/mg/ mathematics general

[Anonymous No.11081694]

/mg/ - mathematics general
John Tate edition
https://en.wikipedia.org/wiki/John_Tate
previous
>>11070543

[Anonymous No.11081767]

first for based graham

[Anonymous No.11081780]

Favorite set of matrices closed by some endomorphism (or operation, like matric multiplication) of your choice?

[Anonymous No.11081812]

>>11081780
C

[Anonymous No.11081819]

So its been a while since iv been in education, my math skills are on a decline, what's a good way to refresh my memory and skills on highschool math. I know this board is more meant for people who know what they are doing and discussing high-level stuff, but this is a general thread, it's not like I bumped off entire thread for this dumb question and I feel you guys would be the most qualified to help me.

All and any advice is appreciated. I hope you all are having a good day or evening, I'm about to head to bed but I'll check this thread in the morning and if need will answer any questions.

[Anonymous No.11081823]

>>11081819
khanacademy.

[Anonymous No.11081827]

anyone know of any recent developments with IUTT?

[Anonymous No.11081828]

>>11081823
I'll check it out tomorrow!

[Anonymous No.11081831]

>>11081827
It's dead. Nobody cares anymore. Mochizuki didn't prove anything. ABC stands open.

[Anonymous No.11081833]

>>11081831
>Mochizuki didn't prove anything.
[citation needed]

[Anonymous No.11081834]

>>11081819
Watch Khan or something.

Once done that get a introductory book on proofs, it will make math so much easier once you get to more advanced topics.

[Anonymous No.11081835]

>>11081834
>Once done that get a introductory book on proofs
proofbooks = memebooks

[Anonymous No.11081839]

Threadly reminder to work with physicists.

[Anonymous No.11082112]

>>11081839
actually true, you can't solve any problem without intuition and most of the intuition comes from physics

[Anonymous No.11082122]

>>11082112
>most of the intuition comes from physics
It doesn't. Just because it comes from the physical world you are experiencing it doesn't come from physics.
In fact physics contradicts about every intuition we have about reality.

[Anonymous No.11082133]

>>11082112
Very little of my intuition comes from physics tbqh.
It's largely geometric.

[Anonymous No.11082159]

>>11082133
>tfw no intuition

[Anonymous No.11082168]

>>11082159
>tfw blind deaf and retarded

[Anonymous No.11082178]

>>11081834
>get a introductory book on proofs
Only read the parts on De Morgan's law and Contrapositives (so like 3 pages). The rest is useless filler.

[Anonymous No.11082347]

>>>/g/73268912
>>>/g/73268941
>>>/g/73268960

[Anonymous No.11082352]

>>11082347
You'll eventually learn to expect nothing from /g/.

[Anonymous No.11082363]

>>11082352
>>>/g/73269034

[Anonymous No.11082377]

>>11082363
>>>/g/73269138

[Anonymous No.11082384]

>>11082133
It's fair to say that geometric intuition is 3D space intuition and 3D space & things in it is physics

[Anonymous No.11082410]

>>11082363
>>11082377
Jesus Christ...

[Anonymous No.11082517]

>>11082384
> geometric intuition is 3D space intuition
Blocks your path

[Anonymous No.11082521]

I have a course on representation theory of finite groups, but the exercises from the tutorials are waaayy easier than those from exams. Does anyone know a book on representation theory with some good useful exercises?

[Anonymous No.11082556]

>>11082521
The exercices aren't enlightening or anything but they're pretty difficult imo.
>pic related

[Anonymous No.11082591]

>>11082384
>and 3D space & things in it is physics
But it isn't. Your intuition doesn't agree with physics.

[Anonymous No.11082635]

Whats the best math textbook for differential equations? I want some comfy self study

[Anonymous No.11082642]

>>11082635
second this,
are there any good rigorous texts on systems of linear differential equations and stuff that explains constants, functions that give solutions, functions that turn constant on solutions etc

[Anonymous No.11082671]

>>11082635
Differential Equations with Applications and Historical Notes Textbooks in Mathematics: George F. Simmons

[Anonymous No.11082674]

>>11082635
but I've heard that anything by Arnold is a fantastic read

[Anonymous No.11082690]

>>11081694
Can someone explain why there is NO LOSS mathematically in the equation:
>E=mc2
Shouldn’t it be:
>E=mc2
————-
0.0000001
To represent packet loss, or do conservation laws really let something transform back and forth without any information escaping?

