/sci/ - Science & Math

algebra is for twinks

http://www.kurims.kyoto-u.ac.jp/~motizuki/news-english.html
2018-11-08
>(Past and Current Research) Updated the webpage containing a report and related documents on discussions conducted at RIMS in March 2018 concerning IUTeich:
http://www.kurims.kyoto-u.ac.jp/~motizuki/[Rpt2018]%20Revisions.txt
>Slightly modified the discussion at the end of \S 3
>Added (UnEx3)
>Rewrote (LVEx)

>(Past and Current Research) Posted the slides for lectures on explicit estimates in IUTeich by Arata Minamide:
http://www.kurims.kyoto-u.ac.jp/~motizuki/Minamide%20---%20Explicit%20estimates%20in%20inter-universal%20Teichmuller%20theory%20(in%20progress).pdf

>>10129492
WhAt KinD of alGeBra bRainLet

>>10129709
I did algebra in middle school you know, I'm a prodigy like that.

bump

>>10129486
Tell me about Hilbert, why did he hate Godel?

>>10130361
'Cause he was a punk bitch

>>10130361
Because Godel was the kinda person who looked at a problem for two minutes and said "can't be solved, I'll prove it can't be solved", then looked at it for another two minutes and said "can't be proved it can't be solved, will prove it can't be proved it can't be solved".

A'ight, so I figured out that Galois's intuition for group theory is actually the best thing since sliced bread and working with groups as permutations on a random set X allows me to do amazingly pretty stuff such as treating isomorphisms and homomorphisms as other permutations on X.
But I've got absolutely no idea how to extend this intuition to multiplication. Thoughts?

Wh-why is the thread so dead?

>>10131829
Remember multiplication is just repeated addition

>>10132268
That does help with abelian rings, but not with non-abelian ones.

What's the smallest Hilbert space you can think of?

>>10132283
Something seems off about this post. Just because you're an abelian ring, doesn't mean every element is an integer multiple of unity.

>>10132314
The vector space {0} over the field {Q} with the norm {0}.
>>10131829
Nevermind, automorphisms.

>>10132314
[math]\{0\}[/math]

>>10132322
you need an inner product
>>10132324
is this smaller than the empty set?

>>10132329
there is no empty vector space
you need a zero vector by definition

>>10132320
Yeah, but it's still a helpful intuition. Consider something like a×b=b^a, where b^a=b(b^a-1). So you do some wizardry and map 'em up to exploit this.
>>10132329
<0, 0>=0. Empty set isn't a vector space.

>>10132329
The empty set contains no identity element you brainlet. also the inner product is just 0*0

>>10132314
well it's definitely not your mother's vagina

>>10132337
A vagina is a manifold.

>>10132339
manifold with a boundry virgin.

>>10132335
So according to you ab = b(ab -1). It seems what their question is really asking is if every element is an integer multiple of unity. This implies abelian, but your condition seems weaker than this.

Actually what I'm saying isn't right either. The real numbers are a counter example.

>>10132347
>According to me
It's an intuition.
And the automorphism one is infintely superior.

Is publish our perish true for maths in most countries? It's the opposite here in the UK, but I was wondering if that's a special case because of how university funding works here.

>>10132283
Start by observing structures of rings without any units, because they obviously have the heaviest restrictions on how their structure can be interpreted. You can still think of every element h instead as a function that sends g to hg because of the associative property of multiplication, but without unity there would be no identity function. But the ring is still closed under multiplication, so of course the functions would have no inverses either, and as such every function must be either not 1-1 or not onto, or both.
What can be said about an element h in the Ring [math]R[\math], such that the function [math] h: R \rightarrow R [\math] with h(g) = hg is not 1-1? Well obviously by definition there are two distinct elements in R that, when multiplied on the left by h result in the same element from R. Is it possible for such a function to be onto but not 1-1? Well obviously not in a finite ring, there need to be infinitely many elements for such a function to exist.
What can be said about an element h in the Ring [math]R[\math], such that the function [math] h: R \rightarrow [\math] with h(g) = hg is not onto? Well by definition the image of R under h is a nontrivial subset of R. That means that there is no element [math]a \in R [\math] such that the function [math] ah : R \rightarrow [\math] with ah(g) = ahg is onto. Looking back at the structure of the ring, that means that multiplying any element on the right by h results in another element that will never reach some of the elements in R if you keep multiplying on the right.
All of this can be extrapolated from the fact that R is a semigroup under multiplication. Keep working along those lines and figure out where the distributive property comes in, because that's where the whole picture comes together.
t. autist

>>10131829
>>10132322
Could you impart some of this intuition on us? I took two semesters of abstract algebra many years ago and think it would be fun to revisit it except this time really going for an intuitive grasp

>>10129486
>True or false? If R is a ring, [math]V \subset W[/math] and [math]V' \subset W'[/math] are left R modules such that [math]W/V \simeq W'/V' \simeq _RR[/math] and [math]V \simeq W[/math], then [math]W \simeq W'[/math]

I'm really bad at intuition for modules. Can someone give me a hint? Or just tips for approaching questions like this? [math]_RR[/math] is the left regular module btw.

>>10132364
publish or perish is true for all research in all universities. I worked at a national lab for my internship and its true there too. there is no escape from it

>>10132364
In the USA/Canada, it depends. If you intend to work at (or are trying for tenure at) a high-ranking research university you absolutely must publish at least decent work on a regular basis.
At lower-ranking universities, the requirements are looser. You can get away with publishing less and/or publishing shit work as long as you don't make all your colleagues hate you. I went to a low-ranking liberal arts school for undergrad and I saw many professors there publish completely pointless meme papers for years and still receive tenure.
At teaching universities or colleges, they don't care at all as long as your students give you good reviews.

Any ideas on how to solve a sudoku puzzle. I feel this one is in reach of our current understanding. Seems like it would be simple, I just don't think we're approaching from the right viewpoint.

>>10131829
Multiplication = composition

>>10132781
Is there a proof that because a 16 clue puzzle doesn't exist, 15,14,13 and so on can't exist either?

>>10132781
What? Sudoku? Let's see it.

>>10132516
Guessing you mean [math] V \cong V'[/math]

>>10132798
Sorry I should have reworded, but this

Sudoku is known to be NP-Complete. There are no known algorithms to solve a general NxN Sudoku in polynomial time

>>10132579
How did that work in practice? Would someone get on your case if you took too long to publish? Or would they just fire you?

>>10132804
Basically, it's saying as we make bigger puzzles it's complexity to solve greatly increases. Instead of being like well the board is 3x as big so plug in a 3 here and baam here's the solution.

I believe this is not np-complete personally, and it's something we can solve with our current understanding of algorithms

I believe it's a problem related to squares and it's a series.

