
What have you tried?

>>10961019
Reading, researching, attending lectures, doing lots of math. Because it's all I have.

>>10960933
When are we getting sushi?

>>10961067
>sushi
That's no way to refer to a lady's private parts, is it?

>>10960172
>CCP Communist programs
>KMT Nationalist animu
Stick to bean counting senpai

>>10961000
Which part of math?

Can algebraic topology have a non-self contractory set of Abelian groups?

>>10961072
what did he mean by this

>>10961261
wut?

>>10961320
>what did he mean by this
I'm not a "he".

>>10961422
It's ok anon, we're all girls here.

Why does this pos calculator give me shit values like this?

>>10961457
Instead of this.

>>10961457
>>10961458
I don't see what the difference is. You dislike precision?

>>10961573
The real issue is that my calculator keeps screwing up long decimals while dealing with integrals. Just trying to come up with some kind of a workaround.

Can someone post the chronological math order you should study meme pic so I can choose my next topic to study, thanks

>>10961608
Your calculator integrates numerically, just like literally any computer system. The only situation in which it will calculate an integral precisely is for a line (no error). Otherewise, whether it's visible in the amount of digits or not, there will always be some error.
Keep in mind that the error is ridiculously small compared to the size of your answer.

Thoughts on this for an anon who has worked 6 years after failing highschool with deep regret and wants to go back into studying?

>>10961640
Lang is a meme.
Probably a good idea though.

>>10961656
Fags who say this are a meme and should not be taken seriously.

>>10961666
Hi Lang. How's the AIDS denial life treating you these days?

>>10961675
Gee i may be a Aids denier but whats that got to do with mathematics.
Tell me based senpai.

>>10961000
don't lmao

>>10960933
what does Higgs mean to them, in this general context?

>>10962138
The section of a Hermitian line bundle [math]V[/math].

>>10962147
What makes it "Higgs"?

>>10962211
Connection 1-forms of the tensor bundle [math]P\otimes V[/math] enter the action just like how Higgs couple to the gauge field strength.

Terry Tao announced today that he has new partial results toward proving the Collatz conjecture!
https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/
we're almost there

>>10962283
>Theorem 2 Let {f: {\bf N}+1 \rightarrow {\bf R}} be any function with {\lim_{N \rightarrow \infty} f(N) = +\infty}. Then we have {\mathrm{Col}_{\min}(N) < f(N)} for almost all {N} (in the sense of logarithmic density). ,
Almost there, and yet infinitely far away

>>10961608
>>10961458
>>10961457
The nature of floating point calculations makes errors inevitable.

If you want exact results you need to consult a computer algebra system, if you want more accurate results you need to use variable precision arithmetic or if you want a mathematically verified bound you might have to dive deep into interval arithmetic.
All of that comes at a computational cost unsuited for a pocket calculator, I recommend that you use an appropriate software package on a desktop computer for anything more then the basic numeric integration your calculator does.

>>10962283
>Little brain Riemann hypothesis
>Normal brain Navier Stokes
>Big brain Goldbach conjeture
>Galaxy brain Collatz conjeture

Can someone explain this one thing to me?
Say I have a space of bounded functions $B\left ( U,V \right )$, where $U$ is a normed vector space and $V$ is a complete vector space. Then one can prove that the $B\left ( U,V \right )$ is complete w.r.t. the sup norm by arguing that a Cauchy sequence of functions $\left \{ f_{n} \right \}$ in $B\left ( U,V \right )$ converges to a bounded function $f$, where $f$ is defined pointwise by setting $f(x)=\lim_{n \to \infty }f_{n}(x)$. But for each $x\in U$ one has to choose $N_{x}$ which may depend on $x$ such that for a given $\varepsilon$ one has $\left \| f_{n}(x)-f(x) \right \|<\varepsilon $. But what guarantees uniform convergence then, if the choice of $N$ cannot be made to suit all $x$ in the domain? Can someone clarify this for me? Here's a cute cat photo in return.

Terry is into abstract art now

>>10961640
Check Khan Academy too desu

>>10961630
>just like literally any computer system.
This is false.

>The only situation in which it will calculate an integral precisely is for a line
This is most likely false, as even basic numerical integration routines exactly calculate polynomials of certain degrees.
See eg. https://en.m.wikipedia.org/wiki/Simpson%27s_rule

>Otherewise, whether it's visible in the amount of digits or not, there will always be some error.
Also false. There might be no error due to you being lucky and depending on the display and calculator internals the error might very well be always visible.

>Keep in mind that the error is ridiculously small compared to the size of your answer.
This is generally false.
Numerical approximations can and do fail in many situations. Even very basic calculations can through elimination produce enormous errors.

I just took the constructivist redpill. Don't think I can go back to seeing math the way I used to. I hope you guys enjoy your fantasy land. It's not for me anymore.

>>10962342
learn agda fag

>>10962342
https://xenaproject.wordpress.com/
https://wwwf.imperial.ac.uk/~buzzard/one_off_lectures/msr.pdf

Say I have three bits of information:
the apple is red (1), round (2), and not rotten(3).

it can be written as 110 in terms of information where each bit represents a resolution of a yes or no "question" "is it red", "is it rotten" etc.

Is there a generalized Mathematical term for these "questions"?

>>10962334
I have tried Khan academy but I got to be honest I really hate his voice and watching his vidfeos are boring as hell compared to reading

Also having to answer the questions in non-latex plain text is just not as nice to me as writing down and checking answer in the back of the book

Maybe I'm just too fucking gay and demanding but I really prefer books

>>10962438
>>10962334
I know what you mean I prefer books too, but I like KA because sometimes videos present the material in a different making it easier to understand. I think it is also a good way to grind exercises, before applying your knowledge to problems.

>>10961640
Precalculus by Sheldon Axler is best

>>10962506
What do you like in it ? I've ordered Precalculus in a Nutshell, it seems more straightforward.

>>10962526
>why do you like it
Because he wrote it.
Axler has been shilling his linear algebra and the Springer GTM series for years now.

>>10962574
You just like the book because Axler wrote it ? Nothing more ?

I'm investigating vibration analysis with eigenvalues and eigenvectors. Can someone refer me examples of a spring-mass system in 2 dimensions? By 2 dimensions I mean the assume the masses are positioned in a plane, not a line and not in 3 dimensions.