[Anonymous No.11082696]

>>11082690
0.9999999999999 not 0.0000000001 sorry. Anyway it seems like some should be loss so the conversion shouldn’t be exactly the square of the constant multiples by the mass, just almost, right?

[Anonymous No.11082706]

>>11082521
http://www-math.mit.edu/~etingof/repb.pdf

[Anonymous No.11082753]

>>11082690
https://arxiv.org/pdf/physics/0308039.pdf

[Anonymous No.11082755]

>>11082690
>packet loss
try /g/

[Anonymous No.11082965]

>>11082960

[Anonymous No.11083002]

>>11082674
Arnold's ODEs is an overrated meme book.
Concepts are introduced without any real sensical structure and with little explanation.
>hurrdurr its intuitive
Maybe it's better in Russian.

>>11082635
>>11082642
Teschl's book (ODEs) is OK.
Hirsch and Smale's book (Differential Equations) is good in the first edition, but that's hard to get hold of. Maybe it's good in the second and third (with Devaney) but I haven't read either of those and have heard bad things.

A book I recently came across and was really impressed by was Differentiable Dynamical Systems by James Meiss.
I thought everything was explained well, and there's a good amount of diagrams etc. It covers stuff other books lack as well.

All these texts are late undergrad +

For more introductory stuff see >>11082671

[Anonymous No.11083181]

>>11082671
Extremely comfy book, seconded.

[Anonymous No.11083217]

I've been studying spaces of non-positive curvature, is there any issue with picturing [math]X( \infty )[/math], the space of points at infinity, in terms of the sphere in the tangent bundle?

[Anonymous No.11083245]

I'm going to go through Zorich's Analysis starting today with zero proofs experience, zero calculus experience, and minimal precalc experience. I just barely finished learning trigonometry yesterday.

Ask me anything.

[Anonymous No.11083333]

>>11083245
how much dick can you fit in your ass?

[Anonymous No.11083339]

>>11082671
>Simmons
into the trash it goes

[Anonymous No.11083392]

>>11081694
Sigma to infinity = - 1/12

[Anonymous No.11083416]

>>11082690
>packet loss
>physics
Try >>>/lgbt/

[Anonymous No.11083427]

>>11083245
Why are you so much of an attention seeking brainlet that you seek approval from fatass weebs on an american anime discussing for. Also does your nail polish match your purse and high heels faggot.

[Anonymous No.11083485]

>>11083245
How many cocks can you fit inside your mouth, retard?

[Anonymous No.11083491]

I really want to be a mathematician and iv'e just finished Calc 2 and I found it mildly challenging.

How much harder do things get? I have a verbal IQ of 134 but I nigger tier working memory. Can I make it?

[Anonymous No.11083493]

>>11083491
>iv'e just finished Calc 2 and I found it mildly challenging.
uhh yeah, you haven't even begun real math yet so strap in

[Anonymous No.11083499]

>>11083493
I'm aware but when do I get to real math? Multivariable/vector calculus? Differential equations?

What's the filter course that will let me know if i'm gonna make it or not?

[Anonymous No.11083500]

>>11083499
Proof based courses. Analysis, algebra, etc.

[Anonymous No.11083504]

>>11083500
So even if I ace differential equations and linear Algebra I still might be a retarded brainlet that has no future?

[Anonymous No.11083508]

>>11083504
No, there's still "applied math" for you. Just don't think you "know" math because you do good in a few first/second year baby courses.

[Anonymous No.11083531]

>>11083485
>>11083427
>>11083333
christ this board is gay

[Anonymous No.11083554]

*flips book to random page*
a-HEM

If [math]S[/math] is self adjoint and [math]U=\exp(iS)[/math], show that [math]U[/math] is unitary. Deduce from this, and from the fact that [math]\sigma(U)[/math] lies on the unit circle, that [math]\sigma(S)[/math] lies on the real axis.

