/sci/ - Science & Math

I took me 6 months just to finish a few chapters in Dummit & Foote.
Should I just give up?

>>15325000
Number theory?

>>15325003
Abstract algebra.

>>15325000
That's an average pace for textbook self-study, if it's not a little slow. That's one reason why I think lectures are so much better than self study, personally. People who read textbooks as quickly as they would read a novel aren't really doing what I would call "studying" them. When I self-study, I am constantly frustrated by the errata that every textbook is riddled with. I'll lose entire days getting to the point where I say, "I'm not wrong, the author is wrong."

>>15325000
Depends. Are you going slow because you are deliberately going slow and deeply thinking about it beyond what is explicitly written in the text, or are you just a midwit taking hours to solve a single exercise or understand a proof.

>>15325011
Same thing if you're still at the first few chapters.

>>15325033
Midwit.

what *is* 1

>>15325000
checkerinoed

is that self study ?

Hey /mg/. New thread new problem. This one isn't all that good honestly. I solved it and it's honestly just a bit of calculation, not much else. I intended to post another problem but I'm still in the process of solving it so this is what I have for now. Do this is as a small exercise I suppose if you wish. It's from a high school exam.

When and if I manage to solve that problem I mentioned, I'll post it too. It's a lot more interesting in my opinion. But oh well. Good luck to all those who attempt! As always feel free to ask for clarification.

>>15325145
Alright, nevermind! I regret posting this because the problem I wanted to post actually isn't all that difficult and I managed solve it. It's definitely a very nice problem, very fun. I recommend solving this one instead of the last one I just posted but of course feel free to do any, both or none.

So yeah. Good luck to all those who want to attempt this one! It's definitely worth it in my opinion and I hope you agree if you solve it.

Let [eqn]a_1^k + a_2^k + … + a_n^k = a_0^k[/eqn] such that [math]n,k \geq 2[/math] and [math]n,k \in \mathbb{Z^+}[/math]. For every [math]k[/math], let [math]n_k[/math] be the smallest possible value of [math]n[/math] such that there exists positive integer solutions for [math]a_0, a_1, …, a_n[/math]. I have two questions regarding this.

1. We know from the disproof of Euler's sum of powers conjecture that for some values of [math]k[/math], [math]n_k \geq k[/math] is not necessary for positive integer solutions to exist. However, does there exist any value of [math]k[/math] for which [math]n_k > k[/math]?

2. As [math]k[/math] increases, what pattern, if any, does [math]n_k[/math] follow? Is it non-decreasing i.e. [math]n_{k+1} \geq n_k[/math] for all [math]k[/math]?

>>15325126
The first successor ordinal.
An abstraction of the first thing you point at when you count a collection of things by successively pointing at a different thing.

>>15324994
What do you have to say to me about ergodic theory

>>15325145
Let f(a,b) be the number of distinct sequences of jumps from (0,0) to (a,b)

f(0,0) = 1
f(a,0) = 1 for a=1,2,3,4
f(0,b) = 1 for b=1,2,3,4
f(1,1) = f(1,0) + f(0,1) = 2
f(2,1) = f(1,2) = f(2,0) + f(1,1) + f(0,1) = 4
f(3,1) = f(1,3) = 7
f(4,1) = f(1,4) = 12
f(2,2) = 10
f(3,2) = f(2,3) = 22
f(4,2) = f(2,4) = 45
f(3,3) = 58
f(4,3) = f(3,4) = 137
f(4,4) = 364

>>15325325
Sorry anon, are you saying the answer is 364? Unfortunately it's not. It's actually bigger. I don't know if you made an arithmetic error or the way you went about it is just wrong, but I believe you can probably get it correct if you try again. Do your best anon! And thank you a lot for your time and effort! You can do it!!!!

>>15325359
The error was right at the start. f(a,0) and f(0,b) follow the Fibonacci sequence instead of being 1.
So the end result is 556 or something liike that.

>>15325375
You're right! Nice job anon, the answer is indeed 556. Brilliant work! Thank you very much. Now, now, what did you think of it? I solved it in a different way, basically listed all cases and added them up, and frankly I thought it was a bit boring. But your solution actually makes it look a lot cooler.

Talking about things not being boring, >>15325171 this problem is certainly not boring. So I doooon't know. Maybe you could give it a shot? So far you're the only one who attempted my problem in this thread so I'm trying to talk to you as much as possible...
You don't have to attempt that of course, you already solved one and that's of course more than what you owed me. Anyway. Thank you again! I hope you have an amazing day!

>>15325171
I'm getting (n+1)/(4n-2) = (2n+1)*2*Sum[k,{k,1,n}]/((2n+1)*(2n)*(2n-1))

Pick the first point and rotate it to (0,1). There are (2n+1) ways to do this.
Draw a line through this point and the center of the circle dividing the points into 2 sets.
Pick the second point of distance k from the first and reflect it across the line (or do nothing) so it is in the top set. There are 2 ways to do this.
Draw a second line from the second point through the center.
The arc between the two points opposite the first and second are the points that will create a triangle containing the center.
There are k points in this arc.
(2n+1)*2*Sum[k,{k,1,n}] gives the number of ways to pick the 3 points in order that give a triangle containing the center.
There are (2n+1)*(2n)*(2n-1) total ways to pick 3 distinct points in order.

Why does taking k-th roots work here? We get \sqrt{k}{(1+logm/logn)n}\sqrt{k}{|n|^{logm/logn}}, but how do we know \sqrt{k}{|n|^{logm/logn}}<=|n|^{logm/logn}?

>>15325424
Your answer is correct anon! Nice job!
I'm sorry but I don't have much time right now so I'll have to read solution later. Thank you a lot for your time and effort! Have a pleasant day, I'll write to you later again.

>>15325475
First do the substitution [math]m \to m^k[/math]
[eqn] \|m \|^k \leq \left( 1 + \frac{k \log(m)}{\log(n)} \right) n \cdot \|n\|^{k \log(m)/\log(n)} [/eqn]
Now take the k-th root
[eqn] \|m \| \leq \left( 1 + \frac{k \log(m)}{\log(n)} \right)^{1/k} n^{1/k} \cdot \|n\|^{\log(m)/\log(n)} [/eqn]
Now take the limit as k goes to infinity

[eqn] \|m \| \leq \|n\|^{\log(m)/\log(n)} [/eqn]


To see that [math]\lim_{k \to \infty} \left( 1 + \frac{k \log(m)}{\log(n)} \right)^{1/k} = 1[/math] you can use the usual Stolz-Cesaro trick to turn the limit of the root into a limit of a quotient.

[eqn] \lim_{k \to \infty} \left( 1 + \frac{k \log(m)}{\log(n)} \right)^{1/k} = \lim_{k \to \infty} \frac{\left( 1 + \frac{k \log(m)}{\log(n)} \right)}{\left( 1 + \frac{(k-1) \log(m)}{\log(n)} \right)} = \lim_{k \to \infty} \left( \frac{\log(m) + k^{-1} \log(n)}{\log(m) + k^{-1} (\log(n) - \log(m))} \right) = 1[/eqn]

[/eqn]

>textbook says Lean will switch goals once the current goal is completed
>it doesn't

Why is this so stupid? This worked in lean 3 and now it's broken in lean 4?

>>15325520
thanks, i'm sick atm and can barely think, missed that log m turns into logm^k

>>15325539
I was flailing at my keyboard and accidentally pressed -

That solved it

Just press -

>>15325570
False alarm. While it did switch the goal it somehow broke the IDE. I have no fucking clue at this point.

>>15325539
What kind of a weird Lean are you using? I cannot recreate your proof because 'define' is not a Lean4 tactic. In Lean3 you certainly wouldn't use 'define' like that at all.
Anyways I don't get your problem. Your current goal is [math]b' \in B[/math]. You haven't completed it.

>>15325601
Here's what the panel looks like.
I have "No goals". I can't switch goals. The textbook says lean will switch goals automatically. I am using lean 4 extension via vscode

>>15325605
That's because you solved a case or a sub-goal but not the whole proof. You'll get that if you have, for example
"cases (a or b) with
|inl h1 => sorry"
or
"have h : blah := sorry"
and then you don't move your cursor from the end of the line to update the state. Cursor position matters when it comes to the Lean state because it automatically updates the Lean state based on your cursor position.

>>15325621
I'm confused at what you mean. My cursor is below the line.

Math is racist and fascist.

>>15325634
This probably has something to do one of those And.intro applications. You probably did one part of one of them and not the other. I would love to recreate this proof so that I can step through it and see precisely what's going on but the tactics you are using do not exist in vanilla Lean4 with mathlib4. What book are you using?

>professor has to pause to review his notes for minutes at a time, makes constant mistakes on the board, writes verbatim definitions, and only gives canned proofs

yikes, what a waste of time. how the fuck do people like this get a phd

>>15325634
>>15325646
Also, you can structure cases explicitly using 'case' or . notation. That might be what you mean by "switch goals." See the beginning of the tactics chapter in TPIL4: https://leanprover.github.io/theorem_proving_in_lean4/tactics.html

>>15325424
>Pick the first point and rotate it to (0,1)
I meant (1,0)

It's important to note that the statement "calculus is not mathematics, it's just arithmetic" is not accurate and is a mischaracterization of both calculus and arithmetic. Calculus is a branch of mathematics that deals with rates of change and accumulation, and is a fundamental tool in many fields, including physics, engineering, and economics, among others.

The statement may be made by people who have a limited understanding of mathematics or who are not familiar with the complexity and depth of calculus. It's possible that they may have had a negative experience with calculus, perhaps finding it difficult or uninteresting, and may be expressing frustration or disdain towards the subject.