>>10962526
Precalculus in a Nutshell is also a great book, but too brief, it's more like a quick review of the most important topics than a textbook that teaches everything you need to know from the ground up the way that Axler's book does. Still, Axler's book is half the number of pages of most precalc textbooks, and still does a great job of explaining everything. You can find both books on Libgen

>Wall-crossing

>>10963196
Dumb squidposter

>>10963350
>cohomogy

>>10962325
Well. since you're considering the space of bounded functions, you know that the maximum pointwise difference between two functions must always be bounded. Using this, coupled with the fact that V is a complete vector space, you should be able to argue that the sup of this maximum variation must go to zero as n goes to infinity. At this point, you can just apply this maximum difference as a crude estimate that applies independent of x, but still goes to zero as n goes to infinity.

>>10962360
CS

>>10963391
I think I sort of get what you're saying, but still I could use some further clarification. What I'm trying to do is basically translate the claim regarding convergence in [math]B\left ( U,V \right )[/math] to a basic claim regarding uniform convergence. So now that [math]f[/math] has been defined as a pointwise limit of a Cauchy sequence, I'm trying to find [math]N[/math] s.t. [math]n>N[/math] implies [math]\left \| f_{n}(x)-f(x) \right \|<\varepsilon [/math] for a given [math]\varepsilon[/math]. Which [math]N[/math] would work here? What confuses me is that the inequality [math]\left \| f_{n}(x)-f(x) \right \|<\varepsilon [/math] holds whenever [math]n>N[/math], but that [math]N[/math] may very well depend on the [math]x[/math] in question.
Here's another cat for you and anyone else willing to help.

>>10963564
Well at one point, you do need to use the fact that [math](f_n)[/math] is Cauchy with respect to the sup norm.
So far, you have only used the fact that it is pointwise Cauchy, but being sup-Cauchy is actually much stronger.
So start with [math]\varepsilon > 0[/math], and let [math]N > 0[/math] such that, for each [math]m, n \ge N[/math], [math]||f_m - f_n|| \le \varepsilon[/math].
Now, for each x in U and all [math]m,n \ge N[/math], we have [math]|f_m(x) - f_n(x)| \le \varepsilon[/math].
Letting m go to infinity, we then get [math]|f(x) - f_n(x)| \le \varepsilon[/math], for all x in U and [math]n \ge N[/math].

>>10963645
That helped. Thanks anon, here's a cat for you.

53^1326 (mod 97)

Fermats little theorem:
53^96 = 1 (mod 97)

1326 = 96 x 13 + 78

53^1326 = 53^96x13 x 53^78 = (53^96)^13 x 53^78 mod 97

53^78 mod 97

now what? If I repeat the same theorem thing it just becomes 53^78 again(?)

How do I prove that if a real sequence (Xn)_n has two limits, a and b, then a = b?

>>10962360
The term in set theory and other such logical platforms is "proposition". In naive set theory at least, a set can be constructed from a proposition by taking all elements for which the proposition is true, and then one could do this for each proposition and then take the intersection of the sets. There are generally more restrictions on valid propositions and creating sets in non-naive set theories.

>>10963876
I'm not a mathematician but that sounds like something that would follow from the axioms of the space/number system you are working with. Unless "real sequence" is one of those weird category theory things.

>>10963876
specifically, well-ordering axioms possibly...

>>10963876
Suppose [math]a \neq b[/math] are such that [math]x_n \to a, x_n \to b[/math] when [math]n \to \infty[/math]. Let [math]\varepsilon = \frac{|a - b|}{3}[/math], and let [math]K_a, K_b \in \mathbb{N}[/math] be such that [math]|x_n - a| < \varepsilon[/math] when [math]n > K_a[/math], and [math]|x_n - b| < \varepsilon[/math] when [math]n > K_b[/math]. Now, let [math]K[/math] be the larger of those threshold indices. What happens when [math]n > K[/math]?

>>10963898
Separation axioms.

>>10963885
Real sequence should just be than n -> R I think. My idea is to use delta-epsilon to find a limit for each one-hand side and somehow show that they the limits are equal to another

>>10963900
I'm stupid so I have some questions. Why did you make epsilon = (|a-b|)/3, and what is K_a and K_b? And why make them elements of the natural numbers?

>>10963914
Follow the advice of the anon above and also note that if [math]\left | a-b \right |<\varepsilon [/math] for every [math]\varepsilon [/math] then [math] a=b [/math].

>>10963935
Those K's are indices, so they are natural numbers. x_1, x_2, ..., x_(K_a), and so on. Why divide by 3? Check what happens with that K.

>>10963950
epsilon becomes bigger or equal to |x_n - a| and |x_n - b| ?

>>10963935
If you do not draw something, you are probably not going to understand it.
The point of taking this value of epsilon is that if you take intervals of radius epsilon around a and b, then they are disjoint (anything smaller than |a-b|/2 would do).
But all terms of the sequence must eventually lie in both intervals simultaneously if it converges to both a and b, which is contradictory.

>>10963982
[math]3\cdot \varepsilon = |a-b| = |a - x_n + x_n - b| \le |a- x_n| + |x_n - b| < \varepsilon + \varepsilon = 2\cdot \varepsilon [/math].

Does anyone know where I can find solutions for this book?

>>10963997
There may be no solutions manual out there. You can always try searching for solutions to specific problems.

>>10963876
Because for any epsilon, there is an N1 for which for n > N1, |a - xn| < epsilon/2 and an N2 for which for n > N2, |a - xn| < epsilon/2.
Then for n > max(N1, N2), |a - b| <= |a - xn| + |xn - b| < epsilon.
Since epsilon was arbitrary, |a - b| = 0.

>>10963900
>>10963950
>>10963988
>>10963989
You're making this much more complicated than it needs to be. Stop confusing the dude.
>>10963982
Just look at >>10964007 , this proof is literally 2 lines.

>>10964007
Fuck, the N2 inequality should say |b - xn| < epsilon/2 of course.

>>10961422
So it is true, yukariposter is a woman. Fascinating. I recall about last year asking you if gym is worth it and you told me its good and you go there with your bf. Either way, man or a woman, gay or straight you seem to know your shit. Thanks for your detailed replies on all of these threads, we're grateful.

is it even possible to do math on temperatures above 20Cº?

>>10963900
what's with some /mg/ posters...
He's talking about real numbers, not some arbitrary topological space. The well-ordering axiom does what the separation axioms do for topological spaces and more. Not everything has to be general.