[Anonymous No.11083598]

*flips book to random page*

Let [math]A,B[/math] be abelian categories, let [math]A^{\bullet}[/math] be an exact complex in [math]C(A)[/math] and let [math]\mathcal{F}[/math] be a right-exact functor [math]A\to B[/math]. Let [math]K^{i}[/math] be the kernel of [math]d_{A}^{i}[/math]. Prove that [eqn] H^{i}(C(\mathcal{F})(A^{\bullet})) \cong \text{ker}(\mathcal{F}(K^{i+1})\to\mathcal{F}(A^{i+1})) [/eqn]

[Anonymous No.11083626]

>>11083598
>category tranny pseud at it again
You dont understand what you are talking about. Just kill yourself already.

[Anonymous No.11083630]

last one
*flips book to random page*

Let [math]R[/math] be a Noetherian ring, [math]I[/math] and ideal of [math]R[/math], and consider the Rees algebra [math]\text{Rees}_{R}(I) = \bigoplus_{\ell\geq 0}I^{\ell}[/math]. It is known that [math]\text{Rees}_{R}(I)[/math] is Noetherian (check this).

>Part 1:
For [math]\ell\geq 0[/math], let [math]J_{\ell}\subseteq I^{\ell}[/math] be ideals of [math]R[/math], and view [math]J := \bigoplus_{\ell\geq 0} J_{\ell}[/math] as a sub-[math]R[/math]-module of [math]\text{Rees}_{R}(I)[/math]. Prove that [math]J[/math] is an ideal of [math]\text{Rees}_{R}(I)[/math] if and only if [math]I^{n}J_{\ell}\subseteq J_{\ell+n}[/math] for all [math]\ell,n \geq 0[/math].
>Part 2:
Assume [math]J := \bigoplus_{\ell\geq 0} J_{\ell}[/math] is an ideal of [math]\text{Rees}_{R}(I)[/math] . Prove that [math]J[/math] admits a finite set of homogeneous generators.
>Part 3:
Choose a finite set of homogeneous generators for [math]J[/math], and let [math]s[/math] be the largest degree of an element in this set. Prove that [math]J_{s+1}\subseteq I^{s+1}J_{0} + I^{s}J_{1} + \cdots + IJ_{s}[/math] (as ideals of [math]R[/math]).

Good luck.

[Anonymous No.11083634]

>>11083626
>sees the word "category"
>spergs out
if you aren't attempting the problem please refrain from replying.

[Anonymous No.11083642]

>>11083634
Why would I want to solve a problem you asked? Just so you can post it on your math instagram. I want to shoot you. No matter how hard you try, you will never be anything more than a failure. You cant even convince anybody that you belong to the gender that you wish you were.

[Anonymous No.11083722]

>>11081827
>anyone know of any recent developments with IUTT?
http://www.kurims.kyoto-u.ac.jp/~motizuki/project-2020-english.html

>Invitation to Inter-universal Teichmüller Theory
>Period: 2020-09-01〜2020-09-04
>Organizers:
>Yuichiro Hoshi (RIMS, Kyoto University)
>Shinichi Mochizuki (RIMS, Kyoto University)
>Ivan Fesenko (The University of Nottingham, UK)
>Yuichiro Taguchi (Tokyo Institute of Technology)

>Inter-universal Teichmüller Theory Summit 2020
>Period: 2020-09-08〜2020-09-11
>Organizers:
>Yuichiro Hoshi (RIMS, Kyoto University)
>Shinichi Mochizuki (RIMS, Kyoto University)
>Ivan Fesenko (The University of Nottingham, UK)
>Yuichiro Taguchi (Tokyo Institute of Technology)

[Anonymous No.11083730]

>>11083722
imagine a bunch of professors getting together for two weeks to do math using a corollary that they know isn't actually proven

[Anonymous No.11083757]

math is boring and gay sorry it's the cuck loser soulless version of computer science

[Anonymous No.11083759]

>>11083554
This should be easy using the spectral theorem and the resulting functional calculus, but that is probably overkill. Is depressing to forget everything you once knew about topics you should really know

[Anonymous No.11083761]

>>11083757
I knew a guy who was the personification of the shirt in your pic. He had a bad birthday so he beat and raped a woman on his way home and went to jail.