>>15325656
ahm how do you have non-verbatim definitions

I saw something about someone saying they wanted to smile but couldn't, which is probably an expression of ideological solidarity.

>>15325832
Calculus uses limits which are *definitely* not arithmetic.

>>15325126
a grammatical form of the indefinite article

Has Erdos written any books?

What are some magic tome like math textbooks.

>>15325646
https://djvelleman.github.io/HTPIwL/

That's the textbook.

>>15325927
"You will also need the Lean package that accompanies this book". That explains it. This book you're using adds a bunch of new tactics: https://github.com/djvelleman/HTPILeanPackage/blob/master/HTPILib/HTPIDefs.lean.. That seems pretty stupid considering that you'll never see those tactics outside of the context of this specific textbook. I recommend using TPIL4 that I linked earlier like everyone else did.

>>15325307
hey amigo, do you have a loicense to post cirno pics??? She already belongs to me pal

The disgusting fact that there is practically a forced meme advertisement structure to shill books from people such as Spivak, Rudin, and Herstein. I wonder what they all have in comon (besides writing terrible books).

Now, Ahlfors, Hildebrand, Lindstrum, Apostol, Lang... now those are names I can trust.

>>15325665
>how to use lean
>first step install vscode
>>15325978
There is no direct replacement for Spivak's DG series sorry freshman

I have invented a new mathematical operator. It allows one to add a number to a number and then subtract the number added all in one operation. It is called the Sneed operator. It's mathematical symbol is §. Behold as I sneed the number 7 with the number 3
>7 § 3 = 7
This is equivalent to
>7 + 3 - 3 = 7
The Sneed operator is part of an overarching field of mathematics known as Sneed Theory. I am currently exploring what's known as a Multisneed which is similar to a normal sneed but using multiplication and division
>7 §§ 3 = 7
The multisneeded number in this case is once again a 7

>>15326173
>7 §§ 0 = ??

>>15326173
I chucked a 9 into a thrembo.

>>15326173
good news, anon
I've been working on a general solution to Birch and Swinnterton-Dyer, but hit a roadblock. however, a couple of iterations of sneeding appear to bypass it, and I'm in the home stretch now. Since I owe this to your breakthrough, what do you say you DM me your contact information so I can see about adding you on as a co-author?

>>15326173
Otherwise called left projection.

>>15326173
1. Sneed has always been an unfunny and forced meme.
2. Your post is not funny in the least.
3. It doesn't belong on /mg/.
4. Go back to /b/.

>>15325942
Sigh...ok anon

>>15326251
I've seen worse

>>15326189
Undefined

>>15326248
No. A sneed

>>15326251
I think you may be underestimating the power of sneed theory. My students and I at my university lab had been sneeding discrete meromorphic subgroups and the modular forms of the upper half sneed plane in an attempt at resolving the unbounded denominators conjecture but were beaten to it by Atkin and Swinnerton-Dyer but only a month ago. Recently we've shifted to sneeding the invariant subspace using a rotated partial eigensneed and are making good progress

>>15326427
Adding buzzwords and well-known math personalities to your joke doesn't make it any less unfunny.

>>15325901
Lol he didn't even write papers, he just left the work for his coauthors to finish up.

i passed calculus 1 with an average mark for my CS degree.
i don't know how to feel because i've always liked math but i feel inferior because of this exam.

>>15326251
i thought it was kinda funny

>>15326475
It's not funny. It's Sneed Theory

[math]\begin{matrix} 10 & 12 // 14 & 24 // 22 & 48 // 26 & 96 // 34 & 192 // 38 & 384 // 46 & 768 \end{matrix}[/math]
Will column 1 ever exceed column 2?

>>15327454
well, can you name the patterns that are occurring in column 1 and column 2?

are there any good, popular physics/math competitive olympiad-style exams at the undergrad level? Yuropoor so can't compete in the putnam.

Hey Anons, I need advice. Im currently reading analysis by abott right now. At the pace Im going, I think I will complete it in the next month or so. My goal is to read folland real analysis. Is abbott enough? Or should I go back to the basic after by looking into zorich (i have both volume lying around) or rudin or measure theory by dohn? Thanks.

[math]\rho A(x)\frac{\partial^2 y}{\partial t^2}+kGA(x)\left(\frac{\partial \phi}{\partial x}-\frac{\partial^2y}{\partial x^2}\right)=0\\ \rho I(x)\frac{\partial^2\phi}{\partial t^2}-EI(x)\frac{\partial^2\phi}{\partial x^2}+kGA(x)\left(\phi-\frac{\partial y}{\partial x}\right)=0\\ \mathcal E_1(H_j,L)=\left\|W_m\left[\omega_m(H_jxL)-\omega_m^{ref}\right]\right\|_{L_r}\\ L_1=\displaystyle\sum_{m=1}^{M_T}|\Delta_m|;L_2=\displaystyle\sum_{m=1}^{M_T}|\Delta_m|^2;L_\infty=\underset{m=1}{\overset{M_T}{\text{MAX}}}|\Delta_m|[/math]

I want all of you to try to grasp what these are used for, I'll provide no context. Something tells me it's hopeless and no one here will be able to figure it out and how to use these.

>>15327633
replace "H_jxL" with "H_j,L", sorry for typos, I wrote this on mobile so I had to rewrite from a CodeMirror(?) textbox into another textbox and wasn't able to copy and paste.

>>15327607
Princeton.

>>15327672
Please elaborate fren

>>15326369
$\partial$

>>15327633
>>>15326369
\(\partial\)

>>15327633
> Eq1: y has to fulfill the wave equation on the support of A, however modified by kG*dphi/dx as a source term
> Eq2: It would be Klein Gordon + dy/dx source term, if I(x)=A(x), but its not ...
> Eq3: maybe you wanted to write L for the norm subscript? It could be an error function, or part of an analysis proof
> L_j are finite dimensional p-norms

These could all be part of some PDE subject.

The two upper PDEs stir something in any field theory enthusiast, but I cannot pin it down. y and phi seem to be a massless and massive field, if I may lend terminology from physics, interacting in a strange way, but I have not thought that one through.

Book Depository will close in 20 days:
>https://www.bookdepository.com/closure
This is your last chance to buy mathematics textbook from them.
>inb4 why not just download from libgen
I do both. I download **and** buy physically.

>>15327942
they are overpriced. you'd think they would make some closeout sale. I can get any math books I need for a tenth of the price, used from abebooks or other used book stores.

Has a course ever gone at too fast of a pace to be enjoyable? I found a cheat sheet online, and every topic has already been covered, with 7 weeks still left to go in the course. I think it might be that my classmates can handle this pace and more is being taught at a faster rate, but I am unable to keep up and wish we would just cover the standard course content. Just because I am low IQ doesn't mean I shouldn't be allowed to enjoy the course at its standard pace.

>>15324994
>/mg/- mathematics general

>>15328312

No one has ever provided me a satisfactory explanation as to why proofs are math. They really aren't. You can absolutely do math without contrived and cookie cutter proof methods which really just teach students to memorize rather than understand. Its a really outdated idea, and I can't picture a serious mathematician sincerely emphasizing the value in modern proof methodology. Don't even get me started on set theory...

Is there a complex Maclaurin series which has radius of convergence 1 but is known to converge at no point on the unit circle?

>>15328616
The generating function for partitions.
https://en.wikipedia.org/wiki/Partition_function_(number_theory)
Prod[1/(1-x^n),{n,1,infty}] = Sum[p(n)*x^n,{n,0,infty}]
The poles are dense on the unit circle.

>>15328644
Forgive me if this question is stupid, but:

How do we see it can't converge anywhere on the unit circle, from knowing its poles are dense on the unit circle?

>>15324994
Hey everyone. I've got a quick question. Let's say you have a fixed collection of vectors on the n-sphere and scale each by some quantity. Is there some projection a that will close the sum of these scaled vectors that *only* modifies the scalars? I'm looking for a projection what minimizes a the change in a certain quantity, but that's besides the point now. I know that it can happen that the only projection is the trivial projection. I'm an undergraduate who has taken a few graduate courses in pure math, but this problem seems more applied and I'm not sure where to start.

>>15328706
I should not that I've tried using Lagrange multipliers, minimizing the change in the cost function and satisfying the closure condition on the sum (magnitude squared of the sum of the scaled vectors is 0). However, I've proven that the only such projection is the trivial one. I'm starting to think this isn't a problem that constrained optimization can solve. Has anyone had experience with such a problem before?

Say there is a rectangle [math]A[/math] of length [math]a,b[/math] and [math]c<a,b[/math].
I want to construct another rectangle [math]B[/math] of length [math]c,d[/math] that lies entirely in [math]A[/math] and maximizes [math]d[/math].
Does anyone have an idea how to do this?

Is "every non-empty set is inhabited" equivalent to excluded middle?

>>15328745
>inhabited

>>15328745
Isn't it equivalent to Choice, regardless of LEM

>>15327607
Abbott is basically equivalent to Rudin minus the multivariable/measure section which isn't that good anyway. I think Rudin's exercises are also slightly more "difficult" but that's about it. You can move on to Papa Rudin or measure theory after Abbott.

>>15327592
Took me 10 secs googling to find this https://www.imc-math.org.uk/
>>15327672
He's talking about stein and shakarchi's 4 volume analysis series. For what it's worth I think if you're interested in the content in Folland you should read it and when something unfamiliar pops up, look it up in either SS's Real Analysis or any book at that level.
>>15328838
No, LEM is enough to prove that statement. I don't think it implies LEM on its own though although I could be wrong

>>15328745
Define
>non-empty
>inhabited

>>15328745
>>15328847
Sorry meant to say:
In the absence of LEM,
"Every nonempty set is inhabited" should be equivalent to Choice

Let [math]M[/math] be a module over a commutative ring [math]R[/math].
Suppose for every [math]a,b \in \mathrm{End}_R(M) [/math] if [math]ab = 1[/math] then [math]ba = 1[/math].