>>10963876
lim x_n - lim x_n = lim (x_n-x_n)=0, by the linearity of the limit operator.

>>10963791
help

>>10964038
I'm not the "I/we're not a 'he'" poster, but I appreciate your gratitude.

>>10964007
thanks!

>>10964162
girl (male)

I need a good lower bound for [math]\log_2 (1-x)[/math] for small [math]x \in [0,1)[/math].

Any ideas?

>>10964257
[math]
lim_{x \to 1^-} log_2(1-x) = lim_{x \to 1^-} log_2(0^+) = -\infty
[/math]

>>10964287
I'm not sure what anon even wanted since that function is decreasing so maximum value is attained for x=0.

>>10964293
I think he meant for x close to 0

>>10964293
anon needs a lower bound for the codomain of his function
It's codomain is clearly [math] (-\infty; 0] [/math]

>>10964287
>>10964293
umm, let me be more specific. I need a simple function [math]f(x)[/math] such that [math]log_2(1-x) \geq f(x) \; \forall x \in [0,1)[/math] (or say, [math]\forall x \in [0, \frac12][/math] or something) and is particularly accurate around 0.

>>10964302
f(x)=x

>>10964302
how simple are we talking?
[math] f(x) = log_2(1-x) - 1[/math]
would do the job

>>10960662
>2019
>still using expressions like "dependent variable"
rly ?

>>10964307
this is not even true, lmao

>>10964302
[math]-x-\frac{x^2}{c}[/math] will do the trick for any c > 0

>>10964302
why do you even ask for this? it seems so weird
what are you trying to solve

what is the correct way of computing equations like
a^-3=8 ?

>>10964382
don't kids do this in elementary school?
I'm not judging you anon, you've got plenty of time to learn

>>10963878
Thank you

>>10964382
take log with base a on both sides
if you have no idea what I'm talking about, feel free to ask

>>10964391
okay maybe something easier for you because the previous one was too hard
1+a=3

>>10964422
a=2

>>10964382
plug it into wolframalpha

>>10964382
That implies [math]a^3=\frac{1}{8}[/math], which has one solution over [math]\mathbf{R}[/math], and three solutions over [math]\mathbf{C}[/math]. No logarithms required.

>>10964309
it's actually really common in experimental physics and just about any of the natural sciences, not everyone in the world is a mathematician

Could someone offer some intuition regarding the second countability requirement for a topological space to be a manifold. The Hausdorff condition and the space being locally euclidean are much more easily motivated, non-Hausdorff spaces are a bit unnatural in a lot of ways, one of them being the possibility of a sequence in the space to have multiple limits and such. Also if the space is locally euclidean you can sort of "locally do analysis" in R^n which is nice since a lot of theory has been developed in analysis on R^n. But why the second countability? I cannot really offer a satisfying reason why would one add that requirement to the construction. Anyone got any ideas?

>>10963997
>Shlomo Sternberg
literally the most jewish name in existence

>>10963997
>>10964500
yeah.. pretty sure that book just gets memed because of the name..

>>10964488
Consider S1 times an aleph one set with the discrete topology.
Absolutely great manifold, innit?

>>10964488
It ensures that your space is of "reasonable" size.
Iirc, you need second-countability to define partitions of unity for any open cover, which are very handy tools to do all sorts of constructions (typically gluing local data)

>>10964488
check out the prufer manifold
>>10964522
someone could say that's not too bad: just a lot of disconnected copies of S1. Also S1 x (countable set with discrete topology) is not much better, and satisfies 2nd countability.

I don't know why, but for some reason I find it extremely pleasant that there is an isomorphism [math]H_*^G( *, M) \cong H_*(G, M)[/math] for any [math]G[/math]-module [math]M[/math]. Is this sensation what people refer to when they speak of mathematical beauty?

Choosing thesis subjects is hard when you have no hard loyalties anywhere.

>>10964500


>>10964505

Do you guys think I should continue reading it? If not, are there any better books for multivariable/vector calc?

>>10965155
Munkres Analysis on Manifolds is the best

How do I find the number of odd ,4th order permutations in the symmetric group 6.

>>10964970
>mathematical beauty
This is a meaningless notion.

>>10965515
>How do I find the number of odd ,4th order permutations in the symmetric group 6.
What have you tried?

>>10965155
>If not, are there any better books for multivariable/vector calc?

>>10965566
I know that the only kind of permutations that satisfy this condition are permutations like (abcd) (that affect 4 elements and keep 2 of them fixed)
I don't know how to quantify the number of these however.

>>10963876
If a =/= b, |x - y| =/= 0. Let this distance be ε, and consider the intervals (x - ε/2, x + ε/2) and (y - ε/2, y + ε/2). these sets are disjoint. Now there exist N, M natural numbers such that for every n > max(N,M), xn is in both of these open intervals, which is impossible.

Generally, a sequence in a Hausdorff (or even T1) space will converge to a unique point if it converges at all.

>>10963935
>And why make them elements of the natural numbers?
Read the definition of sequence convergence

>>10964050
>The well-ordering axiom does what the separation axioms do for topological spaces and more
What do you mean?
>Not everything has to be general.
What's so general about what he provided?

Ok /mg/, two questions.

1. I was allowed to take a lower form of multivariable calculus for my degree, but I know the prof is shit so I want to read a textbook alongside the course. I really want to make sure I get the theory more than anything. Any recommendations?
>The course is intended for students in mathematically rich disciplines. Parametric curves, arc length and curvature. Functions of several variables. Level curves. Partial derivatives, gradient, divergence and curl. Max/min problems. Double and triple integrals, line and surface integrals of functions and vector fields, and applications. Green's, Stokes, and divergence theorems.

2. Are there any fields of pure math left where an undergrad student could do any semi-meaningful research. It could be a nearly trivial result.

>>10965585
This is just combinatorics at this point. How many ways are there to seat 4 people at a circular table from among 6 people?

>>10964488
The way I see it, it fulfills the same function as compactness does to metric spaces in real analysis. It ensures that the space is not crazy big.

>>10962352
Kevin "this would be easy in ZFC" Buzzard

>>10964050
>The well-ordering axiom does what the separation axioms do for topological spaces and more
dude what the fuck are you talking about, the well-ordering principle has literally nothing to do with R being hausdorff

What is one book that if I read I'd be able to mogg 99% of mathfags?

>>10964050
>The well-ordering axiom does what the separation axioms do for topological spaces and more.
What the fuck are you saying man?