[Anonymous No.11083902]

>>11083757
>computer science
not science or math

[Anonymous No.11084020]

I have 7 days to become proficient in pic rel. Rate my chances

[Anonymous No.11084141]

>>11084020
I'd say your chances are pretty good, as long as you don't fuck around between now and your midterm. If you cover two topics daily, you will have covered everything in 4 days, and you will have the remaining 3 days to do more problems to refine your understanding.

[Anonymous No.11084157]

Getting my ass kicked in abstract algebra. Losing motivation and don't know what to do. Just keep chugging?

[Anonymous No.11084209]

>>11081694
what's a 1+1 equal ?? Xd

[Anonymous No.11084212]

>https://en.wikipedia.org/wiki/Fermat%27s_little_theorem
Is there a non-prime p where a^(p-1) - 1 is a multiple of p for all integers a < p?
Carmichael numbers aren't prime but for them a^p - p is a multiple of p for all integers a < p, but they fail this a^p(-1) - 1 being a multiple of p test.

For example let p = 561 now for all a < p it's true that a^p - p is a multiple of p, but the same isn't true for a^(p-1) - 1 because we can pick a = 558 and 561 doesn't divide 558^560 - 1.

[Anonymous No.11084230]

>>11083761
holy fucking shit m8, stuff like this happens...

[Anonymous No.11084237]

I cannot figure out how sqrt(x) simplifies to 2x in this equation. Can someone help?

[Anonymous No.11084249]

>>11084237
oh wait I'm fucking stupid. The first two equations got multiplied by the 2sqrt(x) before getting put into the fraction of the third right? Because they get multiplied by the denominator since it's addition.
How the fuck did I make it to calculus?
It's okay I"ll be back with more stupid questions people. Be amazed that they actually let me take a college calc class!

[Anonymous No.11084422]

>>11083757
CS is the most soulless field.

[Anonymous No.11084423]

>>11083761
He did good :^)

[Anonymous No.11084452]

>>11081767
I used to study outside his office last year

[Anonymous No.11084461]

>>11083598
>>11083626
>>11083554
I'm not fluent enough in those abstractions to solve the problems, but thanks to TeX it out and post some math

[Anonymous No.11084511]

Character Theory is the most innocent, pure and clean theory in mathematics.

[Anonymous No.11084515]

>>11083722
a literal scam, these people should have their funding revoked

[Anonymous No.11084601]

>>11083554
well? the adjoint is obviously the inverse, exp is a power series so adjoint gets pushed in, and then (iS)* = -iS. That does it.

[Anonymous No.11084634]

>>11083554
Well easily [eqn]\left(\sum_{n=0}^N \frac{(iS)^n}{n!}\right)* = \sum_{n=0}^N \frac{(-iS)^n}{n!}[/eqn]
Letting N go to infinity, we get by continuity of the adjunction that [math]U* = \exp(-iS) = U^{-1}[/math]. Hence [math]U[/math] is unitary.
Moreover, if [math]\lambda[/math] is in the spectrum of [math]S[/math], then we have [eqn]U - e^{i\lambda} = \sum_{n =1}^{\infty} \frac{(S-\lambda)(S^{n-1}+\dots + \lambda^{n-2}S + \lambda^{n-1})}{n!} = (S-\lambda) H = H(S-\lambda)[/eqn]
with [math] H = \sum_{n =1}^{\infty} \frac{(S-\lambda)(S^{n-1}+\dots + \lambda^{n-2}S + \lambda^{n-1})}{n!}[/math].
In particular, [math]\ker(S-\lambda) \subset \ker(U - e^{i\lambda})[/math] and [math]\mathrm{im}(U - e^{i\lambda}) \subset \mathrm{im}(S-\lambda)[/math].
Since [math]S - \lambda[/math] is not invertible, either [math]\ker(S-\lambda) \ne 0[/math] or [math]\mathrm{im}(S-\lambda)[/math] is proper.
In either case, it follows that [math]U - e^{i\lambda}[/math] is not invertible, hence [math]e^{i\lambda}[/math] is in [math]\sigma(U)[/math], hence it has modulus 1, hence [math]\lambda [/math] is real.