1x1, 2x2, 3x3, 4x4 (contains 4 2x2), 5x5, 6x6, 7x7, 8x8, 9x9 (contains, 9 3x3)

So at 81x81 (contains 81 9x9, and those 81 9x9, contain 9 3x3)

First # in series must be a 1, only 1 solution and completes puzzle (1x1),

next 2 only 2 solutions exist and can be found via 1,2,3 or 4 digits 4 is completed

As for 3x3, 5x5, 6x6, those puzzles might not be part of the series, but idk if we can eliminate just yet

This one is easy.
21 or 12
12 or 21

>>10132505
Okay.
Imagine an abstract set X and a set S of bijections from X to X that is closed under composition. If the set is closed, f=f^n guarantees the presence of the identity function, otherwise we have to specify it and also inverses are in the set. S is a group.
If I have two groups, R and S a homomorphism maps elements of R to elements of S, but we can also interpret it as a function from X to X. This might sound weird, but imagine X is the Natural numbers, and we have the five symmetric group swapping (1, 2, 3, 4, 5), and fixing the rest of the naturals and the seven symmetric group on (6, 7, 8, 9, 10, 11, 12). If a function φ from X to X maps 6 to 1, 7 to 2, etc, 11 to 1, and 12 to 1, then to each element of the seven group is associated an element of the five group. φ(6, 7)=(1, 2), and (11, 12) naturally is the kernel. Of course, we can still interpret the homomorphism as a function from one group to another, and its cleaner to define it innths old way.
Now I take every element of a group G and associate to each of them an automorphism, writing a=(f, φ), b=(g, δ) and c=(h, σ). We define sum as a+b=fog, and axb=foδ=goφ. Why we intepret a multiplication as an automorphism is really simple: distribution. If a homomorphism gives φo(aob)=(φoa)o(φob), then classical multiplication is a(c+d)=ac+ad!
I've only started formulating this, so sorry if it's a bit confusing or even wrong.

>>10133044
>then to each element of the seven group is associated an element of the five group.
Sorry I'm bit of a brainlet but is this assignment uniquely determined by phi?

>>10133098
Not quite. Looking back at the example, φ can map the rest of the points other than those involved in the permutation wherever. What does matter is the points involved in those two groups.
But I've actually been thinking some more and maybe the identity function has to be the identity function for every group, so instead of splitting X into unrelated groups I should embed them inside each other.
So instead if I have the seven symmetric group, the five group is a subgroup of it and a homomorphism from seven to five is an element of seven, while a homomorphism from five to seven is an element of five.
But I'm starting to confuse myself, so I'll try to formalize it better and come back later.

>>10133137
>Not quite. Looking back at the example, φ can map the rest of the points other than those involved in the permutation wherever.
Just to get this clear, I suppose we could, for example, have phi send (6 7) to (1), (7 8) to (2), ..., (10 11) to (5), with the rest as the kernel. Does this sound right?
If yes, I'm really beginning to lose sight of how this interpretation could be particularly useful... unless, of course, you've got an illustrative example up your sleeve!

If F is an ultrafilter of I on i_0 in I (that is, for all subsets X of I, i_0 in X).

Show that the ultraproduct of A_i is congruent to A_i_0.

Seems like an application of Los's theorem, but feel like there's something missing in my understanding to make that additional jump

>>10133187
It is an application of Los' theorem. The point is that two sequences (a_i) and (b_i) are equivalent iff they agree at i_0 (because F is exactly the set of subsets that contain i_0)

>>10133209
so if a_i and b_i are in the same equivalence class and agree on i_0, does that mean a_i_0 and b_i_0 are equivalent?

>>10133215
yes, and the converse is true

>>10133218
Thanks, can I just ask a conceptual question? When f ~ g = {f_i == g_i} is defined as an equivalence relation, what exactly are the f and g? What are these sequences sequences of? elements in the domain of some A in the model A_i??

>>10133221
Gosh. Here some facts. You have a formula \phi(x) (say one variable for semplicity of writing on 4chan). An element of your ultraproduct (or ultrapower) is a sequence (a_i), each a_i in the structure S_i. In the case of ultrapower (all structures are the same) you have a map S->S^I sending the element a to the constant sequence (a).
By definition, the ultraproduct satisfies \phi(a_i), where \phi is an atomic formula, iff the set of a_i such that S_i satisfies \phi(a_i) is in the filter. Los shows by induction on the complexity that, in the case of ultrafilters, this is true for any formula.

f and g would be elements of the product of the S_i, hence sequences (a_i) and (b_i)
(you quotient by that relation, but notice that you assign an ad hoc structure, not the quotient one; you can show that they coincide assuming small hypotesis).

>>10133221
Okay let's back up. You are defining an ultraproduct of structures A_i wrt an ultrafilter F on the index set I. You denote it by A.
The structure A is set-theoretically the quotient set of the product of the A_i's by the relation f ~ g <=> the set {i, f_i == g_i} is in F.
The f and g in the definition are "sequences" (f_i) where f_i is in A_i for each i, that is elements of the product of the A_i's. The elements of A are equivalence classes of such sequences.
You then make A into a structure by decreeing that the class of a sequence (f_i) has a certain property P if and only if the set of i in I such that f_i that has the property P is in F (I'm being handwavy here).
The fact that this makes sense is basically Los' theorem.

>>10129486
How long does it take you guys to soak up a new definition (at the grad level)? Do you tend to remember it well? This is always seems to be the bit that by far slows me down the most when I'm studying. Like 70% of my time seriously.

>>10133246
(cont. uninteresting case)
Assume now that you are considering the principal filter generated by an element i_0 in I, call it <i_0>.
Then (a_i) ~ (b_i) if and only if {i . a_i = b_i} is in the filter <i_0> if and only if a_i_0 = b_i_0.
This means that your ultraproduct is just S_i_0 (isomorphic, in the strongest sense).
No Los requirred.

>>10133044
Somewhat related, in category theory you can define a group to be a category with a single object, where all morphisms are isomorphisms. This captures pretty well the idea that all groups arise as automorphisms of some object in some category.

>>10132802
Yes

>>10133155
Fucking hell the revelations keep coming. Let's back up a bit.
Let [math]\mathbb{X}[/math] and [math]\mathbb{X'}[/math] be two sets, and [math]S_x[/math] and [math]S_{x'}[/math] be their respective symmetry groups. If [math]\phi :X \rightarrow X'[/math], then for any permutations a and c on X and b and d on X' with [math]\phi (a)=b[/math] and [math]\phi (c)=d[/math] we can write [math]\bod= \phi (a) o \phi (b)[/math], which is the traditional definition of a homomorphism and lets us relate homomorphisms on groups to mappings between the sets they are symmetric on that map group elements to grouṕ elements
>>10133270
Will look into, thanks.

>>10133258
What do you mean by soak up? Like, understanding the words and ordering in your head what you need to make sense of it? Or understanding the concept (which is usually much much more difficult)?
In both cases don't worry, speed in "getting" new concepts is nice but not necessary.

>>10133246
>>10133250
>>10133269
This is very helpful, thank you. I think you've addressed my uncertainties. It wasn't clear to me that these sequences (f_i) in A where just "tuples" or collections of f_i from each A_i for all i in I, represented as a single unit. And to say that i_0 in I is a filter is to say that all sequences in the ultraproduct A where a_i_0 = b_i_0 is actually just A_i_0 because A_i_0 satisfies \phi(a_i_0) for all \phi based on definition of ultraproduct.

>>10133294
More of the latter. I would say you soaked up a definition if you could list 10 examples and 10 counter-examples ranging from those which have obvious disqualifers to those which come deceptively close to satisfying the criteria but fail at some subtle point. More counter-examples if most subtle disqualifiers have not been covered. Extra points if listed examples tie to some other area of math.
At least this is my personal way of learning, but it doesn't change the fact that my comprehension speed is trash

>>10133269
>>10133269
Let me try this on a trivial case:

Suppose we have a filter {I} over I. Then (a_i) ~ (b_i) iff {i. a_i = b_i} in the filter <I>. A filter <I> can be defined as {X \subset I | i in X} then we can see for all sequences a_i, b_i, a_i = b_i and the equivalence relation is trivial.

Thus, ultraproduct = cartesian product i in I for A_i

Seem reasonable?