>>10961640
If you failed high school I'm going to assume you failed algebra. Start with, "Algebra by I
M. Gelfand". "Basic Mathematics", requires a knowledge in a few high school level maths to read.

These are both integrals of the same thing.
How is it justified that they follow different paths depending on whether you opened the brackets before integrating or not?

>>10967023
+c

>>10967023
They are both antiderivatives of h, but they assume different values at 0, hence they differ by a constant. One curve is a translate of the other by 1/6 in the y-direction

>>10961072
>>10961422
Wait wait wait so the physics 2hu anon is also the not a "he" anon? Big if true
Are you a "she", or something else? Cis or trans?

>>10967042
He's a fat chinese gay man.

When one tries to determine all the solutions in the positive integers of a certain Diophantine equation and you got the general formula for x and y, how do I determine the range so that that x and y becomes positive?

whats the deal with commutative algebra?

>>10967066
Take a ring, assume it is commutative, and proceed.

>>10967060
Nvm, I did a mistake which made me doubt everything, I figured it all out. :)

how do you think about rings, ideals, etc...
apparently thinking of integers is a bad idea

>>10967079
purely think about their associated properties and the hierarchy between them

>>10967066
>>10967079
geometry

>>10967083
what do you mean?

>>10967090
commutative algebra is dual to (affine) algebraic geometry. On the geometric side almost all concepts from CA have a simple intuitive description. However, the theorems that say that your geometric intuition is well-founded are generally easier to prove in terms of commutative algebra. Studying one without the other is fruitless imo.

>>10966837
Villani's brick on optimal transport.
>>10967079
Ye.
If someone says "commutative domain", think Z[x].

>>10966998
I think due to highly positive Amazon reviews that book is fairly good, but I've tried it and I couldn't wrap my head around his method of counting and so on. Even tho it's elementary it feels like number theory for highschoolers to me it's hard as fuck

>>10967183
Just keep trying and take breaks anon, sure you can do it. You can skip it if you need to. That part is not really needed for the rest of the book.

>>10967183
fun problem

Marry, Fuck, Kill, operator topology edition.

Your options:

Norm, Strong, Weak

>>10967560
Marry norm, fuck ultrastrong, kill strong.

>>10966026
>I really want to make sure I get the theory more than anything
With the exception of a few theorems like Stokes (which rely on the theory of differential forms to really explain properly, and aren't proved in calculus courses anyway), the "theory" behind multivariable calculus is almost trivial. It's all just extensions of tools you already developed in single-variable calc. There's nothing mechanically new in the course; what's new are the concepts, so it's better to focus your effort on taking the time to actually grasp the geometric intuition of what you're doing.

>Are there any fields of pure math left where an undergrad student could do any semi-meaningful research. It could be a nearly trivial result.
Almost all of them, honestly, as long as your standards for "semi-meaningful" aren't too high.
Unless you are prodigiously talented, you aren't going to be able to produce independent research as an undergrad, but the majority of profs have some small research projects sitting around that are accessible to moderately talented 3rd/4th year students. If you're very lucky you might get a paper out of one of those but it's more realistic to expect that you'll spend a semester proving one lemma in your professor's paper or collecting computational data. Still great to have research experience if you want to grad school, and pretty often you even get paid for it.

>>10967560
Marry weak, fuck norm, kill strong.

Can someone explain to me what are benefits of representing numbers as continued fractions ?

>>10967817
Thanks, that's really helpful. By semi-meaningful, I just meant non-trivial or better, so yeah, lemmas are cool, I'm down for that. I heard that the one pure math prof is starting to do some new research so I think I'll go talk to him about that.

I had a job interview today for a money changing Travel company.

Only realised earlier I was supposed to multiply all the conversions and not divide and I've fucked it all.

Can anyone explain how EM is equal to [math][EA|EB] [/math]
This is from Artin’s first chapter on matrices.
E is supposed to be elementary matrix so if author is multiplying with M which is [math](m)x(n+1)[/math] in dimension then E must be [math](mxm) [/math]. But how is he making a block inside with A being mxn and B being nx1

>>10960203
Saved and thanks, anon.

>>10968272
can anyone verify B’s dimension should be mx1

>>10968306
Will do better than saving meme and laughing at it? Like answer this person’s question?
>>10968272

How rough is linear algebra? I got an A in all my calculus classes so far and I need to take linear and differential equations in order to transfer to a university.

>>10968308
Verified.

>>10967560
Marry weak, kill strong and fuck Mackey

>>10968359
p easy

>>10968272
B is just the vector of results, and you need a result for every equation. think of it like every row is something to the effect of:
3x+5y+8z=9
4x-2y+z=0
x+0y-z=2
is the pic attached, A is the matrix [(3,5,8);(4,2,1);(1,0,-1)] and (9,0,2) is B.

dont u think the dimension of B mentioned shout be mx1?
A is mxn. X should be nx1 so B must be mx1. Is it not a typo?

>>10964038
yukari is a well known tranny and attention whore, most of (her?) comments are not useful and are generally just attention posts. Don't encourage her.

>>10968523
>her

>>10960172
day 1: i just realized my plan before of math history is unironically way harder than pic related, so i am switching to it. it's about time someone finally went through the entire thing. apart from documenting everyday i will also post here whenever i finish a book.

>>10968678
o yeah btw, in case you guys are wondering, i have the original website where the meme spam image was posted. the first book in the series i am working on going through is zorich's analysis because the original russian dude who proposed this curriculum gave it as recommended

>>10968371
>fuck Mackey
who?

Quickly: Need a topic in number theory suitable for an undergraduate honour's project (30-40 pages)

>>10967079
It depends on the king of ring your looking at. General rings are too broad to have an all-encompassing example for, in the same way that topological spaces are too broad to imagine a "generic" space. Commutative rings I pretty much always think of as quotients of polynomial rings over Z or a field.

>>10968767
https://en.wikipedia.org/wiki/Mackey_topology

>>10967884
Making explicit the sequence of good/best rational approximations.

>>10968272
It's effectively by definition, really.

If A is a matrix (say 3 by 3) and B (say 3 by 10), then computing A·B (via the definition of the matrix product) amounts to taking all 10 columns of B as vectors, and applying A to each of them.

I'm interested in the fixed points of the exponential function.
We have [math] \exp(i \pi/2) = i [/math] and of course [math] \exp(i \pi/2 + d) = i \cdot \exp(d) [/math], so they repeat.
Are there other interesting points than these?