Then is [math] M [/math] finitely-generated as a module over [math] R[/math] ?

>>15328854
The definitions are just a google search away: [math]x[/math] is inhabited if [math]\exists z\, z\in x[/math]; [math]x[/math] is non-empty if [math]\neg\forall z\, z\notin x[/math] or equivalently [math]\neg\neg\exists z\, z\in x[/math]

>>15328549
>I can't picture a serious mathematician sincerely emphasizing the value in modern proof methodology
Yet most of published mathematical research consists of articles describing theorems and their proofs.

>>15328930
No, this implication is already provable without LEM. Observe that [math]\exists z(z\in x)\leftrightarrow\neg\forall z\neg(z\in x)\leftrightarrow\neg\forall z\neg\neg\neg(z\in x)\leftrightarrow\exists z(\neg\neg z\in x)[/math] since 3 negations can always reduce to 1 even in minimal logic.

>>15329288
My bad, this is actually wrong because [math]\exists[/math] and [math]\neg\forall\neg[/math] are only interchangeable if we have LEM.

>>15328745
I think yes, like so:
(Using pairing, union and strong Separation)

[math]t := \{x \in \{0\} \mid P\}[/math]
[math]t_\neg := \{x \in \{0\} \mid \neg P\}[/math]
[math]T := t\cup t_\neg[/math]

[math]x\in T \iff (x=0\land P)\lor (x=0\land \neg P)\iff x=0 \land (P\lor\neg P)[/math]

[math]\exists x. x=0[/math]
so
[math]\forall x. x\notin T\implies \neg(P\lor\neg P)[/math]
so
[math]\neg \forall x. x\notin T[/math]
i.e.
[math]\neg (T=\emptyset)[/math]

But also
[math]\exists x. x\in T\implies P\lor\neg P[/math]

E.g.
0 ... your only friend
{0} ... set of your friends
P ... you got evidence all your friends survived the way to your party
neg P ... you got evidence it's not the case that all your friends survived the way to your party
[math]t := \{x \in \{0\} \mid P\}[/math] ... people who showed up to your party
[math]t_\neg := \{x \in \{0\} \mid \neg P\}[/math] ... your dead friends
[math]x\in T[/math] ... x is your only friend and you got evidence he's dead or alive
Membership in T does not just say that someone is your friend, but also carries into by construction.
T being inhabited is a claim about information, but consistent logic (non-paraconsistent logic) rules out that you couldn't know either option

carries information*

>>15329472
You are assuming double negation right?

>>15329512
Nvm not not p or not p is true even without general double negation

>>15329512
I'm not certain what you're asking, but if you're asking whether I assume double negation elimination, then the answer is no.

Actually I think my proof might go through for any Q for which the theory proves not not Q.

[math] q = \{x\in\{0\}\mid Q\}[/math]

[math]{\mathsf{T}}\vdash \neg \neg Q\implies {\mathsf{T}}\vdash (q\neq \{\})\,\land\, (Q\leftrightarrow \exists x. x\in q) [/math]

>>15329521
[math] \neg (P\lor \neg P) [/math] is true by the valid DeMorgan's law, and it has a simple lambda term proof in the BHK interpretation

[math] (\neg \neg P)\lor (\neg P) [/math] is WPEM and not automatically true.

It's equivalent to the not valid DeMorgan's law
[math]\neg (A\land B)\to (\neg A\lor \neg B)[/math]
(From knowing it not to be the case that both Alice and Bob showed up to their date, it does not follow you'd know who of them did not show up.)

The rejection of ¬(P∨¬P) is true, I mean

I suppose you too \neg to be loosely binging.
That's uncommon I think

I have to deal with blowing up stuff and I'm not too bright on the abstract alggeo side. Is every smooth toric variety with corresponding fan having above some threshhold of 1-dim cones a blowup of another toric variety? Cox' Toric Varieties book is kinda hard and I cannot find a statement like that in that book, or any other book/article. Any suggestion for further reading would be nice.

>>15328847
Thank you. I forgot SS is by princeton.

>>15328842
Thank you.

If [math]\dfrac{a}{b} \lt \dfrac{c}{d}[/math], then [math]\dfrac{a}{b} \lt \dfrac{a+c}{b+d} \lt \dfrac{c}{d}[/math]. Can you prove it?

>>15328616
1+z+z^2+z^3+...
>>15328951
Ignore midwits complaining about set theory when it is barely relevant to non-specialists, and indicating that they have no understanding of what actual proofs look like outside of a sterile setting designed to teach retards.

>>15329938
[math]\frac{-2}{1}<\frac{1}{-2}\implies\frac{-2}{1}<\frac{-2+1}{1+(-2)}<\frac{1}{-2}[/math]

Wtf, did /sqt/ get deleted?

>>15329978
It's weird. It wasn't deleted. Threads seems to have archived itself. Sometime has been fucking with the timeouts.

>>15328737
Can someone have a look at this? If a=b I can find a solution but I cannot find a solution for b=a+h.

If P is a false statement, does it vacuously imply not P?

Does anyone have a list of number theory (or something discrete and logical) questions smoothly ranging from ridiculously easy/simple to unsolved/too hard? Is there some book/textbook that has this?

the fact that there's such complexity in the relations between the natural numbers is wonderous. i'm certain the complexity of the universe originated in a similar way, from a simplistic logical base.

>>15330741
Are you good at math?

>>15330807
not very

>>15330829
That's a coincidence because I agreed with that post, and I don't think I'm very good at it either.

>>15330643
There's several ways to read the question.

If "P is false" is supposed to mean that "not P" holds, then we can say two things:
Firstly, "not P" does hold.
Secondly, if the theory proves the common theorem "A->A" (usually quickly derivable from the logical axioms), then indeed '"not P" implies "not P"' in the sense of
(not P) -> (not P)

The other reading of the statement is that "not P is false" is supposed to be read as "not P is False" where False is the/a probably wrong proposition.
Something interesting happens here:
Of course if the theory proves explosion/ex falso, then since P is true means False holds, anything follows (and in particular "not P")
The interesting thing is that "not P" follows from False also when explosion is not adopted, simply by implication introduction
A->(B->A)

Now the latter rule is a bit strange. For example, the rule validates
"If I'm wearing red pants right now, then if my name is Bob, it implies I'm wearing red pants right now"
The somewhat strange nature of such a rule is addressed in relevant logic
https://en.m.wikipedia.org/wiki/Relevance_logic

>>15330911
In the second reading I meant to say
"P is False", in the first sentence.
I also make the use of defining "not B" as "B -> False"

Getting a proper buggering in this course.
There is a story that has been posted on this site many times, that persons of a certain IQ are incapable of nesting relations past a certain point. Think mine is 3. This problem is a real 3 to me.

>>15330978
a good step 1 is usually to assign new variables to prevent clutter
e.g. make Φ^{-1}(x)=g and Φ^{-1}(y)=h

>>15331000
I think, that for some reason clears things up. For S to be a group I need to just clear some checkpoints like well defined(one to one?), associativity, identity element, and check for inverses.
Its a bijection so it should be self -evident that its well defined, but I'll still go through the annoyance of proving it.
Anyhow, I'll start there, maybe.

but something like:

x,y,z elements of S, x = x', y = y', --> x*y = x'*y',

x*y = F(F(x)^-1 # F(y)^'1)
x'*y' = F(F(x')^-1 # F(y')^-1)

so as F being a bijection, F(x)^-1 = F(x')^1, same for y.

so F(F(x)^-1 # F(y)^-1) = F(F(x')^-1 # F(y')^-1)..... it just seems circular and I think I'm missing some sort of trickery, identity, rule, or theorem I must apply. On my third week of this course...

I do not fucking get it, pic related.

So then [math]\rho_{u,X}(f, g) = [ \int_{\!R^k} |f(x)-g(x) \,dX(x) ]^{1/u}[/math]
Or what? How the fuck does this make any sense?

X is not a measure, it's a set.

> Spend 15min applying the Gauss method
>Get the wrong answer
I HATE MATRICES, I can hardly solve a fucking 2. degree polynomial without misplacing a fucking minus sign somewhere and now I have to juggle all this shit AAAaaaaaa

I do not fucking get it, pic related.

So then [math]\rho_{u,X}(f,g)=[∫_{\!R^k}|f(x)−g(x)|^u \, dX(x)]1/u[/math]

Or what? How the fuck does this make any sense?

X is not a measure, it's a set.

I do not fucking get it, pic related.

So then [math]\rho_{u,X}(f,g)=[∫_{\!R^k}|f(x)−g(x)|^u \, dX(x)]^{1/u}[/math]

Or what? How the fuck does this make any sense?

X is not a measure, it's a set.

>>15325000
Did you take notes with the purpose of explaining the textbook to yourself? If not go back and try that, see if it speeds up your time.

>>15331049
Based actual mathematician. God gives his hardest matrices to his most stubborn mathematicians.

>>15331055
In [math]C(X)[/math] you usually measure the distance between functions with the supremum so
[eqn]\rho_{u,X}(f,g) = \sup_{x \in X} |f(x) - g(x)|[/eqn]

>>15326173
>Sneed operator

>>15330692
bump

>>15328659
since the poles are dense you cant find a radius of convergence, because any such radius has a pole(s) in it.