[Anonymous No.11084636]

>>11083554
Well easily [eqn]\left(\sum_{n=0}^N \frac{(iS)^n}{n!}\right)^* = \sum_{n=0}^N \frac{(-iS)^n}{n!}[/eqn]
Letting N go to infinity, we get by continuity of the adjunction that [math]U^* = \exp(-iS) = U^{-1}[/math]. Hence [math]U[/math] is unitary.
Moreover, if [math]\lambda[/math] is in the spectrum of [math]S[/math], then we have [eqn]U - e^{i\lambda} = \sum_{n =1}^{\infty} \frac{(S-\lambda)(S^{n-1}+\dots + \lambda^{n-2}S + \lambda^{n-1})}{n!} = (S-\lambda) H = H(S-\lambda)[/eqn]
with [math] H = \sum_{n =1}^{\infty} \frac{S^{n-1}+\dots + \lambda^{n-2}S + \lambda^{n-1}}{n!}[/math].
In particular, [math]\ker(S-\lambda) \subset \ker(U - e^{i\lambda})[/math] and [math]\mathrm{im}(U - e^{i\lambda}) \subset \mathrm{im}(S-\lambda)[/math].
Since [math]S - \lambda[/math] is not invertible, either [math]\ker(S-\lambda) \ne 0[/math] or [math]\mathrm{im}(S-\lambda)[/math] is proper.
In either case, it follows that [math]U - e^{i\lambda}[/math] is not invertible, hence [math]e^{i\lambda}[/math] is in [math]\sigma(U)[/math], hence it has modulus 1, hence [math]\lambda [/math] is real.

[Anonymous No.11084675]

How do you cope with the feelings of inadequacy you get when you read about talented mathematicians?

[Anonymous No.11084678]

>>11084675
Why read about them? They are merely vessels.

[Anonymous No.11084729]

>>11084461
the least I can do. just want to see some actual math in the mathematics general you know.

[Anonymous No.11084906]

>>11084675
Rethink why you do math

[Anonymous No.11084948]

I feel like I can keep up with Calculus until euler shows up and fucks up everything

[Anonymous No.11084991]

>>11084948
Study more.

[Anonymous No.11084995]

>>11084675
"yep, me and other 7 billion people are drooling retards compared to this dude. Might as well just keep doing what I enjoy."
That's how I cope.

[Anonymous No.11085002]

>>11084991
you're right sorry

[Anonymous No.11085015]

This is a stupid question but I don't care.
Lately I've been trying to watch type theory talks and I understand it all right until these symbols come up. Can someone tell me what these are? I've never studied logic before, is that where these are from? And also, I would appreciate it if someone could suggest some text that I can use to fill in the gaps.

[Anonymous No.11085020]

Is it a bad idea to go into (applied) math PhD as a /smart but not genius/ tier student?

I'm double majoring in EE so I could choose to settle being a code chimp or pursue an engineering PhD instead. On the other hand I enjoy studying math, but I don't know if I'm cut out for it. Solving open problems and creating new knowledge seem very daunting.

Is this just imposter syndrome talking or is it a legitimate concern? Would appreciate thoughts/experiences from fellow 13x IQ people.

[Anonymous No.11085025]

>>11085015
natural deduction

[Anonymous No.11085049]

>>11081694
how is this even possible

[Anonymous No.11085061]

>>11085049
who needs a year for one grad text

[Anonymous No.11085065]

>>11085020
No, everyone in every PhD program is a genius

[Anonymous No.11085067]

>>11084675
well I know nifty things that most mathematicians don't so in some ways I'm better than them, in some ways I'm worse, feels good. For example I think I realized what a sum and a prod operators formally are. Like, for example Sum[0,2] ai = a2 + a1 + a0, Sum[0,1] ai = a1 + a0, Sum[0,0] ai = a0 but what is the sum Sum[0,-1] ai and what should it be equal to?

[Anonymous No.11085080]

>>11085067
formally they are folds. you didn't figure out shit retard.

[Anonymous No.11085089]

>>11085067
congratulations, you figured out integration with respect to counting measure (such a braindead obvious topic i can't believe you'd waste our time with it)

[Anonymous No.11085091]

>>11084675
i don't feel inadequate
if you feel inadequate to an extent that it shakes your enthusiasm for mathematics, you should be worried.