Also, I tried looking at it via a plot (absolute value, though). Is there a way in which I can parametrize that curve? I.e.
[math] t\mapsto \gamma (t) [/math] such that [math] |\exp( \gamma(t) )| = |\gamma(t)| [/math] . That might be the same as [math] \exp( \gamma(t)*\cdot \gamma(t) ) = \gamma(t)*\cdot \gamma(t) [/math]

And then maybe even without the absolute value.

I'm interested in the fixed points of the exponential function.
We have
[math] \exp(i \pi/2) = i [/math]
and of course
[math] \exp(i \pi/2 + d) = i \cdot \exp(d) [/math],
so they repeat.
Are there other interesting points than these?

Also, I tried looking at it via a plot (absolute value, though). Is there a way in which I can parametrize that curve? I.e.
[math] t\mapsto \gamma (t) [/math]
such that
[math] |\exp( \gamma(t) )| = |\gamma(t)| [/math]

That might be the same as
[math] \exp( \gamma(t)^*\cdot \gamma(t) ) = \gamma(t)*\cdot \gamma(t) [/math].

And then maybe even without the absolute value.

I meant

[math] \exp( \gamma(t)^* + \gamma(t) ) = \gamma(t)^* \, \cdot\, \gamma(t) [/math]

I meant

[math] \exp( \gamma(t)^* + \gamma(t) ) = \gamma(t)^* \, \cdot\, \gamma(t) [/math]

and whether the pi/2 is in the exp or not is not too important to me

>>10969103
0 is the only fixed point of exp

>>10969159
nvm

Ok I'm bad at proof but I think I managed to make one that holds - please tell me if does or not:

to prove:
Establish the statement: Every integer of the form n^4 + 4 with n > 1, is composite.

My proof:
Since n^4 + 4 = (n^2)^2 + 2^2 , and (n^2)^2 and 2^2 is both always even, then (n^2)^2 + 2^2 is always even too. Since it is always even, you can always divide it by 2, and must thus be composite. (2 is an even prime but the formula can never be 2).

>>10969191
3^4 is not even

>>10969203
fuck. but 9 is it divisor, so it's not a prime? i guess i could go from there

>>10969191
As anon >>10969203 said, there is a gap in your reasoning.
Maybe try using induction. The base step is easy to check:
If [math] n=2[/math] then [math]n^{4}+4=2^{4}+4=20[/math] which is composite. Now assume this holds for some [math]n>2[/math] and proceed from there.

>>10969207
I don't know. My first guess would be induction, but it doesn't look completely straight forward

>>10969208
>>10969213
Okay, what about:

Since (n^2)^2 and 4 always have a divisor (n^2 and 2) that is something else than 1 or itself, and the sum of two values that are divisible is also divisble by something else other than 1 and itself, n^4 + 4 is composite

>>10969224
fuck that is not even true. reeee

>>10969224
You have to prove that they have a common divisor (other than 1). You're claiming that since a is composite and b is composite, then a+b is composite, which isn't necessarily true. Look for example at 10+9=19, both 10 and 9 are composite but 19 isn't.

>>10969229
yeah i just realized

>>10969191
Try completing the square, i.e [math]n^{4}+4=(n^2+2)^{2}-...[/math]

>>10969233
-((4n)^(1/2))^2, but so what? I cant see what to do from here

>>10968678
That would be an interesting experiment, maybe you should make a blog if you're really serious about it, instead of posting on 4chan, where the messages are eventually going to disappear.

>>10969247
Notice what you get by completing the square is a difference of squares.

>>10969251
oh, so a difference between two squares is always composite? why? because x^2+y^2 = (x+y)(x-y) (unless x-y = 1), but i dont see why they neccessarily are composite

>>10969261
x^2-y^2* ofc

>>10969261
[math]x^{2}-y^{2}[/math] is always composite (assuming x,y are integers), because it is equal to a product of two integers.

>>10969267
oh. yea, ofc. lol, thx :)

>>10969268
No problem anon, we're here to help and encourage.

>>10968678
dude, if you actually manage to go through one of these meme guides and you document it, you will be the absolute hero of /sci/ and /mg/, trust me

Is Anki good for learning maths?

I want to solve a very large, very sparse system of linear equations Ax=b, where A's entries are either 0 or 1. How can I exploit this fact to solve the system faster?

>>10969318
>very large
What size are we talking here?

>How can I exploit this fact to solve the system faster?
The sparsity can be exploited by using an appropriate data format.
Together with a decent enough iterative solver, BiCgStab is a commonly used one, you should be able to approximate your solution.
The chance is also pretty good that it's condition number is rather small.

>A's entries are either 0 or 1
I do not think that it can be exploited that it has just 1 as non-zero entries. You might get to save a single floating point computation per iteration per matrix element, but the effort of optimizing is most likely not worth it.

>>10969267
> because it is equal to a product of two integers
One of those integers might be 1. In fact, every odd number (prime or composite) can be written as the difference of consecutive squares. If x=2k+1, (k+1)^2-k^2=x. So you need to exclude this.

In this case, n^4+4
=(n^2+2)^2-(2n)^2
=(n^2-2n+2)(n^2+2n+2)
=((n-1)^2+1)((n+1)^2+1)
This is composite provided that neither factor is 1. The first factor is 1 iff n=1, the second iff n=-1. So both factors are >1 (and thus n^4+4 is composite) if n>1.

>>10969318
Reduced row echelon matrix?
Read Artin frist chapter

>>10969445
You're right, but if applied to the problem at hand the method yields the correct result because it forces both factors to be different than 1.

How important would you guys say is Sylow theory in field and Galois theory? I find it pretty tedious and I'm willing to skip it if it's not necessary to produce results in Galois theory. I'm doing calculations of Galois groups of fields right now and so far I didn't need any Sylow theorems, not even once. So can I skip it for now?

>>10969250
oh i totally am. though i wont post it till i get passed some of the more babby bits cause i dont want to out myself as some plebian

>>10969564
Wouldn't that take years? Like, 10 or more? I'm 4 years into properly learning math and I'm still waaay of.

>>10969499
>Reduced row echelon matrix?
Lol.
Using Gauß is retarded, learn numerical linear algebra.