>>15326173
>a Multisneed which is similar to a normal sneed but using multiplication and division
That could get pretty complicated if you don't define division as multiplication by the inverse.

>>15330978

(a).

[closure] Phi is a bijection. g,h are elements of G. As G is a group, then gh is an element of G. As a bijection, it follows that Phi(g#s) is an element of S.

[Associativity] can only exist by inspection if # and * are the same operator. This assumption will be needed for part (b)

G is a group and by definition has an identity element. As a bijection, it follows that the unique [identity element] is preserved and an element of S.

Phi is bijective, G is a group, inverse of Phi will map S to G including [inverses].

(S,*) is a group iff *=#, as we can see it holds closure, associativity, identity, and inverse by inspection.

(b)
For phi to be [isomorphic], the function must be bijective and a homomorphism. We are already given that phi is bijective.

To be a homomorphism it must be structure preserving. This can only happen if the groups have the same binary operator. So as shown earlier in the description of associativity, * = #

this is the best I got. its been 3 hours.

>>15330692
R.P. Burn's "A pathway into number theory" is the first half of what you're asking for. As for unsolved problems I would just search arxiv for "list of unsolved problems in X" where X is number theory, class field theory etc

>last term in a master's program
>have a 4.0
Will this look suspicious to PhD admission committees? How often do people graduate with 4.0s? Should I get one A- to make my As look more legitimate? I realize this is a retarded problem to have but this program just isn't very hard compared to my undergrad.

>>15331384
No retard, they want to see 4.0s. But if you're asking this question you should probably get at least one F so admissions knows you're retarded.

Bros, how can I better my mathematical reasoning. Im an undergrad, I understand the topics, can work through out the examples and counterexamples, but when it comes to doing exercises i can't do the hard ones, (or the ones that require sketchy arguments) for example, those who requiere to write a function a way that helps achieving the demonstration, or those who uses a lot of basic algebra, i need to get better at this, as i said is not like i can't do anything, so a basic book will not be useful.
Ive heard that combinatorics are a good way to practice mathematical reasoning.
Any books recommendations?

>>15331389
There's no way of practicing mathematical reasoning outside of doing a lot of problems. Just choose a book about something you're interested in and which isn't too hard for you and do as many problems as you can stomach

What is the "usual dual space argument"?
I'm so fucking confused...
I have to learn functional analysis and fourier analysis to write my bachelor thesis.

I hate math, it's the worst fucking thing. We would've been better of as just hunter gatherers.

>>15330978
it was a pretty straightforward calculation

>>15331429
>we would have been better off as hunter gathers
Can you write a proof for this?

>>15331397
Any interesting book with problems?

What was the name of the combinatorics book that is always been shilled here?

>>15331616
Applied combinatorics by Tucker.

>>15331429
I assume that this squigglyN_k(psi) is a linear subspace. If it is not dense in C^m,p, then it clearly it's not dense in L^p. If it is not dense in L^p, then it's closure, call it Y, is a closed, proper, subspace of L^p, so L^p\Y is nonempty and it contains some function x_0.

Now there is a corollary of Hanh-Banach theorem (proven here for example: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwiygPLGmZb-AhVOrosKHe7sBG8QFnoECDcQAQ&url=https%3A%2F%2Fwww.ucl.ac.uk%2F~ucahad0%2F3103_handout_6.pdf&usg=AOvVaw0HT02VGysWGBLSkYWJltwa)) that says that there will be an continuous linear functional (call it y) such that y(x)=0 on Y and y(x_0)=1.

The standard duality that they have in mind is that, for 1<p<infy, any continuous linear functional on L^p (and therefore y) can be written as y(x)= integral gx dm, for some g in L^q, where 1/p+1./q=1.

Now, I imagine, that whatever the operation squigglyN_k() does, it a) commutes with taking derivatives (D^alpha(squigglyN_k(psi))=squigglyN_k(D^alpha(psi))), b) If squigglyN_k(psi) is not dense, so is D^alpha(squigglyN_k(psi). If that is the case, repeat the argument for each alpha and you get you g_alpha's

Given [math]\mathbb{R}[x,y][/math] and the ideal consisting of functions for which [math]f(0,1)=f(1,0)=0[/math], what is the factor group [math]\mathbb{R}[x,y]/I[/math]?

>>15331769
I forgot all the terminology, but do the factors
(x^2+(y-1)^2)
(y^2+(x-1)^2)
help with the problem?

>>15331769
I hope you know the first isomorphism theorem for rings.
https://en.wikipedia.org/wiki/Isomorphism_theorems?useskin=vector#Theorem_A_(rings)
What you have to do is construct a ring homomorphism whose kernel is I.
So take
[eqn]\varphi: \mathbb{R}[x,y] \to \mathbb{R}^2 \\
f(x,y) \mapsto (f(0,1),f(1,0))[/eqn]

It's clearly surjective since for any [math](a,b) \in \mathbb{R}^2[/math] you can take [math]\varphi(ay + bx) = (a,b)[/math].

>>15331387
I will add a note to my essays informing them that I am retarded, thanks for the advice

>>15331611
yes

>>15329938
[math]a(b+d)/b<a+c[/math]
[math]ab/b+ad/b<a+c[/math]
[math]a+ad/b<a+c[/math]
[math]ad/b<c[/math]
From the first afirmation we can get that [eqn]ad/b<c[/eqn]
Same thing for the second part
[math]a+c<c(b+d)/d[/math]
[math]a+c<cb/d+cd/d[/math]
[math]a+c<cb/d+c[/math]
[math]a<cb/d[/math]
From the first condition we can get that [eqn]a<cb/d[/eqn]

>>15328892
I don't think so. Any left invertible map of Z[x] to itself should also be right invertible (I believe these are precisely the maps that send x to +/- x + b for some integer b) but it's clearly not a FG Z-module.

>>15329890
can't answer your question yet, but am glad to see someone else reading this book

>>15332259
I actually found something related to my problem with regards to the resulting cohomology, see
https://mathoverflow.net/questions/394161/whats-the-cohomology-ring-structure-of-a-blow-up
The mentioned article by Okonek and Van de ven is, lets say, not easy to read. No, seriously, I hate it when proofs go like
>Obvious, see [citation of 900 page long book].
This is for some reason more common in alggeo than in diffgeo.

I'm still a bit unsure about the Cox book. One of his other books, "Ideals, Varieties, and Algorithms", is really nice. But so far I'm really really confused by the toric book.

>>15328706
>>15328723
Does anyone have any clue where I might start with this? Not trying to spam the thread or something but even a book recommendation would be helpful.
The problem boils down to being given a set of unit vectors (supports) [math]\{v_i\}[/math] and assigning a bunch of scalars (masses) [math]\{\overline{m}_i\}[/math] to them. Then you have to manipulate the scalars such that the sum of all the vectors times their scalars is 0. I'd like to minimize the change in some simple cost function while doing this, given by [math]\sum_{i=0}^n((\sqrt{\overline{m}_i} - \sqrt{m_i})^2)[/math] where [math]m_i[/math] is the new mass assigned to each support and [math]\overline{m}_i[/math] is the original mass. I've tried lagrangian multipliers but the gradient of the cost function is always 0 where the constraint is satisfied. Any help at all would be appreciated :(

>>15331055
>>15331142
the "u" in "u,X" means "uniform norm" and the X is indicating the set you are maximizing distance over, namely X. This is independent of any underlying measure mu on X you might take, hence the notation.
>>15330978
>>15331467
You should draw a couple pictures and try to internalize the relationship between the picture and the symbolic proof. It sounds like Buddhism but it's actually a very important skill to be able to intuitively understand these things.
>>15328659
>>15331272
Just because it does not converge/has poles at roots of unity does not necessarily mean the power series does not converge anywhere on the unit circle. The set of roots of unity is a countable union of closed sets, hence F_sigma.
By the theorem of Herzog and Piranian, there is a power series with precisely those points being convergent:
https://math.stackexchange.com/questions/288765/convergence-power-series-in-boundary
https://mathoverflow.net/questions/49395/behaviour-of-power-series-on-their-circle-of-convergence

>>15332365
>The problem boils down to being given a set of unit vectors (supports) {vi} and assigning a bunch of scalars (masses) {m¯¯¯¯¯i} to them. Then you have to manipulate the scalars such that the sum of all the vectors times their scalars is 0

Nigga, you just described the problem of solving a homogeneous system of linear equations. If the vectors linearly independent, the only solution will be all m_i are 0. If the vectors are dependent, then, m_i will lie in some hyperplane passing through the origin of the space with coordinates m_i. In any case, your problem of minimizing the change is the same as finding the distance from the initial masses to the space of solutions or your system, in the metric given by your cost function.

Is there a sequence of positive rationals [math] a_n [/math] with [math] \sum_{n=1}^{\infty} [/math] rational and [math] \lim_{n\to\infty} a_n/a_{n+1} = 1 [/math] ?

>>15332739
yes

>>15332739
[eqn]a_n = \frac{1}{(n+1)^2 - 1}[/eqn]

>>15332743
>>15332744
Thx

If [math] p(x) [/math] is a degree ≥2 monic irreducible integer polynomial, then is [math] \sum_{n=1}^{\infty} 1/p(n) [/math] irrational?

>>15332786
I don't know and I am retarded. But what I would do is, I would write a function f(x), such that it's taylor series around 1 is sum 1/p(n) (x-1)^n, then I would try to differentiate it couple of times, add/subtract multiples of the derivatives till i get a differential equation (that somehow should be feasible, since you have this polynomial P there ). Then I would try say something about the solution of that equation, maybe it always maps rationals to irrationals?