>>10133302
To be even more clear, to build said isomorphism consider the map
prod A_i /~ - A_i_0 sending (a_i) to a_i_0,
this is clearly compatible with operations and it is bijective by the argument given. To finish up you need to show that it preserve and reflects relations, so if you have a binary relation r for example,
you have to show that r((a_i), (b_i)) if and only if r(a_i_0, b_i_0) for every (a_i) and (b_i), that is phi(x, y) = r(x, y) an atomic formula.
But this is just by definition + particular case of principal filters.

>>10133337
I lost myself a bit. You are now considering an indexing set I, and the trivial filter {I}. This is not an ultrafilter, but in this case the relation is (a_i) = (b_i) if and only if {i . a_i = b_i} = I if and only if they are identical. So in this cae you have not an *ultra*product, because you are not considering an ultrafilter (it is called reduced product for general filters I think). But I agree with your final conclusion, in this case you obtain just the usual direct product.

>>10133307
I really recommend not comparing yourself to others too much, in particular since you have no in any way proper measurement how fast they arctually understand new stuff.

>>10132516
>>10132516
If one of V or W is finitely generated, then it is true, so you're gonna have to think non-noetherian... Good luck

I'm on the bus so I can't get pen and paper and work it out, but perhaps think of the ring Z[x_1,x_2,...] as a Z-module In infinitely many indeterminates and the submodule generated by all ideals of the form (x_i,x_j,...,x_k), that is, in finitely many of the indeterminates, and the ring of countably many copies of Z and the submodule consisting of "infinity minus 1" copies of Z, Hilbert-hotel style. Both quotients should be Z (the second one is obvious, the first one because any polynomial in infinitely indeterminates only consists of finitely many terms), however, the second ring is isomorphic to Z[X]

>>10131829
Multiplication as in the group operation [math]G\times G \rightarrow G[/math]? Think of [math]G[/math] as a category with [math]1[/math] object [math]\ast[/math] then the group operation are just arrow compositions [math]\ast \xrightarrow{g} \ast \xrightarrow{f}\ast [/math].
Now do the same thing but think of the category as acting on some set and viola.

>>10132806
They will refuse to renew your contract.

>tfw can't figure out if there are non-obvious embeddings or not

>>10132792
That's addition.
>>10133741
Will learn category theory then come back. Nice Touhou.
>>10133757
Of course you'll struggle some, they aren't obvious.

>>10133757
Update: there are. New question, are all embeddings conjugate? I suspect so.

>>10133796
Conjugate in what sense? Topological conjugates between embeddings are not in general well-defined.

>>10133808
Embeddings of groups, not topological spaces.

>>10133817
>not topological spaces.

>>10133817
Why not just say "semi-subgroups" or something then?

>>10133824
I don't know any topology desu, I can tell you the basic stuff you use in analysis i but other than that I'm clueless. I'll read munkres someday I promise.

>>10133817
The cheap and easy guide to finding subgroups:
Pick a random element.
Close up the set containing it.
Pick an element that's not in any previous subgroup and close that up too.
Once you start accumulating groups, take unions and close them up too.

>bachelor's degree in math
>three years experience as math tutor
>working as janitor at hospital
>can't teach at high school/middle school level because of muh "certification"
>friends who studied math education have decent jobs straight out of undergrad
>friends who took education theory classes and passed the praxis on the second or third try are more qualified to teach than me
>friends with shittier gpa are more qualified to teach than me
I don't get it.

>>10133830
Sorry we don't need any prison topologists.

>>10133855
still sounds comfy anon. just live minimally for now and pick a direction you want to go. why not shoot for those certs?

>>10133828
All the algebraists I talk to just say embeddings

>>10133892
This. Sounds like you're living a lower stress life and have time for hobbies.

>>10134155
How did you infer that? Dude's a janitor

>>10133855
It's like that movie with the math dude who's really good at maths whatsitsname.

>>10134212
No, no, the janitor dude who's really good at maths.

>>10134217
Good will hunting

>>10133855
>Bachelor's in biochem
>Go through a 2 month alternate certification program right after graduation
>Immediately get a job as a high school physics/chemistry teacher

Just get certified, it's really not that hard. You just need to sit through a bunch of garbage "how to make flipbook" style lectures, they need math/science teachers.

>>10133258
One of the best pieces of advice I got from one of my professors was to apply definitions repeatedly and you'll find you understand them.

This probably sounds obvious to many, but it bothered the hell out of me because pretty much the only way I can understand shit is abstractly, so when I can't have that understanding, I get frustrated.

But he told me that when he got to grad school, he basically threw the idea of "understanding" out the window; he learned the material as well as he easily could at first, and then used it.

He applied it to problems; he read theorems (even if he didn't entirely understand them) that used these ideas. He just exposed himself to them in anyway he could.

He told me his understanding basically increased immensely after adopting this approach, and frankly, I agree. He was one of the clearest lecturers I ever had and after taking his advice, I really found myself much more confident in my understanding.

Using concepts without understanding them might get you called a brainlet here, but for me, it's the fast track to understanding.

>tfw supposed to talk to freshman math majors about how to have a successful undergrad
>tfw I'm a lazy bum who's just failed upwards for four years
this feels unethical

>>10134227
Shit movie.

>>10134460
Anon, don't call people's lives trash like that. Think about how janitoranon feels.
>>10134372
Make some light-hearted jokes about how you failed upwards. I do mean light-hearted, things like:
"I'll be giving tips to you guys. Not things that I tried and worked, but things that I was told to do and got screwed over for not doing."

https://arxiv.org/pdf/1811.02418.pdf
>Non-trivial zeros of Riemann's Zeta function via revised Euler-Maclaurin-Siegel and Abel-Plana summation formulas
>Xiao-Jun Yang
>(Submitted on 3 Nov 2018)
>In this article, we prove that the real part of every non-trivial zero of the Riemann's Zeta function is 0.5 with the aid of the critical strip, Todd type functions and revised Euler-Maclaurin-Siegel and Abel-Plana summation formulas. The distribution formulae of the prime numbers and the twin prime numbers are discussed in detail.

>>10134495
>GM

>>10134495
>China
Not even checking if it makes sense.

>>10134495
>Todd function
So Atiyah was right all along? Kek, get rek'd incels. Old guy still has it!

>>10134495
Why do these nerds always have the lamest fucking titles for their papers. If you legitimately thought you solved RH you'd use some flashier.

>>10134545
>If you legitimately thought you solved RH you'd use some flashier.
t. reddit

>>10134555
sorry I forgot this was a no-fun board

>>10134545
Using a flashy name might turn people off from reading your supposed proof, even more so than the usual "proofs" of the RH do already.

fug I just solved p=np

>>10134638
no you didnt

>>10134466
All I'm saying is that the movie was shit.

>>10134639
yep, np complex problems can be broken down and solved where they lie on pascals triangle, and some manipulations

>>10134643
take a 1x1 latin sq there is only 1 latin square, and at 0 clue it's 1 board, 1 clue it's 1 board, 1,1.