>>10969543
i'm pretty sure you can skip it

>>10969543
Idk, Sylow theory is obviously relevant to the study of finite groups in general. If you are trying to list the subgroups of a finite group (to find intermediate extensions), then you might be interested in knowing the Sylow subgroups and how many there are. But obviously, this is only interesting if you are working with relatively large complicated groups

>>10969564
I think making some kind of journal where you document everything math related you do, even the less important thing, could be beneficial to you so that you can see what progress you've made.
You don't have to go into the details, but make sure you write what you do each day, your thoughts, remarks, copy part of articles you find interesting etc. I'm thinking about doing something like that myself.

>>10969568
How old are you ? I wasn't interested in math in high school, but I started to look into it 2 years ago without ever taking a serious commitment to learn it. I really got into it a few weeks ago but it's kinda depressing to think that I've lost two years of learning (I'm 21 btw), but eh better late than never I guess.

>>10960933
Why is it always high-energy physics

>>10969543
Knowing the basic too Sylow theorems and their applications is essential to study finite fields.

>>10969688
I'm 26, soon to be 27. I couldn't go to uni after highschool because we were broke and I had to find work. I did learn a solid bunch of math during that time, so as I started uni in 22 I didn't really have that much trouble with the uni work. I'm doing masters now, I'm primarily focused on commutative algebra and algebraic geometry.

It takes a lot of work, you really have to learn and relearn the material a lot of times. I went through 5 different books on algebra for example. Only after a couple of reruns I've started to really get comfortable with the ideas. And it's also frustrating, you are basically making mistakes all the time.
Anyway good luck anon, ganbatte

>>10969743
Inspiring story thanks, I hope you succeed in your endeavour mate.

How do I start reading papers? Do I just pick a field I'm interested in and look for the classic papers? Is it normal to not understand most of the material at first, or does that mean I should read some prerequisite text?

>>10969791
I think the proper time to start is when textbooks you're using start referencing specific papers.

>>10969308
Depends, for problem solving? No, unless you can't look up definitions and have trouble remembering them.
In general it's decent for learning a number of definitions or proof sketches, for example in preparation for an oral exam.
Specifically if u understand everything but have problems remembering details.

>>10969797
>pick up literally any algeo textbook
>it references Serre's Faisceaux coherents

>>10969857
Literally any textbook tangentially related to measure theory or applications references Rudin too.

Jesus christ, algebraic topology is a fucking mess, is it really useful? I have no idea why I'm learning all of this. Interested in mathematical physics and my profesor wants to get into field theory, but I can't see why is this useful at all.

>>10969894
When Einstein theorised relativity in the 1910's, differential geometry was mostly seen as something amusing but without too much practical impact, because for all the problem we had so far, a classic, rigid 3D space was more than enough.
Then we realised we were wrong and today differential geometry is integral to our understanding of modern physics. Same could be said for functional analysis and QM.

Point is, you may never know when a mathematical tool is going to be used until someone has a stroke of genius and applies it. As for algebraic topology, topology in general seems to be a trendy topic (if the PhDs offered talking about it are any indication), and the algebraic approach is immensely powerful.

>>10969894
what exactly are you learning ?

>>10969964
Do you have particual exampkes of algtop used in physics?
>>10969969
Basic homology and homothopy. Starting Cohomology next week. I understand that the topological data I get with constructions are usuful for clarification theorems, which in itself are interesting, I just don't see why this data is important in field/gauge theories or whatever.

>>10969975
https://www.findaphd.com/phds/project/topological-mesoscopic-superfluidity-of-3he/?p104533
https://www.findaphd.com/phds/project/applications-of-topology-to-soft-matter-physics/?p107820

Topology is also useful for the study of fluids, as it describes "global" properties of the fluid. V. Arnold wrote a book on it.

>>10968678
>introduction to quantum mechanics (Kostrikin-Manin)

>>10970008
Thanks lad.

>>10968828
never done topology before.
currently doing LA.

Is there an infinite dimensional version of a manifold? How does that look like?

>>10969894
Topology problems are hard, in the sense that a naive approach would get you nowhere. You should think of algebraic topology as a collection of easy to compute gadgets that help you tell two spaces apart, among other things.

>>10969975
Eberhard Zeidler wrote 3 huge books about QTF, using algebraic topology as core for many theories.
Quantum Field Theory
I: Basics in Mathematics and Physics

Has anyone had success using a theorem prover like Coq for writing notes and doing exercises when going through a maths text?

>>10970083
it's the linear algebra book that's in the pic. the creator of the list is a massive mathematical physics fag so the books he recommended usually try connecting rigorous mathematics to physics from time to time. this one in particular had sections dedicated to intro qm stuff. im going to read that one after zorich

>>10969334
>What size are we talking here?
On the order of a few billion rows/columns.

>>10970737
use agda

How do you study /mg/? Just paper and pen?

>>10970916
>Just paper and pen?
LaTeX

I'm taking topology and real analysis at uni this fall what should I expect

I am reading Intro to commutative algebra by Atiyah.
In pic related, G is an abelian topological group.
Could someone explain me what is the topology on the completion G^ of G ?

>>10968359
>linear algebra
Possibly some of the easiest math i had to learn at Gymnasium.

How the fuck do I solve mixing problems, I feel retarded

Is there a way to do this with algebra and not using the quadratic formula or factoring? You can get 13/5 but how do you do it to get the -1?

>found a breakthrough matrix preconditioning algorithm
>lets me solve giant finite element simulations on my six year old laptop
>not going to tell anyone about it
>going to use it to become iron man instead
And nobody can stop me.

>>10971428
Iron Man isn't the suit, honey. He's the brains, the money, the personality, the man.

>>10971425
(5t-4)^2 = 81
=> sqrt((5t-4)^2) = sqrt(81)
=> |5t-4| = 9
=> 5t-4 = 9 or -(5t-4) = 9
=> 5t = 13 or -5t = 5
=> t = 13/5 or t = -1

>>10971445
Why did you do abs val instead of +-sqrt?

>>10971452
>Why did you do abs val instead of +-sqrt?
sqrt(a^2) = |a|

Math noob here taking a comp sci course.

Explain this please (n is subscript) for the function S

Use a program to evaluate and tabulate both
Sn(1)
and
e−Sn(1)
for n = 7,8,9, . . . ,16.

What does subscript mean and what is this asking me to do? I thought the parentheses meant the value being passed in like f(x). But it's saying to solve this for s(1) with a subscript. I just don't get it. Where is the subscript being used in the function?

>>10971485
edit i think it might be saying to extend the infinite series they gave us to have up to 16 additions to it.