>>15332786
It's hard to tell with n going from 1 to infinity.
Let the sum be from -infinity to infinity.
1/p(x) = Sum[1/(p'(r)*(x-r)), r root of p].
Summing this over Z gives:
Sum[-pi*cot(pi*r)/p'(r), r root of p]
https://en.wikipedia.org/wiki/Trigonometric_functions#Partial_fraction_expansion
Something tells me you would need a miracle for this to be a nice value for an irreducible p.

Let [math]T=\bigcup_{n\in{\mathbb N}}\{0,1\}^n[/math] denotes the countable set of all finite binary sequences.
What's a nice injection of [math]f\colon T\to \{0,1\}^{\mathbb N}[/math] into the unending sequences, such that each [math]t[/math] is the initial segment of [math]f(t)[/math].
One way I thought about is taking another bijection [math]b \colon T\to \{\mathbb N\}[/math] and letting [math]f(t)[/math] be [math]t[/math] , followed by some delimiter, followed by [math]b(t)[/math] in binary and then zero's. Still feels a bit ugly and I'm not 100% sure how to define a delimiter so that the end of the initial segment is computable.

ignore the {} around N

>>15332744
What was your process of finding this?

>>15333268
Use partial fractions, it telescopes

>>15333283
I don't mean validating that it is a solution, I mean finding it in the first place.
Of course a_n would have to be something that falls more than 1/n (to get a converging sum) and something not to grow to quickly so that the ratio stays 1.
But even then, or even if we search for a quadratic function, it's not so clear how we find something fitting rationality. Not to me anyway.

43qterq345r, werq4rq4trqwe.

> siaforjowe4rqwe5r4, qwrwe

>>15333294
>I don't mean validating that it is a solution, I mean finding it in the first place.
Just look for something that telescopes lol. Anything of the form 1/n - 1/(n+k) should work

>>15333328
What's telescopes?

>>15333505
the partial sums collapse like a telescope becsuse all the middle terms cancel

>>15333505
Like 1/n - 1/(n+1) . The sum is
(1/1 -1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ...
= 1 + (-1/2+1/2) + (-1/3+1/3) + (-1/4+1/4) + ...
= 1

universal primitive notions.

what are some examples of universal primitive notions?
Here is one: When you use a symbol and then use the same symbol again, they refer to the same thing, not different things.
Is there a list of all universal primitive notions? How many are there?

>>15333672
I do not agree that that is a universal primitive notion. I also do not believe there is any such thing. You can always regress further and assume more primitive axioms that imply the ones you want.

>>15333702
naming/symbols and definitions would be universal primitive notions wouldn't they

>>15333745
No. Those things are fairly critical for communicating information, but they are not information itself. You need only look to category theory for a degree of abstraction that does not require symbols or names, everything is uniquely defined by its relationship to everything else.

>>15333757
category theory isn't even a foundation
and if everything is uniquely defined by its relationships to everything else, then you have the universal primitive notions of things, relationships, and a grammar that relates them.

Bros, how much of mathematics am I supposed to remember and have perfect recall of?
I remember someone here advised me that to keep your maths skills fresh, you should practice occassionally the concepts you've learned. It worked out for me in the beginning, but now since I'm gonna have further and much more difficult maths courses, I can't do it anymore. I can't practice all those aths courses, while also dealing with my other college courses. I plan on going to a maths intensive field after college, so its not like I can discard certain maths topics. I can literally feel all the maths leaving my body. I am becoming dumber by the day.

>>15333766
My point is exactly that you can't decide on one foundation, it is a core result of category theory that understanding objects and their properties is equivalent to understanding purely formal relationships. Neither is "more primitive" and you can keep "zooming out" forever. That's the whole point, you can always get more primitive so you have to just choose a starting point. The axioms of Euclidean geometry aren't the most foundational axioms possible, they're just the ones that line up with our intuition and encode all the properties we want flat geometry to have. Ditching the parallel postulate doesn't make your theory more or less primitive, it's just a different theory and you have to decide if it's interesting to you. You can't even call names and grammar the most primitive because a blind/deaf math savant might develop Euclid's axioms independently in some form that is entirely incommunicable because they don't have any concept of language and grammar at all, surely that suits your intuition for "more primitive" since it requires "less structure" to develop.

How can you calculate theta for an archimedean spiral if you're given the length of the spiral, gap between turns and starting radius?

I know the formula for an archimedean spiral is [math]r = a + b\theta[/math] but I don't know where to go from there.

I'm very slightly familiar with basic calculus and it feels like integrals would somehow be involved for this.

Can't believe linguists and philosophers hijacked math and made it some vocabulary definition nonsense. I hate this. What can I specialize to avoid this pozzed nonsense. PDE's?

>>15333806
You can literally just not think about it. No one is making you use category theory.

>>15333809
I want you to fucking die, that's the problem. None of that garbage should exist and should be called something else entirely.

>>15333792
how can someone come up with a foundation if they don't even have a vehicle with which they can parse their thoughts
to even refer to anything at all, reference must be a primitive notion, so even a blind/deaf man would still need that. otherwise, if he did not reference anything at all, he wouldn't even be thinking of anything.
and of course more things are needed aswell. things are needed.
and there needs to be a way to relate those things, or there's nothing thought of only the thing itself.
and you need a representation of these concepts whether in the mind, on writing, or wherever, and it needs to have continuity, where it can be referenced later on and refer to the same concept.
and of course, since there's relationships, there's grammar, no matter if its in entirely incommunicable to others.
There are still the same basic primitive notions needed in these situations as there are in others.
And just because you mightn't agree with some primitive notion that i give like symbols, doesn't mean it is right to dispense of them at all. I haven't done much research into this topic, and i doubt many people have. But it is definite that some exist. otherwise you couldn't even have an active thought at all. you couldn't even have a foundational framework for anything you come up with and it would all be meaningless.

>>15333815
My point is that there is no one most fundamental way to represent things or relate them and if you give me a system of primitive notions I can add a layer of abstraction that is "more primitive." It isn't a helpful game to play, the use is realizing that because you can regress forever, you might as well just pick the primitive notions that seem most natural and go from there. My whole argument is: There's no foundational bedrock so don't think about it too hard.

>>15333827
it is not a game. what is more primitive than reference. Please answer that. what does it reduce to. Right now I am thinking of something, how is thinking of something not a universal primitive notion of mathematics. A rock couldn't come up with a mathematical system. It can't think at all. There is a foundational bedrock. That is obvious. The universe itself stems from a foundational bedrock. Objective reality exists, and so does objective truth.

>>15333806
maths is lacking an objective foundation. and relativists are stopping it from gaining one. proper structure and methods of deduction are needed. Otherwise maths will be stuck in a perpetual subjective realm where axiomatic systems can be decided on a whim, and none can be objectively tied to the physical world. At the moment, all that can be done, is an axiomatic system can be decided, and then it is compared to the real world to see if it is an accurate reflection. Which isn't an objective proof that it is a reflections, its only a subjective scientific theory. It's like solving the Goldbach's conjecture by using the scientific method and gathering evidence tons of the numbers its true for, and saying that since there's no counterexample, it would be a sufficient solution. It would be objective. It wouldn't even be a proof.

>>15333866
meant it wouldn't be objective, in the second last sentence

>>15333866
Cool word salad.
Arithmetic is the base.

>>15333852
You answered your own question. Math doesn't need you to think about it for it to exist, the universe exists anyway and the level of abstraction is dealer's choice. "Reference" is not a primitive notion. That's just how your mongoloid ape brain interprets information. I wouldn't be so hard on rocks when you clearly can't come up with a mathematical system either. Does a baby imagining a cube meet your requirements? Does an animal watching water flow understand enough about fluid dynamics to call it mathematical reference? A higher being with a greater understanding of mathematics might argue that the difference betweem them and you is the same difference as between you and a deer. Language is a communcation tool that doesn't actually change mathematics. Math exists and the notions you use to understand it can always be regressed to a more primitive state. The level of understanding does not change how primitive a notion is, "primitive" is a subjective human evaluation based on intuition. You're just cutting the qualification for "primitive" off at a certain level of intelligence, which is a dangerous game for someone like you to play.

>>15333881
you can have arithmetic as the base, but only if you sell out the actual foundations to linguists and philosophers. you can do that if you want. but do know that linguists and philosophers suffer from relativism and lack of rigour and it will lead to arithmetic have a shaky foundation since you will need to use assumptions. And assumptions aren't objective fact.
But arithmetic is definitely important. it's the foundation of what ordinary think of when they think of maths. And having it as the foundation of the entirety of maths is fine, but the underlying logic and reasoning behind it would just be swept into other fields.

>>15333902
You are not doing math.

>>15333902
Imagine writing all that and then this idiot responds with that little thingy

>>15333928
>more word = better
laughable

>>15333885
you are completely straying from the point.
The truth is that to have a mathematical framework written by a human, you need things like reference. You are stuck in your own cycle of infinite regress, you say
>"Reference" is not a primitive notion. That's just how your mongoloid ape brain interprets information.
which is just using semantics to try to avoid the word reference. the actual word doesn't matter, its the idea. And you bring up information which of course according to you wouldn't be a primitive notion either, so using it to explain reference would be pointless.
A human's mathematical framework of course needs to be thought of for it to exist. Since all it is is a form of communication.
If universal primitive notions didn't exist, how would you have language? If meaning followed infinite regress then it would be impossible to parse because everything would need an explanation. But it exists, and so do universal primitive notions.

>>15333672
>>15333766
>category theory isn't even a foundation
It's not used as a foundation but you can write down a first-order theory of [math]\circ[/math] and go down the topos theory approach to get sets and hence anything else

>>15333910
what is it then

>>15333943
So you concede that your "universal primitive notions" are human constructs designed to make the universe comprehensible to you?

just how much commutative algebra do i need to know before taking an AG course?