Take a 2x2 latin square
1,4,6,4,1. Note that the 0 clue can solve x solutions, so in a 2x2 0 clue can solve 2 boards, like wise the n-1 clue is how many squares, and the nth clue is 1 because 1 full board can only have 1 solution

>>10134651
One 3x3 board is 1,9,......9,1
Added manipulation comes into play because take for example shidoku, a 2x2 (4x4 latin sq), sudoku is 3x3, (9x9 lating sq), now when you look at a
12
34, it is 16 solutions for one quadrant, 288 solutions for one shidoku complete board, so you'd need a 299 line pascal triangle, 9x9 will be difficult, but within the solutions you can solve how much overlap there is within each solution of the triangle, take a 2x2 latin sq, we see at 0 clue you have 2 solution, 3x3 latin sq at 0 clue we have 12 solutions, etc, etc

essentially the more you scale the complexity of the problem, the higher line it will lie on a pascals triange, this might only be applicable to certain p=np problems tho like sudoku

congrats, you showed that p is indeed a subset of np
this new information is sure to shock the math community

>>10134495
I can't fucking stop cracking up at this reference style
>cites 7 papers for the definition of zeta

>>10134659
no you brainlet, you can solve any np problem by just knowing where it lies on the pascal triangle, which pascal triangle is not np

Do you guys really think snails and dandelions are doing some multi millennial guess and check when they make their seeds/petals/shells. Do you think bacteria is doing some insane # of process when they're folding their proteins.... Nature will rely on the simple pattern to create complex structures Fibonacci, which is surprise within pascal triangle

>>10134664
this is illogical
if an algorithm lied in pascals triangle, it would be in polynomial time

something like the subset sum problem though runs in O(N*2^N) (with the brute force algorithm) which doesn't lie in pascal's triangle

your proof boils down to
>assume an NP problem can be solved in polynomial time
>it would be solved in polynomial time
>therefore P=NP

I dunno if this is the right thread to post this, but i need help.
I am mathematically illiterate (a result of brazilian education and no interest in the subject as a dumb teenager).
But now that i'm older (21) i see that math is basically a translation of the real world into ideas and logic.
I would love to learn it, but i really need to start with the basics, i can't even sum numbers without using my fingers.
Is there any course or website that you guys can suggest?
thanks :v

>>10134691
https://www.khanacademy.org/

>>10134643
>np complex
>mfw people still fall for this bait

>>10134689
this is the line triangle times summation

N=0, so 0
Line 0 * sum 1 = 0

N = 1 so 2
Line 1 * sum 2 = 2

N=2, so 8
Line 2 * sum 4 = 8

N=3 so 24
Line 3 * sum 8 = 24

N=4 so 64
Line 4 * sum 16 = 64

N=5 so 160
Line 5 * sum 32 = 160

>>10134736
so now we can take the subset problem, and find a pattern in relation to its pascal line

>>10134744
all you did was show that the line times the sum of the line is the same as N*2^N, this doesn't show that the problem can be reduced to an algorithm that can be solved in polynomial time

>>10134756
Just playing around we can say if 3 integers {123}, line 3 on triangle

they'll always be a multiple of 8.

123 can form
6 1 time
5 1 time
4 1 time
3 2 time
2 1 time
1 1 time
so that is 24

{222}
6 1 time
4 3 time
2 3 time
is 24

{392}
14 1 time
12 1 time
11 1 time
9 1 time
5 1 time
3 1 time
2 1 time
is 56

Ill keep looking to see if it means anything

>>10134809
i have no idea what you're even doing here
other posts i had a vague idea of what you were trying to convey but here i'm at a loss

also i don't think you know what the subset sum problem is

>>10134815
ye it was wrong, but subset is just trying to find subsets of a sum so {1,2,3}, and you have to check every sum they can form, but sometimes you can cut it off if you have a select value you particularly are looking for

>>10134817
https://en.wikipedia.org/wiki/Subset_sum_problem

>>10134818
Ignoring the far left box (triangle sides of 1)
There a # of solutions within that box. When given 3 integers, there exists 3 solutions in box 2, 3 solutions in box 3 and 1 solution in box 4 for a total of 7 solutions. We know that box 4 is simply the summation solution and there can only be 1 sum of all three integers. We know in box there exists 3 solutions, which is the 3 separate #s as themselves. Box #2 holds 3 solutions of "two integers".

Given 4 integers. We have 15 solutions. This is seen in the 4+6+4+1, remember we ignore the 1st box. The last box, and second box are the most obvious, the solution is all 4 summed, and each integer on its own. In box 3 we have two integers six times, and in box 4 we have three integers 4 times. So the difficult part is figuring out the solutions contained inside the boxes that aren't the last box, and the 2nd box in poly time

Anyone know of a proof of the prime number theorem that goes via Siegel's lemma/determinant method or something similar?

>go to my uni's library to find something on functional analysis
>Brazil
>find baby Rudin
Nice.

>>10135341
Which uni?

>>10135344
UFSC. There's still another copy there.
Please don't somehow dox me from this.

>>10135348
I won't, don't worry.
t. fellow esaguiano

>>10135348
I promise I won’t go to your house and fuck your girlfriend. There’s no reason for you to go to the library when you can download books from libgen.

>>10135484
Paper helps me focus.

>>10135484
Enjoy your shitty eyesight in your 30s.

>>10135484
>There’s no reason for you to go to the library
how about studying from your recently downloaded books

Can anyone recommend me a book about differential geometry?

>>10135789
>>10135377

>>10135789

>>10135789
Read every book

>>10135789
Tu

>>10135789
Milnor: Topology from the Differentiable Viewpoint (absolute must)
Lafontaine: Introduction to Differential Manifolds
Bott-Tu: Differential Forms in Algebraic Topology

>>10135789
If you are serious about your question, you should specify your background (how much analysis and geometry you know, if this is your fist contact with the subject, what you expect to learn).
>>10135864
disturbing

>>10129486
Math brainlet here. Can anyone explain this 1/12 thing?

>>10136763

>>10136763
Look at the wikipedia page for "1 + 2 + 3 + 4 + ⋯".

Basically, you use techniques which calculate the sum of a convergent series, and apply them to divergent series.

The interesting thing is that there are half a dozen different approaches which end up with -1/12.

>>10136635
I like how you changed the phrasing, but there are basically no jokes in it anymore.

>>10132781
Since sudoku is NP complete you can't have algorithm to solve it directly but using an artificial intelligence which tries trail and error it can be done but it can't be solved in one step

>>10134638
Show us the proof

>>10137523
That's not what it means to be NP complete, you can just code a solver that does bfs and finds a solution that's efficient enough for small N

>>10136763
it's actually analytic continuation of Riemann sum

horrible abuse of notation, the same symbols is used for different thing, the misunderstanding can be easily solved by inventing a new symbol

Let V be a vectorspace, W linear subspace of V, while H is a hyperplane of V.
how do i show that dim(H intersec W) = dim(W) -1?

>>10137592
oh and [math] W \not\subseteq H[/math], otherwise its obviously dim(W)

>>10136763
https://www.youtube.com/watch?v=YuIIjLr6vUA

I have to do an essay about mathematics and existentialism, any ideas /sci/?

>>10137592
Pick a basis B of W. There is a w_k in B not contained in H. Since H is a hyperplane, every other basis vector is contained in H. Hence B\{w_k} is a basis for H intersec W.

>>10137598
Both together?

>>10137592
That's wrong. For example, a line is a hyperplane inside R^2 and is also a vector space. If W=H, then their intersection is one-dimensional, which contradicts your equation, even if the usual case is the intersection being zero dimensional.

>>10137633
>>10137596

>>10137623
thanks
>>10137633
yea i added another condition in >>10137596

>>10129486
Does the theorem that every non empty subset [math]X\subset\mathbb{R}[/math] that has an upper bound has a least upper bound dependent on AOC?

>>10137780
No. https://en.wikipedia.org/wiki/Least-upper-bound_property#Proof_using_Cauchy_sequences

what is the relevance of quadratic forms to critical points and linear algebra?