>>10961675
>>10961695

> reads the wikipedia page once
> repeats the slander by the media

Why did you fucking do this? The guy dedicated his life to math, wrote tons of books, but because he asks questions about a controversial topic, the media makes sure that's the only thing you know about him.

"aids denier" is a label intended you to think he is a bad guy. It is not a criticism.

Please fuck off and stop quoting wikipedia pages.

>>10962352

> believing math is essentially a formal system.

Computer formalized proofs are less convincing because no person can understand them. It's an excruciating amount of detail that you just have to trust both the computer, and the translation process. How do you know it was typed in correctly, doesn't this double the possibility of error?

Just give it up. Math is a social process. Proofs are mostly about the insights behind them, not the formalism.

>>10968678
this is the biggest meme. Galois theory in high school is absurd unless you attend a Massachusetts prep schools and your parents are professors.

if you did linear algebra in high school, you are well on your way to being a mathematician.

>>10970834
Okay, but much more important is the NNZ, if they are in the range of a few billion too you should be able to just use the already existing LAPACK BiCGStab for sparse matrices.

Roughly guessing one matrix vector multiplication takes a minute to an hour or so (if it really is just a *few* billion) so I would assume it will take maybe a couple of days to a week to get a somewhat decent residual.

If you want more you have to look into how you can parallelize the problem and get your hands on some decent hardware.

>>10971485
>>10971497
They are saying to write down a table of the values S7(1), S8(1), S9(1), ... S16(1). And then those numbers subtracted from e.
What is S9(1)? It's the sum from n = 0 to 9. What's S14(1)? It's the sum, but now from n = 0 all the way to 14.
So right there you do have the correct expression for S7(x).
Note that these are all just polynomials. Also, when you see how close to e these get, you'll see a bit better why we care about them.

>>10971535
Oh look, Lang's back.
Don't you have another undergraduate level introductory text to slave over? The adults are trying to discuss real mathematics here.

>if you did linear algebra in high school, you are well on your way to being a mathematician.
can you imagine someone posting this unironically
lmao

>>10971734
I don't get it. What is Linear Algebra even?

ok /mg/, i've posted somewhere else but here it's again.
I wanted to learn about abstract algebra since first year of uni, tried many times and dropped because i couldn't understand some proofs, and shit is so deep i can barely get what it is all about. I can't see the implications of what i'm learning.

I've been trying to learn this stuff for the Galois theory and it's implications on solving polynomials by radicals.
>why?
Don't know, just for the sake of knowing stuff.

any advice for this?

>>10971732
Why the hate on people who write undergraduate textbooks ? It is an important job and somebody has to do it.

>>10971951
JUST DO IT.

Read a book and do the exercises?

>>10971959
any recomendations for a brainlet?
I've tried starting reading Herstein book but man, wth

Im a brainlet, how do I solve this? My answer keeps coming out wildly different from the book.

>>10961000
Feel the need to understand.
Do you remember law of cosines by heart?

>>10970973
Sorry I wish I could help. I am topologylet right now.

>>10971971
You have to use ODE.
First differential is gram per litre.
Second is litre per min.
Just try to form an identity and solve. I have forgotten ODE.

>>10971988
I have no idea how to set up the problem. I get to dA/dt = 30 - A/200+t but then i solve the standard form and i my number is way off.

How do you deal with this shit, when there's nothing to work with in the first place?

>>10972096
farts

Is it possible for someone with a 2.5 GPA in an unrelated subject to get into a top Math PhD program (say Princeton or Harvard) by writing a really impressive paper and submitting it to a professor there?

>>10972096
Fourier transform

>>10970973
bump

>>10972144
>submitting it to a professor there
Make it three or four good papers, submitted to journals, plus not being old as shit.

I really like Lax's Multivariable Calc, but I'm not sure whether it's an actually good textbook or just my autism acting up about how he explains linear functionals, matrices and the rest through the long route, so I'd like a second opinion before I start recommending it to people.
Has anyone here read it? What did you think about it? Do you think it's a solid introductory text?

>>10972096
Mathematica tells me

[math] \int x^2 \sin(k x) \mathrm{d } x= -\frac{1}{8k^2} \left(-\frac{\cos (2 k x)}{8 k^2}-\frac{x \sin (2 k x)}{4 k}+\frac{x^2}{4}\right) [/math]

Suggests to me you use those rules about powers of trigonometric functions and their frequency, i.e.
cos^2(x) => cos(2x) + ...
and those things

>>10972096
Mathematica tells me

[math] \int x \sin(2kx) dx = -\frac{\cos (2 k x)}{8 k^2}-\frac{x \sin (2 k x)}{4 k}+\frac{x^2}{4} [/math]

Suggests to me you use those rules about powers of trigonometric functions and their frequency, i.e.
[math]cos^2(x) [/math] to [math]cos(2x) + ...[/math]
and those things

∫ x sin(kx)^2 dx
on the left hand side I mean

>>10972096

Notice the bounds are symmetric about the y axis, so your going to want to appeal to symmetry. [math]\int_{-2}^{2} x\sin^2(x) \,\mathrm{d}x = 0[/math] since [math]f(x)=x[/math] is an odd function and [math]\sin^2(x)[/math] is even, so the entire integrand is odd.

>>10972225
>>10972096
Express the square as double argument and integrate by parts.

>>10972164
>>10972225
>>10972237
Y'all are being calculation monkeys. Just think about what the graph of the integrand looks like. The value of the integral is zero.

>>10972000
> I have no idea how to set up the problem. I get to dA/dt = 30 - A/200+t
I get that, where A is the total amount of chemical in grams.
Solving gives A=(15t^2+6000t+C)/(t+200). A=1600 at t=0 => C=300000 => A=(15t^2+6000t+300000)/(t+200) which equals 35700/13=2746g at t=60. The volume is 3(t+200) = 780L at t=60, giving a concentration of 35700/10140 = 595/169=3.521g/L.

>>10970907
What's the difference?

>>10972144
delusional

>>10972295
>>10972144
Isn't that Ed Witten's story? He was some nobody studying history then somehow got into Math/Physics and ended up with a Field's Medal.

>>10972293
style points and mixfix notation

>>10972144
Possible? If you're being pedantic then yes. Otherwise, no.

>>10972323
>some nobody
No, he was a jew with the phenotype and connections, completely different.

>>10972144
You wouldn't be asking this question if you actually saw the admissions process. No dude, you have no chance whatsoever. 0%.