>>15333957
are you conceding that they exist?

>>15333965
Of course they exist, you just invented them. I'm telling you that they are in no way primitive nor universal.

>>15333986
now you are are denying the existence of primitive notions. you are too far gone.

>find the order of each element in dihedral group 4 and find the cayley table.
>don't want to just copy the answer from the book and try to derive them for the last 4 hours
>fuck it up
>Proved quartenion instead.
I despise this course. Am considering suicide.

>>15334148
Genuinely how do you fuck that up

>>15333772
bump, I really need to know

>>15334166
I am 3 weeks into this course and I don't think I can truly solve a single problem the way they're asking me to solve it.
For dihedral group tables for any n-gon, I just solve for the pattern in the top left quadrant, the bottom left quadrant.
Then the dihedral group table is just a 2x2 matrix of matrices[A B; C D], where A and D have the same elements, and B and C have the same elements, but A and B have the same pattern, and C and D have the same pattern.
Somehow we've already covered most of Beachy and Blair's textbook, so I'm not sure what the hell we are supposed to cover for the rest of the course. I also don't know why we can't just read the book in order.

>>15333772
>>15334171
Certain things should be immediate. A lot of linear algebra, common inequalities, basic group theory, definitions and first properties of other algebraic objects, stuff like that which can rear its head out of nowhere. When you're computing some homotopy homology cohomology topological thing, you want to immediately be able to go "oh this is a group/ring/module" and immediately have a decent idea what kind of structure you're working with. Groups and vector spaces are the ones you want to be able to whip off proofs about the easiest, in my experience, because they really show up constantly. Though if you're doing number theory that'll be more rings and modules. Oh and basic set theory is absolutely critical, if you find yourself trying to remember which one is one-to-one and which one is onto you'll be dead in the water. Assuming you don't have a problem with that unless you're in high school, though. For good measure, some combinatorics is always useful but you can generally bootstrap that shit up on your own.

>>15324994
Sorry to shit up your thread, but my math doesn't seem right. I double and triple checked and I think I could be retarded. Please confirm:
Picrel claims there is 1200lbs of DS cartridges, mostly "iCarly 2: iJoin the Click" there are many claims that there is just no way. The dimensions of the box are unknown and mostly not important. Without taking the pallet or box into consideration, I thought I'd try to calculate the volume of 1200lbs of DS carts.

Ds cart: 3.5g (0.12oz)
133.3 carts = 1lb
1200lb = 160,000carts
Dimensions: 35mm x 33mm x 3.8mm (0.27in3)
160,000carts = 43,200in3/3600ft3

A box with the dimensions on 6ft x 6ft x 100ft would weight 1200lbs.

>>15334148
[math]a^n=e[/math]
[math]b^2=e[/math]
[math]ab=ba^{-1}[/math]
literally all you need to know for any dihedral Cayley table

>>15333772
You should just not worry about it and keep learning new stuff all the time. You'll end up remembering what you need to.

>>15333258
T maps to (T,1,0,0,0,0,0,0,...)
easy to see it is injective (find the last 1 and strip it off)
>>15334311
everything until the end is correct with some error
but
43200 inches^3 is not 3600 ft^3
it's 25 ft^3
so 5x5x1ft
Although, potentially up to half the space in the box is empty, so maybe it's really like 50 ft^3

I'm graduating high school this 2024 and I'm taking engineering in college because I like maths. Maths has been a serious hobby for me in the past 2 years and right now am stuck in what to self study. I REALLY REALLY want to self study real analysis but studying applied maths would be more beneficial to me as an engineer. What do? shpuld I gk with what I want or with what is more practical?

>>15334749
Take an upper level pure math course when you get to uni. Be disappointed that its nothing like the math you actually enjoy, and the departments are hijacked by English majors. Find out actual math that you'll enjoy is in engineering disciplines or applied math.

>>15334184
You are overcomplicating the fuck out of this. It's a multiplication table. Just multiply the elements and reduce to standard form. It is then super easy to determine the order of each element by just hopping around the table until you get to e and counting your steps.

where do I find a guy who can help me with my math test

>>>32056972

where can I find a guy who'll help me with math

>>>32056972

>>15334777
>>>/soc/32056972


hehe

>>15331285
To perform a sneed operation, one changes the number and then does the inverse to change it back again.

Recently I have been researching sneed calculus. Such as in this simple case of integrating from x to sneed

[math]\int_{x}^{§} 2x[/math] [math]d§[/math] = [math]2x[/math]

The sneed operator is an operator, but it's also a value, but it's also neither. Because when you sneed something it doesn't change. But you still sneeded. So there is a sneed state which is instantaneous and is not undefined but is unknown. As if you went to infinity and back again, it happened but you can't describe it, but it did happen. The sneed operator is like that

>>15334790
Hmm, consider a sneed space [math]S[/math] with sneed operator [math]§[/math]. Given a subset [math]A[/math] of [math]S[/math], we can introduce the following idea of a sneed cover: A sneed cover [math]\mathcal C[/math] of [math]A[/math] is a countable collection of sneed balls [math]\{B(S_i,r_i)\}[/math] such that [math]A\bowtie_§\bigcup B(S_i,r_i)[/math]. The idea is that iterated sneed-invariance from chaining integral operators [math]\int_x^§[/math] gives rise to a unique global structure that we can patch together from the individual sneed operators locally. So it seems possible to develop a global theory of sneed operators, though there is a lot of work to be done since it is rather nontrivial to have a useful system of axioms that really captures the theory.

>mash '9876543210123456789' into really big number factorizer
>result is:

>(3333333333 - 1111111110)(3333333333 + 1111111110)
>The factors of 9876543210123456789 are 2222222223 and 4444444443

is anything cool happening here or am i just easily impressed? because it looks kind of neat to me

>>15334908
Its cool to find patterns. This is what math is all about. Ignore the word salad autists.

How do you find the area of the common overlap of three or more circles given their radii and coordinates of their centers?

>>15334932
draw the triangle
compute its area
add the three arc things, each area is easy, use each circle

i just like plooting functions ok

>>15334932
You could always use coordinate systems and calculus albeit that'd be painful

>>15334749
Study linear algebra (theoretical) and real analysis.
Most of engineering boils down to those two subjects.
They're also foundational for studying anything applied: optimization, numerical methods, differential equations, probability and statistics.

Abstract algebra is nonsense to me. Absolutely going to fail this course. There is no math in it, and I'm not allowed to even do math, everything must be vocabulary definitions at all times. Try solving something geometrically? Uh, no sweaty, you have to use the definitions of a group subgroup homomorphism structure preservation "Well ordered" automorphism cyclic generation, or you won't get any credit, it doesn't matter if your answer is correct. Losing my mind.

Just found out about this (very natural) concept in Roman's Advanced Linear Algebra. I personally haven't read about it elsewhere, but I'm not familiar with more advanced literature so I'm asking if "complexifications" of vector spaces and transformations are used frequently in some field, maybe useful for some functional analysis/spectral theory problems? I just wonder what can you get from this

>>15336564
>Roman's Advanced Linear Algebra.

>>15336564
looks like its describing I on an XY plane, pivoted diagonally with the 0's of the identity matrix as the pivot line. 1's would move into the Z axis.

>>15336535
>it doesn't matter if your answer is correct.

>>15336593
I don't understand from your picture. What I'm getting at is that the course learning objectives are the pedagogy of abstract algebra. If I start using methods to sidestep the course learning objectives, he will not give credit, but also doesn't want to design problems are unsolvable unless you use those methods, so its rather frustrating.

>>15336564
Differential geometry with the study of complex structures vs real structures
Algebraic geometry: this is a special case of the more general "base change" (to field extensions or even just to ring/whatever extensions)

>>15336564
>maybe useful for some functional analysis/spectral theory problems?
Since you can formulate things this way, it is striking (in a bad way) that I have never seen general relativity treated as a problem in complex analysis. Even if there was nothing new to be shown (dubious), it would be instructive in the best way to make a survey of what all the fundamental theorems of complex analysis mean in GR.
>you know this irrelevant stuff about holomorphic functions?
>well actually if we add the units of meters then...

Also, I hate it. and it is profane in the worst way. that I'm laying here as a homeless slave in Antarctica getting my balls electrocuted by some stalker in the adjacent room and you all are pretending like you're not reading my paper. I really hate that. My anger over this is more than enough to fill a cup.

>>15336603
Atiyah once said asking if you'd rather be an algebraist or a geometer is like asking if you'd rather be blind or deaf. Make a real effort to engage with purely algebraic arguments, anon, it will give you new insight into geometry you thought you already understood. I had a similar experience and hated algebra as an undetgrad but it really is worth your time. Maybe you'll find this enlightening: Most of modern algebra was developed because people noticed they were all proving results about seemingly disparate things in similar ways, and upon investigation (including a fair bit of trial and error), determined that the axioms of now fundamental algebraic objects were satisfied by all these "unrelated" objects that had come up naturally in other subjects. That's how things like groups and modules were defined. The abstraction is necessary because that's what makes them so universal, once you get the intuition for it you can bring it with you everywhere you go without necessarily having to start from square one working out the details every time. You could very easily find yourself in a situation where your geometric intuition fails completely, but because whatever you're looking at still has a group structure, your roadmap can carry you through and reveal surprising connections in the process.