>>10137821
Every quadratic form uniquely defines a symmetric bilinear form.

>>10137780
Nah. It can be derived from Dedekind's axiom for the completeness of the reals or often enough placed as an axiom.
Since anon already posted the Cauchy-sequences proof, I'll just say the Dedekind cuts proof literally instantly follows from the definition of a Dedekind cut.

>>10132781
every row and column are linearly independent.

>>10137523
I'm working on subset of a sum now. Latin sqs/sudoku is hard to write on paper past 4x4. But with subset of sum I'm putting it in a Pascal triangle which subsets itself down many Pascal triangles. Which is still an exponential function. A lot of it seems to be noise (repeat sums), so I'm trying to see if there's away to remove all the repeats, and solve all the solutions depending on how many integers each sum requires.

>>10132516
Yes, because of the 5-lemma.

>>10132516
>V is a subset of W isomorphic to W
>a ring V
Ey mate I'm pretty sure V is literally W, then again ring isomorphisms are weird stuff.

Is there any way to prove completeness of a metric space if it is topologically equivalent to a complete metric?

For instance, if I have two metric spaces in R2, one of them being the Euclidean metric which is obviously complete, and my other metric space is topologically equivalent, can I use that information to say anything about the completeness of my other metric space?

>>10138092
Yeah.
In fact, since they have the same topology, every Cauchy-sequence in one is trivially a Cauchy-sequence in the other, and since limits are entirely determined by topologies you can stack the results to get that.

>>10138092
No.
In fact R and (0, 1) are homeomorphic, the first being complete and the second one not.
Completeness is a strong-metric property, not topological.

>>10138109
Pretty sure anon means "If I have a set, and two different metrics that induce the same topology on it, does one being complete imply the other being complete?"

>>10138101
And that is a result of them both being metric spaces on R2, right?

From what I’ve read, this wouldn’t necessarily be true if they were topologically equivalent on different sets. The example given is that (0,1) and R are topologically equivalent but R may be complete and (0,1) is not. In that example, the same metric is induced on different sets, where as in mine, I have different metrics induced on the same set.

>>10138119
In that case he should employ a different word instead of 'topologically equivalent'. He should read this: https://en.wikipedia.org/wiki/Equivalence_of_metrics

>>10138127
>And that is a result of them both being metric spaces on R2, right?
Just say "metrics on R^2" unless you actually mean a subset of R^2 alongside a metric.
But yeah, if I have two metrics on the same set that induce the same topology, and one of them is complete, both are.
You'll see in functional analysis that R^2 has basically one topology that matters, but that's later.

Fuck probability and fuck statistics! Fuck rhetoric and ethics of STEM, too! I am counting the weeks (4) to cozy winter break. I just want to read my own damn books (NOT probability).

>>10138127
In fact you should find the right generalization on maps between metric spaces, so that when you are considering two metrics over the same set the equivalence becomes that property on the identity map. You could pull back (or push forward) the metric of R on (0, 1), so that you have two non strongly-equivalent, but topologically equivalent, metrics on the same space. Chop chop.

>>10138141
Your text is basically illegible so I'll try to translate what I think you're saying to the original questioner.
If you have the sets R and (0, 1), it's fairly trivial to see they're homeomorphic, but that one's complete and the other isn't. However, you can define a metric by d(x, y)=d(f(x), f(y)) which makes it easier to work with some things.

>>10138101
>since they have the same topology, every Cauchy-sequence in one is trivially a Cauchy-sequence in the other
>>10138136
>But yeah, if I have two metrics on the same set that induce the same topology, and one of them is complete, both are.
No. Consider the two metrics on R:
[eqn]d_1(x,y) = |x-y|\:,\: d_2(x,y) = |\arctan(x) - \arctan(y)|.[/eqn]
These define the same topology on R: a subset U of R is open in [math](\mathbb R, d_2)[/math] if and only if [math]\arctan(U)[/math] is open in [math]((-\pi/2, \pi/2), d_1)[/math], ie. if and only if [math]U = \tan(\arctan U)[/math] is open in [math](\mathbb R, d_1)[/math] since [math]\tan[/math] is a homeomorphism from [math]((-\pi/2, \pi/2), d_1)[/math] to [math](\mathbb R, d_1)[/math].

Then, [math](n)[/math] is Cauchy with respect to [math]d_2[/math], but not with respect to [math]d_1[/math]. In particular [math](\mathbb R, d_2)[/math] is not complete

>>10138196
Does that even satisfy the triangle inequality?

>>10138211
Actually, d2(1, -1)=0, so it doesn't satisfy jack fucking shit.

>>10138215
No wait shit.

>>10138215
>Actually, d2(1, -1)=0

>>10138232
Right, I figured out the problem.
https://en.wikipedia.org/wiki/Metric_space#Open_and_closed_sets,_topology_and_convergence
d(50, y)<pi/4 is the set [something something, infinity), which is open under the second metric, but closed under the first one, thus demonstrating they don't induce the same topology.

>>10138246
>which is open under the second metric
is it ?

>>10138246
>d(50, y)<pi/4 is the set [something something, infinity)
why is it closed on the left ?

>>10138256
Yup.
>>10138260
Forgot about the >. Anyways, (-inf, something something) is a closed set, which is even worse.

>>10137598
Just do a rehash of Tegmark's mathematical universe. Or if you want, you can use the flat lander analogy to talk about any number of metaphysical concepts, explain Plato's cave, or whatever.

>>10138271
>Anyways, (-inf, something something) is a closed set
But why ? I insist that this is not going to happen

>>10137922
Thanks fren

redpill me on fractional operators

>>10138286
You're right. I'm sorry, I've been kinda stupid today.

>>10138297
so you come up with a general form for any n as an integer, and then think "heh...what if i put in a half...." and then bam there you go, a useless but mildly interesting operator is born

>>10138313
>useless
really?

>>10138323
well not completely
sometimes it can make some nonlinear systems easier to model, but if something else affects the system you could fuck yourself over real easily by using fractional operands

>>10138286
No problem ! The neat thing about this is that it clears up a lot of confusion about the topology of metric spaces. Basically most metric properties (boundedness, completeness, Cauchy-ness) are not topological and are not preserved by passing to a distance that induces the same topology and this offers a simple counterexample to all three. So it also motivates the introduction of stronger equivalence relations (equivalence of metrics / bilipschitz / isometry) that do preserve these properties

why are there so many french posters here

>>10138335
Pourquoi pas ?

>>10137973
Huh, so that's good for something after all...

>>10138335
¿Cómo lo sabes?

>>10138331
>Completeness
Let A be an infinite sequence of open sets of X, such that the previous contains the next. If for any A the intersection of every element of the sequence is non-empty, the sequence is complete.
>but anon, you can't do infinite descending chains of sets
I know. Fucking set theorists ruining topology.

>>10137973
How do you know that there is a map that makes the diagram commute ? I agree that if such a map exists, then it has to be an isomorphism but there is no guarantee

>>10138354
>>10138358
lol

>category theory

>>10132516
In this particular case, yes because R is a free R-module.
Basically, if x lifts a generator of W/V and x' lifts a generator of W'/V', then W is isomorphic to V (+) Rx and W' is isomorphic to V' (+) Rx' in a way that is compatible with the isomorphisms V ~ V' and W/V ~ W'/V'.
Without the assumption that W/V is R, it would be false. For example, if R = Z, W = Z/4, V = <2>, W' = (Z/2)^2, V' = <(1,1)>, then V ~ V' and W/V ~ W'/V' but W and W' are not isomorphic.