We have that

[math] \mathrm { det } ( \exp(A) ) = \exp( \mathrm { tr } (A) ) [/math]

for matrices A where

[math] \exp(A) = \lim_{n\to \infty} \sum_{k=0}^n \tfrac {1}{k} A^k [/math]

Can we say something nice already about

[math] \mathrm { det } \left( \sum_{k=0}^n \tfrac {1}{k} A^k \right) [/math]

?

>>10970973
i'm not sure but you'd probably want that that Φ is a continuous group homomorphism. so my bet is that you just take the finest topology such that Φ is continuous

>>10965562
spotted the heretic

>>10972576
i highly doubt it. consider the very simplest case, when A is a 1x1 matrix A=[a]. Can you say anything nice about [math] \sum_{k=0}^n a^k/k! [/math] , where a is an arbitrary real number?

>>10972815
Good point, thx.

Also, sad.

>>10971734
the insecurity on this board is incredible. guess what, that age you took XYZ won't make you a successful person. most high achievers who get 4.0s, triple major, graduate early, etc, proceed to work an unremarkable office job for the rest of their career.

>>10971821
differential equations and linear algebra are the last math classes undergraduate engineers are required to take. its typically a sophomore level class for non-math majors.

if you mastered calculus and linear algebra in high school, you are ready to jump into analysis and abstract algebra as a freshmen, which as I said, is a great start to being a mathematician.

larpers and insecure fags on this board like to pretend they are hyper geniuses who invented analysis from first principles.

>>10972815
>>10972879
Btw. I got to that question by wondering what conditions on A we must impose to guarantee that that finite sum is invertible.
That question is sort of natural, I think, because exp(A) is always invertible while it's not so clear when the sum is. If A is a diagonal n times n matrix, then

[math] \sum_{k=0}^n \tfrac{1}{k} A^k = 1_n + P(A) [/math]
where
[math] P(A) := \sum_{k=1}^n \tfrac{1}{k} A^k [/math]

so this is just asking for a complex value [math] a [/math] where [math] P(a) = 1 [/math].

>>10972910
-1 I mean

>>10972905
some sources (books or vids or whatever) reccomendations anon? i'm the anon in >>10971951

>>10972884
yes, and so do most imperfect students. in fact, a far far far higher proportion of them.
doing anything worthwhile takes a bit of luck and a bit of persistence. but one can maximize their chances by practicing a modicum of foresight.

>>10971956
>write worse version of *insert existing classic here*
>wow i just spent my time so wisely!
lmao

>>10972144
If you are black women or jewish women, transwomen.

>>10970944
Thanks for all the replies guys

>>10973354
I actually suspect this is even truer for these early academic performers. they are good at following rules and staying on track, not doing new or risky things.

>>10973203
sure anon. there is a free and open source text book called Abstract Algebra by Tom Judson. it works up from basic groups to galois theory and is well written.

Why am I reducing -10 and 4 by 2, and -260 by 4? Why not -260/2 ?

Answer was provided, I just don’t understand why we should reduce by some arbitrary multiple. Shit, why not -260/5 ?

>>10973647
are you dense?

>>10973647
I'm not sure I understand what you're asking here :

[eqn]

x = \frac{-10 \pm \sqrt{-260}}{2}

= \frac{(-5 \cdot 2) \pm 2i\sqrt{65}}{2 \cdot 2}

= \frac{-5 \pm i\sqrt{65}}{2}

[/eqn]

>>10973779
Maybe I should have added that :

[math]\sqrt{260}=\sqrt{4 \cdot 65} = \sqrt{4}\cdot\sqrt{65}=2\sqrt{65}[/math]

>>10973542
Stop asking dogshit questions if you want replies.
>>10972910
Working over the product strikes me as easier than with the sum, but I know absolute jack shit about infinite products.

>>10972144
Who in their right mind wants to get a Math PhD?

>>10973605
thanks anon, i appreciate it

>>10968523
butthurt much?

>>10973542
>Thanks for all the replies guys
What did you expect as an answer?

If you want to know what either of theses topics is, read the Wikipedia article, if you want to have book recommendations ask for book recommendations.
But LITERALLY no one here knows what you should expect.

>>10970944
Maybe you should check out the Infinitely Large Napkin, it's a free PDF book written by an Harvard student that aims to give an overview of many higher math topics :

http://web.evanchen.cc/napkin.html

I'm starting my first year in a couple of weeks. What are some practices I should adopt right now in order to maximise my chances of getting into a good grad school?

>>10974856
Hang out with the smart guys who'll still be there in 1 year.
Don't get friends with 5 people, 4 of which will drop the major, leaving you at square one.
Talk with people smarter than you - there's gonna be some higher year people who can and like to talk.

Dedicate 2 hours on reading stuff you don't understand and that's not needed on Friday afternoons. Find out what the actual research in the fields are, not the uni curricula that you'll learn in the next few years. Two hours per week isn't that bad and getting to know what's out there helps even if you don't get it, really.

Start a notebook, or wiki, or blog if that's your thing.

Ask stupid questions when nobody dares to.

Go to events when you're invited.

Show women that you actually like women, don't be a nice guy.

is the sequence (Xn)n>=1 = (n)n>=1 bounded from below? I mean, it will never become negative?

>>10974949
Not being max-lazy in asking your question would help. It's hard to decipher.

>>10974007
Thanks, I came to the same conclusion.
(Sidenote: The finite product is of course also a sum, but det(X^n)=det(X)^n helps a lot)

>>10974949
>(Xn)n>=1 = (n)n>=1

>>10974960
>>10974965

>>10974973
It seems you're actually asking what the expression [math] (n)_n [/math] means.
Otherwise I don't see how you could think it would become negative.

>>10974976
le 1+2+3+...=-1/12 face xD kappa lmao

>>10974976
But I see similar examples where the sequence is not bounded, even when it doesn't become negative. Like n^2 is supposed to be unbounded

>>10974984
Yeah, it's unbounded because it can get arbitrary big. But not negative.

n^2 is bounded from below but not from above.

Start by writing down the formal definitions of each and work with them.
And when you read this, I mean write down the formal definition, not just go back and read them. I know you cunt.

>>10974989
I do read the formal definitions, that's almost all I do, but it doesn't help. How is it unbounded when it can't become negative? How is it not bounded below? How is (n) different from (n^2) when they're both always n>= 1 either way?