>>15336603
Yeah, I get it. That's like when my papers are "unscholarly" because I don't use the jargon established by a bunch of people whose papers I never read and whose work was irrelevant to mine. You wouldn't think the lack of upvotes on science Twitter would nullify the independent merit of my work, but it seems to. In fact, you wouldn't think my colleagues would refuse to communicate with me anywhere but on 4chan while pretending like they didn't read my papers, but that's how it is. Sometimes it feels like I'm the only one who doesn't work for the jews and by that I mean, "When you're pretending like you didn't read my paper, you are doing what the jews want you to do."

>>15336641
>Atiyah
Atiyah called his function the Todd function because my Hebrew name is God.

>>15336661
can you schizopost somewhere else

I solved the Riemann hypothesis years ago and I'm supposed to be able to collect $600k (after taxes) for that but I can't because everyone who reads my paper works for the jews and pretends like they didn't read it and doesn't call bullshit on us all being in Antarctica.

Hairer plagiarized my MCM unit cell and won a Field's medal for it in 2014, and he got the $3M Breakthrough prize for it while I was living in my car driving from town to town trying to find some place where I wouldn't getting constantly raped and tortured. When Hairer's colleague Quastel saw that he had obviously plagiarized me, he could have said, "Hairer ripped of Tooker," but since he works for the jews he said, "Hairer's work was done by aliens."

Martin Hairer takes $3m Breakthrough prize for work a colleague said must have been done by aliens
https://www.theguardian.com/science/2020/sep/10/uk-mathematician-martin-hairer-wins-richest-prize-in-academia-breakthrough

>>15336683
In fact, I have heard it said, "If you solve the Riemann hypothesis, you get tenure anywhere you want," and that is 100% bullshit because everyone works for the jews.

>>15336641
Thank you for your post. I will continue to try harder and stop being so petulant.

>>15336683
>I solved the Riemann hypothesis years

A little bit of a breakthrough. I think its solved. Just a basic homework problem, but it took me a long while, and I don't think I could have made it without the gap-bridging textbook example of cyclic groups of the same order being isomorphic.

Mood: https://www.youtube.com/watch?v=frKbks4IB9Y

The main purpose of the facility in Antarctica is to further the cause of Satan. Satan has deceived many people about that and they will adamantly deny the truth of what I say, but what I say is true.

>>15336770
>if I see a sign, how will I know?
https://www.youtube.com/watch?v=LsYvuxmzxX4

>stupid Tooker
>it's not because I work for the jews
>it's because I work for, uh...

>>15336746
Reasoning all looks solid. [math]<g>[/math] (which is the subgroup generated by [math]g[/math]) is nontrivial, so [math]<g>=G[/math] which means [math]G[/math] is cyclic with the same order as [math]g[/math]. Then [math]n[/math] must be prime because if [math]n=ab[/math] for natural numbers [math]a,b>1[/math], then you get nontrivial subgroups of order [math]a[/math] and [math]b[/math] generated by [math]g^b[/math] and [math]g^a[/math] respectively. [math]a<n[/math] and [math]b<n[/math], so those are proper, which contradicts your assumption. Nice job. Once you get more comfortable with this kind of reasoning, these arguments can get very concise and satisfying. Everything clicks nicely into place.

Do distributional derivatives have a natural generalization to multivariable? I see H(curl) and H(grad) spaces brought up, but am having a hard time finding a reference for their actual definitions and properties.

Are you having a hard time finding a reference in the paper whose whole purpose is discuss work which ought to be carried out in the future, shit cunt?

Are you having a hard time finding a reference in the paper whose whole purpose is to discuss work which ought to be carried out in the future, shit cunt?

>this work that one homeless guy did in his car in the spare time when he wasn't running away from his rapists isn't as polished as this other work that resulted from the collaboration of tens of thousands of people, possibly hundreds of thousands of people, across more than 100 years
>into the trash it goes

Do third order linear PDEs have a discriminant the same way second order linear PDEs do?

>>15336821
Awesome, thank you....
This course has been a rude wake up call, since until now I have been able to solve every homework problem on a homework assignment or test after just reading the chapter and a little bit of practice. Spending hours banging my head on one problem is pretty intense.

>>15336865
Why would it be different than for single-variable. You still move all the derivatives to the test function.

[eqn](D^\alpha T)(f) = (-1)^{|\alpha|} T(D^\alpha f)[/eqn]
[eqn]T \in \mathcal D'(\Omega) \\
f \in \mathcal D(\Omega) \\
\Omega \subset \mathbb{R}^n \\
\alpha \in \mathbb{N}^n
[/eqn]

If C is small and D locally small then [C,D] is also locally small right

>>15336945
Is it that simple, just plug in distributional partial derivatives? What about the function spaces, the multivariable Sobolev spaces won't just be Cartesian products of 1D Sobolev spaces will they?

>>15335250
Hey, is it possible to ploot something inspired by
the gif? (ignore the at&t part)

wait, is tensor-hom adjunction literally just currying that respects algebraic structure
It's so trivial then

>new techniques of far-reaching importance

>>15337649
fuck off retard
take your meds, touch grass and stay off 4chan

How do I solve this? I tried polar coordinates, I tried to use parameters but I can't get it to work, answer should be e, and I got far enough to kinda see it(limx->inf(1+1/x)^x = e) but that's still not correct since it's going to zero. I don't know, assistance would be appreciated

>>15337928
My first thought would be to set x=y but maybe the rate of convergence in the xy^2 term might mess that up. Second thought would be to let a=(x^2 + y^2) and b^(-1)=xy^2, and then take the limit as (a,b)-->(0,0). Maybe you get some familiar form for f^b(a).

>>15337934
I tried the first 2, couldn't get it to work. Doing your recommendation yields the limit being 1, but the textbook says the answer should be e. Maybe I messed up, I'm not sure, this one was difficult

>>15337940
I think the limit being e comes from setting x=y, doesn't it?

>>15337958
really, you don't even need to set x=y since the limit comes out the same if you take x->0 or y->0 first.

>>15337958
That's what I was thinking, but doing that gives me (1+2x^2)^1/(2x^2+x^3) which I don't see how I could get e from that

>>15337928
You can use the squeeze theorem.

[eqn] - \left( \frac{2}{3}(x^2 + y^2) \right)^{\frac{3}{2}} \leq x y^2 \leq \left( \frac{2}{3}(x^2 + y^2) \right)^{\frac{3}{2}} [/eqn]

So by going to polar coordinates
[eqn] \lim_{r \to 0} \left(1 + r^2 \right)^{\frac{1}{r^2 + \left( \frac{2}{3} \right)^{\frac{3}{2}} r^3}} \leq \lim_{(x,y) \to (0,0)} \left(1 + x^2 + y^2 \right)^{\frac{1}{x^2 + y^2 + x y^2}} \leq \lim_{r \to 0} \left(1 + r^2 \right)^{\frac{1}{r^2 - \left( \frac{2}{3} \right)^{\frac{3}{2}} r^3}}
[/eqn]

The limits on the LHS and RHS can easily be proven to be equal to e with L'hospital by using
[eqn] \lim_{r \to 0} \left(1 + r^2 \right)^{\frac{1}{r^2 \pm \left( \frac{2}{3} \right)^{\frac{3}{2}} r^3}} = \lim_{r \to 0} e^{\frac{\log \left(1 + r^2 \right)}{r^2 \pm \left( \frac{2}{3} \right)^{\frac{3}{2}} r^3}} [/eqn]

Is it worth reviewing and getting good at calculation based calculus before reading Spivak’s Calculus? I have a bachelors in physics but it’s been ten years since I did anything with that. Doing this because video games feel like I’m wasting my life away

>>15338005
Squeeze theorem is a major filter for me, in single variable I never got it, gonna have to try and understand it from now on so I don't miss out on too much. I'll try to understand what you've posted, thanks

Are there any books on Order Theory that present the subject from a modern and maybe more computer scienc-y POV?

More concretely (obviously doesn't have to deal with everything listed):
- Acknowledging the entire order theory <-> category theory stuff (peorders <-> categories, closures <-> monads, galois connections <-> adjunctions, pre-fixpoints <-> endofunctor algebras, ...)
- A focus on constructive mathematics
- Some stuff about domain theory
- Induction, coinduction, mutual induction and all the related fixpoint theorems

How much math do I need to understand the hodge conjecture? I have a BS in math so I took 2 semesters of algebra and 2 of analysis as well as 1 of complex analysis and 1 of differential geometry. As far as I can tell I'll still need

- Point set topology
- algebraic topology (up to cohomology classes)
- algebraic geometry (which I'll need to learn some commutative algebra before I start studying that)
- differentiable manifolds
- complex geometry

Is there anything else I'd need?

Is there a correct way to LEARN and understand abstract algebra, within the confines of a semester system and timed exams? The topics are interesting, but the pedagogy is build upon building up a bag of varied tricks rather than taking it slow and digesting broader concepts, and what meaning you can extract from certain problems. Yes yes, I could just memorize a corollary that applies to a very specific type of problem, which seems to be what is desired of students, but that is ironically more plug and chug than any even basic calculus. Should it not be "here is a structure, analyze its properties as it relates to concepts covered so far in the course". I'm probably wrong, but it would be way more fun, interesting, and less stressful that way, over getting textbook problems that require a very specific computation.

>>15338065
Paul Taylor's practical foundations of mathematics. Properly formated djvu's are online somewhere

Is doing a Ph.D in pure math worth it (I already have my masters)? I have a few offers and need to make a decision at the end of this coming week. Otherwise, I'm thinking of just doing A.I/Machine Learning.

>>15338065 #
Paul Taylor's practical foundations of mathematics. Properly formated djvu's are online somewhere..