>>10138360
What ?

>>10138362
What about it?

>>10138380
It's bad.

Using the concept of infinity. An almost near 0 value that you can keep flipping a coin and always land on tails. Why couldn't we say that you'll certainly land on heads. Or another example. You roll a 10,000 sided dice. As you approach infinity you're always bound to roll each number which would contradict that you can keep rolling without hitting a specific #/flipping a certain coin side.

>>10138457
Your question is somewhat confusingly worded. I think what you are trying to ask is
>why does (T,T,T...) have probability zero when it's a valid outcome?
The answer to this is that the elementary school idea that "probability 0 = cannot happen" and "probability 1 = always happens" is not necessarily true when your sample space is infinite.
So when someone says "you are certain to roll each number at some point", that means it happens with probability 1, which as explained above is not the same thing as saying every infinite sequence of rolls MUST include every number.

Do you guys ever get bored doing math? I don't know if I'm burned out or something but I don't have the motivation to learn anymore. But it's not like I can think of something better to do, either. Do you ever have such moments? I'm currently applying to grad schools and got a couple of nice offers but I'm scared of starting it and remaining in this state of apathy.

>>10138497
https://www.helpguide.org/articles/stress/burnout-prevention-and-recovery.htm

>>10138335
knowing french is a requirement to do algebraic geometry, after all...

>>10138489
Thanks that's what I was asking. Still a bit confused but I understand that they're different things at least. I understand everything but the last sentence. I know why an infinite long sequence doesn't have to include every number rolled, I just don't get how you'd relate that with the probability of 1 which is that you will roll each number at some point.

>>10138497
>I'm scared of starting it and remaining in this state of apathy
I am there, guy. It's bad.

>>10138497
I took a five year break between under grad and grad. Well, I failed out actually. After five years I came back and my interest was renewed. I probably just got burnt out from doing school Kindergarten to Bachelor's. Some people don't get burnt out though, so if you don't need a break go for it.

>>10138815
grug need help

I have just under 24 hours to learn most of this stuff, what's the most efficient way to spend my time /mg/? I was going to just marathon exercises but I don't know how a lot of the theory works since the lecturer basically didn't cover it at all.
Wut do? Basically I feel like I should be able to learn this in the time I have left, but I just don't know how to actually use the time.
cheers

How is it that the limit is equal to [math]\int_{E}f dm[/math]?

>>10139206
Because that's the definition of a direct limit.

>>10139214
I was more so interested in a more explicit answer.

>>10139206
>How is it that the limit is equal to [math]\int_{E}f dm[/math]?
Read the part after "Indeed, we have".

>>10139196
>I have just under 24 hours to learn most of this stuff, what's the most efficient way to spend my time /mg/?
Spend it shitposting in /mg/.

In a perfect world, we’re all number theorists.

>>10139196
lol ur fucked oh god hahahahahahahaha

>>10139345
any thoughts on this?

>>10139498
yeah i got a few thoughts myself
not many, but a nonzero amount

>>10139498
If you integrate it, won't you just get a finite sum?

>>10139206
Observe that [math]\int_{E_n} f dm = \int_{\mathbb R} f 1_{E_n} dm[/math].
If f is nonnegative, then the sequence [math](f1_{E_n})[/math] is nondecreasing and converges to [math]f[/math], hence you can use the Beppo-Levi theorem to get
[eqn]\lim_{n \to \infty} \int_{E_n} f dm = \int_E f dm.[/eqn]
Then, in general, write [math]f = f^+ - f^-{/math]

>>10139206
Observe that [math]\int_{E_n} f dm = \int_{\mathbb R} f 1_{E_n} dm[/math].
Since [math](f 1_{E_n})[/math] is nondecreasing and converges to [math] f 1_E[/math], Beppo-Levi yields [math]\int_{E_n} f dm \underset{n \to \infty}{\longrightarrow} \int_E f dm[/math].
In the general case, write [math] f = f^+ - f^-[/math]. Both [math]f^+[/math] and [math]f^-[/math] are in [math]L^1[/math] and you can apply the above.

>>10139206
Bumping, how long does it take to learn the following? I have 12 hours (going to put 4-5 aside for sleep) but I'm absolutely shitting myself, please help /mg/ I can't afford to fuck up this exam
>Signalling games
>Incomplete/imperfect information
>Bayesian equilibria
>Bargaining, finite horizons>Multistage and repeated games

>>10139674
Just curious are you a CS major?

Why is there no equivalent to this for math books? Can I have an anime girl holding Munkress or D&F please?

>complex integral took hours to solve
>check answer
>my answer is wrong and the real solute takes entire 3 A4 pages of steps

alright I've had enough of this shit

So onto Pascals triangle. It's looking like I may have found something. Take Fibonacci sequence. Set the lines as 0,1,2,3,4 etc. When given 3 integers you can eliminate the noise by cutting out boxes that would be within the pascal sequence. So imagine You have 3 integers block 1 is 0 digits, block 2 is 1 digit, block 3 is 2 digits and block 4 is 3 digits. Well we know that when 3 integers you can form three single pairs/three double pairs. Take the double pairs, and make 3 corresponding 2nd level pascal triangles to each pair. Then use fibonacci to eliminate redundant sums. The sums we want will always be in the last box in a 2nd level pascal triangle. Of course given 4 digits, all the 3 integer pairs will form 3rd level pascals and those will form 2nd level pascals. So maybe fibonacci can bulk eliminate. But who knows.

So imagine this.

You have one block of the pascal triangle. That block contains three pairs of two digits.

Inside that block each of those three pairs make more blocks (2nd level pascal triangle). The sums you want are the furthest right block of the 2nd level triangle. You want to grab those sums but there's too many. So we have a laser gun and we shoot into the block at an angle that makes a "fibonacci" line, and separate all the bad blocks from our good blocks, the problem lies in the fact that we have three triangles, so we would have to shoot all three blocks simultaneously

This where I'm at. If you figure out the solution plz cite this ty.

>>10139718
International relations, hence I am struggling

>>10139722
Fuck that, I want a math equivalent to this.

>>10139722
Yukariposter has dozens of those pics, but they're usually math applied to quantum mechanics.

>>10139722

>>10139966
based

>spend hours on simple problems, can't solve shit
>sleep
>brain is working at full throttle again

>>10140157
That's how sleep usually works, yes.

Can I get some tips on how to effectively draw and solve triple integrals? I'm having a hard time visualising the figure enough for me to draw it and correctly write the iterated integrals as a result of said figure. Any tips?

>>10140276
give me one and ill walk you through

Does algebraic geometry have application to solving geometric problems or is it basically just algebra?

>>10140817
It's just solving polynomial equations. Geofags like to pretend the Nullstellensatz connects ring spectra to "space" so they have an excuse to draw their little pictures.

>>10140844
Thanks, I can now completely give up on algebra and embrace the whole rest of mathematics.

>>10139832
How do I git gud at reasoning with all these abstractions?
I got completely lost in Graph theory's class today cause I couldn't figure out the huge amount of definitions and properties we relied on for our proofs.
If I only had graphs Ii wouldn't really have any problem, but there are so much other subjects and courses besides it!
Generally exercises are a good way to master a subject, but there are no exercises in the textbook. Tbh I don't see what kind of exercises one could make, it's literally just a bunch of definitions, properties and their proofs.
Could it be I'm just not cut out for this and I should seppuku?