[Anonymous No.11085096]

>>11085049
flip through all the pages, reading the theorems and definitions, quickly doing proof sketches in your head
there might be a few tougher ones so you can give those a read
should take 10 to 12 hours at most

[Anonymous No.11085118]

>>11085080
what?
>>11085089
what are you talking about?
I guess you misunderstoo, what I figured out is that the Sum and Prod operators (and for example repetitive compostion of n functions : f1 circ f2 circ f3 ... circ fn) are essentially the same operator that takes a group and if upper index is lower than low index it outputs the neutral element of this group hence if are summing numbers Sum [1,0] ai = 0 if we are multiplying them Prod [1,0] ai = 1, and if we take compostions of n functions that I will denote the same symbol as Prod then Prod[0,2] fi = f2 circ f1 circ f0, Prod [0,1] fi = f1 circ f0, Prod[0,0] fi = f0, Prod[0,-1] fi = id_X and don't you dare fucks tell me that somebody discovered this before me, I have not seen a single source having that covered

[Anonymous No.11085147]

>>11085049
avoid complex exercises.

[Anonymous No.11085155]

>>11085089
My friend please be respect. This mans are work hard learn summation are student mathematical exchange knowledge practice alway...

[Anonymous No.11085168]

>>11085065
I don't feel I've "mastered" any material beyond about linear algebra II whereas I see freshmen kids acing manifold analysis or take graduate core courses in 2nd year. I think I would have a much easier time in EECS postgrad. Just don't want to insta regret one semester in to the program is all.

[Anonymous No.11085230]

>>11085049
I can read in English (not my native language) at 2 pages per minute. A 300 page textbook takes me 10 hours to "read" which doesn't mean I understood anything or learned anything.

[Anonymous No.11085233]

>>11083757
>Computer science is boring and gay sorry it's the cuck loser soulless version of math
Ftfy. Imagine believing what you wrote lmao

[Anonymous No.11085236]

>>11085118
You write incomprehensibly.
Try formulating a theorem, like any mathematician would.

[Anonymous No.11085260]

>>11085118
sum and product notations are catamorphisms, you're not discovering anything new here. Your "Prod[0,-1]" seems to be referring to the concept of an "empty product" which is indeed the identity element for products, similarly there is an "empty sum"

[Anonymous No.11085275]

>>11085118
It's a fold over a monoid retard. Function composition isn't even a group.

[Anonymous No.11085301]

>>11085067
This sort of recursion is explicated even in Lang I think.
More broadly, on ordinals, you can look at transfinite recursion. The uparrow used to inductively define objects via a function appears very early in Jech.

[Anonymous No.11085336]

>>11081694
I want to get started on statistics and probability to use on trading. Any good books for beginners?

[Anonymous No.11085365]

>>11081694
Is there a general theory of "classifying object" for some data in an arbitrary category? The only examples I know of are classifying spaces for principal G-bundles and subobject classifiers in elementary topoi. The idea is just so novel and fascinates me. Has anyone written substantially on this?

[Anonymous No.11085366]

>>11085236
This is not a theorem,
it's just an observation that we have already abstracted any similar sets into algebraic structures (rings/fields etc) but I find it amusing that no one seems to notice that the same thing can be done with "recursive" operations such as Sum and Prod. But not only those, basically any operation that takes as an input elements of the set and applies to all of them one single operation (addition, multiplication, composition etc). These operations can all be abstracted as functional symbols that take as an input some group (X,*,e) and also some finite sequence a of elements of this group, a:{m..n} to X, where m,n are any integer numbers. Now if m <= n, they recursively compute and return a_m * a_(m+1) * ... * a_n and if m > n they return e, the neutral element of this group.
For example let's denote this abstact symbol as "Prod" (Product)
If we take a group G = (M,*,E) where M are all 2 x 2 matrices over C, and * is a matrix multiplication and E is a 2 x 2 identity matrix and give a sequence a:{3..5} -> M then Prod[G](a) = a5 * a4 * a3. If we give it a sequence a:{3..4} -> M then Prod[G](a) = a4 * a3 and if we give a:{3..3}->M then Prod[G](a) = a3, now if we give it an a:{3..2} -> M (basically an empty sequence) it will return Prod[G](a) = E, the 2 x 2 identity matrix.