>>15338296
you can read algebraic geometry as necessary, the complex geo is more relevant (they're the same by gaga anyway)
you also need some pde but most treatments will black box it

>>15338296
Me thinks you doth read too much. Just start playing around with the conjecture. Do simple calculations to see why it might be true. Dont read the literature, but do USE the literature. The easiest way to learn about a theorem in my experience is not to first learn all the background of every part of the theory, but just use the theorem in simple cases. In a lot of ways working with a thing makes its proof and place in the surrounding theory quite obvious.

>>15331769
assuming by R[x,y] you mean a polynomial ring, the factor ring is R^2. this is a standard algebraic result, if K[x_1,...,x_n] is your polynomial ring and you take the ideal m=(x_1-c_1,...,x_n-c_n) for a point c=(c_1,...,c_n) in K^n, the ideal m is maximal and the factor ring K[x_1,...,x_n]/m is naturally isomorphic to K^n

Does there exist an equation for any arbitrary closed shape? And by closed shape I mean like an island where you can walk from one point to any other point on that island.

I mean, if you put the shape on the Cartesian plane, does there always exist a corresponding equation for it (excluding parametric equations and other fancy equations)? I heard that you can't have an equation for a triangle but what if we exclude shapes which have straight lines or sharp corners.

>>15339215
The ideal is not m=(x-a,y-b) though. Also in your result, K[x_1,...,x_n]/m is isomorphic to K, not K^n.
The answer is R^2 as already explained by >>15331805

>>15333806
Yh, PDE is probably safe

>>15339500
Yes. The term you're looking for is a simple closed curve, and they all have equation representations if you accept an infinite series. You can center it at the origin and express it as a radial Fourier series.

>>15339833
But can you express is as an equation between X and Y? In the same way as you can express a circle as X^2+Y^2=1. What if instead of being allowed to use infinite series you were only allowed to use any real numbers.

>>15339852
You're probably looking for algebraic curves. Unless you're willing to also accept stuff like sin, e^x, etc. You should define the class of functions you care about.

>>15339875
Right, I think it's an interestig problem that way. Not allowing trigonometric functions. Only an equation involving a finite number of repetitions of addition, subtraction, multiplication, division, extraction of roots, and raising to powers.

>>15337235
yea but that was probably made on something like blender opposed to just plooting functions in c++. like a 3d modelling software which im not very good at

Thoughts on become a professor of history of mathematics ? Seems even harder to get in than regular mathematics

>>15340273
becoming*

>>15339926
>>15337235
That's interesting. Even if it's C++, it'll still be good.

what are some unsolved maths problems that are easy to understand but extremely difficult to get anywhere near solving

>>15340390
well, there's the obvious answer of Collatz
but for a slightly more varied one, a lot of graph theory problems fall into that category, because usually there's not a good way to prove your solution besides brute force

>>15325145
This is literally just a memoization leet code problem kek.

>>15340390
Riemann hypothesis in it's Goldbach type formulation. Looks something like
For all n in N, H_n > exp(H_n) * log(H_n)
where H are the Harmonic numbers.
I'll look it up 4u

>>15340602

>>15340273
That sounds like something fun to do on the side if you're already a professor of mathematics, but otherwise its kind of beneath the normal role of a math professor, isn't it? I don't mean to insult you or what you want to do, but I don't think history of mathematics is something that serious. Maybe if you're a professor already at a mainly teaching college where no one else does research, then you can still be a hotshot in your department and scale. Just the opinion of an anonymous guy, so don't take it the wrong way, I'm just very bad at phrasing things without coming off bad.

>>15340678
No I disagree - it's a field of study. I mean just like History of Gothic architecture is a field of study. There's conferences and books being written.

>>15340706
Okay, sure. But you should take care to separate it out from mathematics. If actual core math is called applied mathematics, and near philosophical bullshit is part of "pure" mathematics, by adding more things like history of mathematics into it, people will just lose respect for "pure" mathematics in general. I understand that you want to have the title "mathematician", but for me personally, its getting to the point that I am warry of it and don't believe until I see what their specialty is. In the case of a mathematics historian, they are not a mathematician anymore than an accountant is a mathematician. Analysis, topology, PDE's, those are real mathematicians. Logicians are already on the fringes with probability theory, you would be squarely outside of it.
Sorry to be so wordy, but what I think you should do, or what I would do if I were you, is just be a regular mathematician and also do math history on the side, that way you can keep your dignity.

>>15340779
It's a bit vague to me where the "near philosophical bullshit" starts or what it is (especially if you consider academic pure math as bullshit too - I suppose you mean super high dimensional topology questions etc.?)
But I seem to have a much higher view of academic history than you, if you think this study would pull down math. Working out the history of math is non-trivial academic effort as well.
At the same time, I don't think mathematics is something which has it necessary that we defend it's status. Math is sort of inevitably useful and valuable in this sense - and it's also "fun" at the same time.
As for logicians - those weren't viewed as part of mathematics before 1980, say, and it's fair to separate it as a discipline. Not that I could argue either of those to be higher value than the other.

>you should do
I'm not the guy asking the original question.
Just someone who read a few great math history books. Also helps understanding the math, as a side effect.

>>15340678
>>15340779
lol you sound like such a faggot.
>nooo your math isnt real math reeee
>u will lose ur dignity hurr
You know relatively few mathematicians even do research past their PhD, right? Plus history of math is an actual subject, studied by actual mathematicians. I mean pretty much every math class is a history of math class, since you typically study things already proven. The only difference is the historical trends are ignored for brevity.

can someone enlighten me?
I'm supposed to find the right sequence in which functions are put in their asymptotic growth rate (from slowest to fastest), but how do I evaluate them correctly? The right answer is supposed to be D

how do i show that the category of directed graphs (and graph homomorphisms) has coequalizers?

>>15340584
Well anon you're not supposed to use computers and calculators for it. Why did you say kek, what makes it funny?
>>15325424
Sorry about not getting back to you anon. I fought demons. I have read your solution. It's quite good! Really good! So great job to you. Thank you again for this. Appreciate it! HAVE A NICE DAY, I'M GOING BACK TO FIGHTING DEMONS.

Are there any definitions of absolute value for [math]C[/math] that don't involve the Pythagorean Theorem ?

>>15341145
Depending upon how strict you want to be, there's the polar form [math]|re^{i\theta}|=r[/math]

I'm fucking retarded. I have that [math] \sum_{k=0}^p {p \choose k}x_{p-k}=p^n [/math], where [math]x_p=S_{n,p}[/math] is the number of surjective mappings from the set [math] \{1,2,\dots,n\} [/math] to the set [math] \{1,2,\dots, p\} [/math]. How do I show that [math] S_{n,p}=x_p=\sum_{k=0}^p (-1)^k {p \choose k} (p-k)^n [/math]? The first part of the problem was proving the first identity, and you're supposed to deduce the second identity from it, so I don't think surjectivity should enter into the argument at any point. It should just be a proof by induction using basic algebra, but I can't figure it out. Help a brother out.

>>15341145
Yes. For a complex number z=a+bi you can define the absolute value of z to be sqrt(a^2 + b^2). This definition doesn't depend on the pythagorean theorem in any way, and is consistent.

>>15336535
Do you not feel satisfaction from proving something while making as few assumptions as possible? Also, as that other anon said, you'll most likely eventually gain respect for its universal applicability, even if it may be poorly motivated for now. Learning the exact statement of some definitions and theorems, even if it's fairly akin to learning vocabulary, will equip you with the ability to recognize these very general structures in more specific things, and rigorously formalize them on the fly.

Abstraction is present pretty much everywhere in mathematics and I'm sure you'll learn to appreciate it once it doesn't come off as arbitrary as it may appear to you now.

>>15340815
>read a few great math history books
Any recommendations? I'm specifically looking for something regarding geometry, as that is a rather old field that sprung off into a billion fields in the past century. I also don't know much about modern synthetic geometry.

>>15341210
My interest in history typically don't go as far back.
My fav. book is, I think, Modern Algebra by Corry. (2nd edition)
https://www.amazon.com/Modern-Algebra-Rise-Mathematical-Structures/dp/3764370025
Covers the perception of the discipline around 1800-1960 (Galois, Hilbert, Noether play a big role.)
For some casual short bios, this is nice
https://www.youtube.com/@moderndaymath/videos

>>15340815
>It's a bit vague to me where the "near philosophical bullshit" starts or what it is
I'm with you on that. I've no idea where people get these notions from nor can I figure out where the think the line is. Doesn't help I never find anyone explaining it.
>>15340779
>and near philosophical bullshit is part of "pure" mathematics
Echoing other people I am genuinely curious where you think that line is and why.

>>15341162
Use generating functions.
Or it is called binomial inversion/binomial transform

>>15331757
Thanks for this.
Spent a few days with self-pity.

But.
> it a) commutes with taking derivatives
Isn't the D for derivatives?

I'm not sure how this would work...
The squigglyN_k(psi) is a collection of sums of linear functions, for n = 1, 2, ...
I don't understand what you mean with "commutes with".

For [math]f:X\to Y[/math] and an [math]A\subseteq X[/math] we call the set [math]\{\:y\in Y\mid\exists x\in X\;(y=f(x)\land x\in A)\:\}[/math] the image of [math]A[/math] under [math]f[/math].
Is there a name for the "other image" that tends to pop up from time to time: [math]\{\:y\in Y\mid\forall x\in X\;(y=f(x)\to x\in A)\:\}[/math]?

>>15342540
It's [math]Y \setminus f(X \setminus A)[/math].

>>15342559
It looks more like something like f[A]

Fellas, should infinite products be covered in algebra or topology?

>>15342682
Infinite products of numbers are rather covered in analysis.

new thread, please

>>15342965
I made one
>>15343107
You're welcome.