How do I git gud at reasoning with all these abstractions?
I got completely lost in Graph theory's class today cause I couldn't figure out the huge amount of definitions and properties we relied on for our proofs.
If I only had graphs Ii wouldn't really have any problem, but there are so much other subjects and courses besides it!
Generally exercises are a good way to master a subject, but there are no exercises in the textbook. Tbh I don't see what kind of exercises one could make, it's literally just a bunch of definitions, properties and their proofs.
Could it be I'm just not cut out for this and I should seppuku?

>>10140856
Graph theory is fucking cancer, don't take it too hard.

>>10140817
Yes, obviously. It *is* a sort of geometry that studies geometric objects defined by polynomial equations (eg. vector spaces, conics, cones, elliptic curves...)
Now because these objects are defined by algebraic equations, all of their properties (connectedness, smoothness, dimension, etc.) are encoded in some way as algebraic properties of that system of equations. That is why algebra is such a central part of algebraic geometry.
But of course, this goes both ways: You can then interpret some purely algebraic problems in terms of geometry, which allows some powerful transfers of intuition (see the incredible analogy between the theory of Riemann surfaces and that of number fields)

>>10139722

>>10140572
Sure.
D is the region between z = x^2 +3y^2 and z = 8 - x^2 - y^2
Write a triple integral in accordance with this D.
The thing is, when I plug the equations into GeoGebra for example to a 3d drawing of it, I can easily solve this and other problems, for obvious reasons. My issue is translating this 3D graph onto a 2D paper, and knowing how to accurately and correctly shade the D so that I can start answering.
Thanks for your help, anon.

>>10140817
It does solve geometric problems
For example, at a basic level, you can attach a local ring at every point on a curve. Then the ring is regular iff the point is not a singularity.

>>10129486

i have to write about the brouwer hilbert dispute

Are there any merits to formalism or can i completely shit on it

>tfw too brainlet to instantly understand linear algebra in a couple of days before my test

>>10140844
>he only knows classical alg-geo

Are you all math majors/grad students? This looks fun but I can't understand most of it

why

>>10141544
why can you just suddenly add sums of ncubed. Don't you have to do that to both sides?

>>10141556
That's what I'm wondering. It all follows the normal analytic continuation rules for infinite series until the last step

>>10141561
Did you do that yourself or copy it from somewhere, You subtracted c-2c. incorrectly. 1-2(1) = -1, not 1.

>>10141544
>circles instead of periods
Spotted the woman. No wonder its wrong

>>10141544
>>10141556
>>10141561
>>10141568

>>10141568
It says c-2c = -c

obviously me

>>10141574
c = 1,2,3,4,5,6....
-c = -1,-2,-3,-4,-5...
They'd cancel out no?

>>10141575
Not in the methodology used for the -1/12 result. The odds would stay

>>10141575
-c=1,3,5,7,9,11 is incorrect.

You'd get 0 = all positive n sums + all negative n sums

>>10141578
Is it C^2, and not 2C, and then you you could choose the even sqs to be + or -.

>>10141579
If there was an implied time component or an implication that we can't reorganize by index, that would be the only answer. For divergent infinite series I'm not sure what the rules are

>>10141594
Z = 1 - 1 + 1 - 1 + 1 - 1 ......
1 - Z = 1 -[ 1 - 1 + 1 - 1 + 1 - 1 ....... ] = 1 - 1 + 1 - 1 ..... = Z
1 - Z = Z
2Z = 1
Z = 1/2

Adding the evens, or 2c = (-2/12) to Ramanujan's result here (1/4) also gives +1/12. Equal to -c -(-1/12)

>>10141617
The idea behind this one is that ((-1)^(n+1))/n will give a result of 1/(n+1) clearly for all n>1. The result is assumed for n=1 but the steps aren't usually shown because it's basically weaker than induction

https://www.youtube.com/watch?time_continue=6&v=w-I6XTVZXww

lol this is fucking retarded. Just take the average, wtf

>teacher gives an exercise, between many of them
>says B follows easily from A
>spend the whole night trying to figure out, can't sleep
>itdoesnotmakeanysense.jpg
>finally get the guts to think that maybe the claim is false
>outline a proof of that, use the whole arsenal to complete
AHHHHHHH.

>>10141822
what happened?

>graduating in the spring
>applying to masters programs
>all my friends know what they want to study
>I'm a part of a research project but don't really know what I want to study yet
>mfw

>>10142087
>all my friends know what they want to study
your friends _think_ they know what they want to study
provided you're talking about pure programs, I would bet you money that 100% of them actually know fuckall about the topic they're telling you they're going to work on

>>10141578
-1/12 is dumb. A summation with no alternating sign can’t have two values.

>>10140844
But, it does.

>>10141822
need details

>tfw you know that vectors are actually just magnitude and direction

>>10142364
direction is also a vector

What are the mathematics behind labyrinths? (building & resolution)

>>10142504
Graphs, I'd imagine.

>>10142364
Shut up EMH

>mathcucks think induction is a legitimate means of proof

>>10142902
Why wouldn't it be?

>>10142902
Induction in general is not considered acceptable in mathematics, i.e. "examples are not proof". Arithmetical induction is an exception because the natural numbers are "simple enough", although it is still an assumption. I wonder if induction works for directed sets with more than one initial elements.

>>10142902
>mathcucks who think that sets exist
>mathcucks who do math
>mathcucks who comunicate
aha.

>>10142902
>finitismfags reject a fundamental technique in mathematical proof just because they can't grasp a domino analogy
lmaoing@ur.life

Holy shit I just dreamed in 5D, my mind is still blown.

In the dream, people's bodies were going to get an upgrade. There would be a new spatial dimension that the new body can access.

Perception was kind of distorted and moving around was hard. I walked like a cripple in 5D, can't get used to it.

>>10143370
What makes you think your dream really was in 5D?

how do I count the edges of an n-dimensional (unit) cube?
I get that edges are all shifting exactly one bit, so an edge exists between (0,0,0,0,0) and (0,0,0,0,1) but not between (0,0,0,0,0) and (0,0,0,1,1)

>>10142902
Mathematical induction is not the same as inductive reasoning.

>>10143377
Hint: two vertices are connected iff their coordinates differ only by one entry.

>>10143405
yeah I mean that's kinda what I said that I already get. I guess it's a pretty quick step to counting them from there but I need to set up how to count it.

>>10143377
>how do I count the edges of an n-dimensional (unit) cube?

Are you asking how many edges or the value of each edge?

Total# is just n*2^(n-1)

Anyone know a good book (or books) on intro to functional analysis? preferably at the advanced undergraduate level, not graduate.

>>10143475
>advanced undergrad
Is that code for "I don't know topology and/or measure theory"?
Baby Rudin goes through the general definitions of that stuff in the beginning of his Functional Analysis.

>>10143497
no, it's code for not graduate.

>>10143475
This book ===> Infinite Dimensional Analysis: A Hitchhiker's Guide
It's the one with the least prerequisites (only calculus and linear algebra)

>>10143497
I mean advanced level undergrad would be senior level classes if you just take the scope of undergrad classes.

Maybe I'm just dumb it's shocking to me how isolated most research seems to be, like they'll be a group of 20 to 30 people in the world working on a topic that no one else has even formulated. Then once results are found they're published and only read by people in the original group and maybe some more that are on the fringe, maybe from collaborating with someone in the group or they're a former student of someone in the group or something. The result is a lot of progress is made, but in no unified direction and to the knowledge of none of the community at large. It makes me think we could have some major breakthroughs (like whole new fields emerging) if people could just sit down for say, a decade, and JUST work on synthesizing the developments of the last quarter century (or longer). But because the modern university system is a huge fucking meme no one will ever keep a position long enough to accomplish it.

>>10140953
Thanks. If you guys have more please post.

>>10143498
>>10143505
You do know that different universities have different curriculums, right?

>>10143420
The total, but more interested in how to count that.
I love combinatorics, it's really comfy to me, but I am pretty shit at it (so far)

speaking of texts with only linear algebra and calc prerequisites, does anyone have recommendation of texts with only that as prereq but that have good solutions manuals?
any topic really, and I have some diff eq under my belt too but really I just love problems, and hate when I can't check my solutions (not in university)

>>10142995
Induction works fo partial orders which are well-founded (no infinite decreasing sequence)

>>10143548
also I'll submit one such text is Gil Strang's Differential Equations And Linear Algebra, love that dude, most all solutions on a corresponding webpage and quite a lot of worked examples in the text

>>10143537
http://www.math.brown.edu/~banchoff/Beyond3d/chapter4/section05.html

This is a good read. Essentially starting with a
square has 4 vertices, 4 edges, 1 square face. Then you compare that to a cube, then a 4n cube, and so on.

But really I just remember the formula
vertices is 2^n
and edges is n*2^(n-1)

>>10143537
You can prove the formula by induction. For n=1 you have a segment which has 1 edge. For a n>1 a n-dimensional cube is made of 2 (n-1)-dimensional cube connected vertex by vertex, and there are 2^(n-1) vertices in a (n-1)-dimensional cube. It follows that—if K is how many edges has a (n-1)-dim cube—a n-dim one has 2K+(2^(n-1)).

Thus, starting for n=1 and counting, the number of edges is:
1=1*2^0
2+2^1=2*2^1
2(2+2^1)+2^2 = 3*2^3
etc

>>10143567
Thanks
>A square has 4 edges, and as it moves from one position to the other, each of its 4 vertices traces out an edge. Thus we have 4 edges on the initial square, 4 on the final square, and 4 traced out by the moving vertices for a total of 12
This is how I started looking at it, but I guess I have very limited ability/interest in visualizing higher dimensions spatially so I suppose I'm pondering on how to work from the fact that all edges are connections of vertices with one bit flipped, and how to make sure I'm counting systematically. And I really have no need or interest in memorizing, but I'll read this more thanks.

>>10143509
Improvement makes strait roads, but the crooked roads without Improvement, are roads of Genius.

>>10143576
>>A square has 4 edges, and as it moves from one position to the other, each of its 4 vertices traces out an edge. Thus we have 4 edges on the initial square, 4 on the final square, and 4 traced out by the moving vertices for a total of 12
That's where you use induction. Nobody visualizes 198471-dimensional solids, at least not me.

>For a n>1 a n-dimensional cube is made of 2 (n-1)-dimensional cube connected vertex by vertex, and there are 2^(n-1) vertices in a (n-1)-dimensional cube.
So for n=2, a square, is made of 2 n=1 segments, connected vertex by vertex
For n=3, a cube, is made of 2 n=2 squares, connected vertex by vertex

This feels messy to me, but I'm probably dumb.

>>10143595
is there a better way to work from (x1,x2,...,xn) for xi in 0,1

>>10143597
>better
don't focus on that, I regret it. But surely there's a combinatoric solution to this that makes sure I've counted the correct number of edges for the n-cube.
I guess I could look at the formula given n*2^(n-1) and try to figure out the combinatoric solution. It's more fun to try and think about it though, but I'm weak at combinatorics despite loving it.

Does anyone have a text recommendation on specifically combinatorics btw? It always seems to be an aside in discrete

>>10143503
this is cool, thanks

>>10143597
Dude, you're counting edges there's no need for coordinates. Induction is your friend one you get to know each other better

If you insist though, I guess take the standard n-cube whose vertices have "easy" coordinates (all 0 or 1). An edge always connects two vertices who differ only in one coordinate—for instance for n=3 there's an edge connecting (0,0,1) to (1,0,1) (an edge parallel to x axis) and an edge connecting (0,0,0) to (0,0,1) (parallel to z axis) but none connecting (0,0,0) to (1,0,1).

There are n coordinates. For the first coordinate x1, edges parallel to the x1 axis all connect a vertex (0,?,?,...,?) to the corresponding (1,?,?,...,?). How many combos are there for those n-1 question marks? Each can have value 0 or 1, so it's 2^(n-1). So there are 2^(n-1) edges parallel to the x1 axis.

By the same token, you can see that there are 2^(n-1) edges parallel to the x2 axis, and so on. Thus the total is n times 2^(n-1). The end.

>>10143636
Sorry if I sound condescending. It's a regional accent.

>>10143636
Thanks, this is a great explanation.
I did enjoy induction and its various forms but I'm quite a bit out of university and in my specific case, a counting solution such as this is more straightforward. I appreciate it, thank you.

>>10143647
/dpt/'s solution btw

>>10143692
I don't mean to foster any kind of competition or superiority I just think it's cool to see different approaches and attempts

brainlet here, is it possible that ZFC is inconsistent or is it pretty clear that it's not, even if we can't it in-theory?

any of you worked on the mathematics of music?
speak on this if you would

>>10143700
https://sites.math.washington.edu/~morrow/336_09/papers/Ada.pdf

I would love to know what smarter people think of this

>>10143700
The picture? The green lines are minor thirds, the red are major fifths, and the blue are major fourths. I don't know what the embedding on the torus is supposed to mean.

>>10143739

I'll believe it when I hear it

>>10143695

it's obvious that ZFC is nonsense and all useful results obtained using ZFC are coincidental

>>10143749
there has been a lot of work on mathematical music theory, you can seek it out if you'd like

>>10143746
>I don't know what the embedding on the torus is supposed to mean.
I'm not sure how it could be more clear

>>10143772
Do tell

>>10143374
sorry, i meant 35D and i was dreaming about anime tiddies

>>10143548
why not try "putnam and beyond"? Or perhaps some IMO books?

>>10143374
Because that was what the authority figure in the dream said. Our bodies will get upgraded to access another spatial dimension. What happened was movement just got awkward, like I can't go in a straight line, moving wrongly actually feel painful, and my vision has something like fisheye lens distortion (never really saw the other dimension).

>>10143783
it wraps around the x and y my boy

>>10143790
haven't heard of either
well I've heard of putnam and that's a great idea because they publish solutions correct?
But I'm also not wanting exclusively challenging stuff, I like to grind problems. It feels GOOD.
I have been going through Stewart's Pre-calc and honestly gotten a bit stumped now and then.
I'd like a solutions manual for this.

There's nothing wrong with grinding problems and some of you fucks don't understand that with your imposter posteur bullshit.

>>10143823

>>10143823
I don't know if they publish solutions, but most of the problem books like Putnam and beyond let you work on the problem, then give the full solution.

>>10134245
>>Immediately get a job as a
>>high school physics/chemistry teacher
ehhhhh

Quit your job
Make Videogames

baby
https://www.youtube.com/watch?v=A_MjCqQoLLA

https://www.youtube.com/watch?v=j6UoYi1EADs

/agdg/ - The Amateur Game Development General
Thank you, good night

https://www.youtube.com/watch?v=sYMHBN4xQr0

>>10144354

>>10142492
More like an element of projective space.

Some masters degree student claims that this function is only definied for x[-pi/2,pi/2], but I would argue for it being defined for all real x.

Who's right?

>>10144738
I should add that we're looking for real solutions.

New thread when

Can this be our last thread? Thanks.