[Anonymous No.11085398]

>>11085366
retard I just told you they are folds which are catamorphisms and they are well known

[Anonymous No.11085422]

>>11085398
in CS?

[Anonymous No.11085466]

>>11085366
Anon, this is literally what fold does.

http://zvon.org/other/haskell/Outputprelude/foldl_f.html

[Anonymous No.11085494]

>>11085398
>>11085466
now I'm most confident that mathematicians should work with computer scientists too

[Anonymous No.11085497]

>>11082112
Try using physics in module theory!

[Anonymous No.11085530]

>>11085275
This

[Anonymous No.11085531]

>>11085422
They're mostly of interest to CS people, yes.

[Justin Ramsey No.11085540]

Being born male and female is completely random! Honk honk.
Infinite I could have been movies now.

[Anonymous No.11085564]

>>11085365
How about
https://ncatlab.org/nlab/show/classifying+topos

Also that woman which did work on brigdes and had this controvery some years ago, I think she went in that direction.
Just search for "classifying" on nLab, or maybe moduli space.
I mean I suppose all things Teichmüller also go in that direction, and the Japanese guy does a "big theory" of this. Although it may be too arithmetical for you.

I'd also like to know more about it btw.

>>11085398
I think I wrote the bulk of the catamorphism Wikipedia article years ago. On that not, look at F-algebras and the general theory of formal arithmetic of types.
https://arxiv.org/pdf/math/9405205.pdf
If you're explicit about your bijections, you can do cool stuff like "taking the square root" of data types X (that is to say, give concrete meaning to power series of stuff like X^(1/2) or 1/(X+1))
On that note again, there's a recent cool set of talk slides by Bauer on explicit bijections in combinatorics and the question what they should be (and he's shilling dependent types and hott)

http://math.andrej.com/wp-content/uploads/2019/07/What-is-an-explicit-bijection-FPSAC-2019-slides-with-presenter-notes.pdf

[Anonymous No.11085651]

>>11085494
>>>/g/73285731
Every single advance math thing for CSgger got unused.

[Anonymous No.11085853]

>>11081819
>I know this board is more meant for people who know what they are doing and discussing high-level stuff,
lol

[Anonymous No.11085858]

>>11081819
>I know this board is more meant for people who know what they are doing and discussing high-level stuff
you must be new here

[Anonymous No.11086118]

>>11085494
This is math. Math doesnt end at calculus.

[Anonymous No.11086157]

Redpill me on representation theory. I'm self-studying DiffGeo and Analysis with a long term goal of MathPhys. Should I learn some rep theory?
Just looking at some grad school curriculum it seems like an essential topic in modern algebra, with lots of applications.

[Anonymous No.11086175]

>>11086157
https://arxiv.org/abs/math-ph/0005032
This guy made whole book about it.
https://www.springer.com/gp/book/9783319134666

[Anonymous No.11086308]

>>11086157
Representation theory is among the nicest theories to work with, and has many applications in combinatorics, (linear) algebra, and geometry - and many groups form smooth manifolds in themselves (Lie groups).
All that said, they aren't prerequesits of DiffGeo (let alone Analysis). In fact it might even help to know some diff geo to then get into Lie groups more enlightendly. The reason is that diff geo is taught w.r.t. less rigit requirments than their group theoretical counterparts and even if you got a metric, early on you tend to not need anything than groups over R.

[Anonymous No.11086369]

Three girls at uni begged me to help them understand the simplex method lmao. I was the only one of 150 people in my econ major to understand it and the only one to ace the linear algebra course. Tfw popular for 20 minutes in my life. In the end i couldn't teach them squat because they're dumb as rocks. How the hell do you teach someone a whole course in less than an hour, one day before an exam?

[Anonymous No.11086379]

>>11086369
>How the hell do you teach someone a whole course in less than an hour, one day before an exam?
You don't. In this case, you're supposed to have passionate sex for the entire hour and say good luck on the exam.

[Anonymous No.11086411]

>>11086369
Easy examples and solve trivial problems, change complex math language, for yes, multiply this numbers, yes if you look this problem use this method, this plus this multiply this equal to ...

Avoid theory and explain practical problems.

[Anonymous No.11086527]

>>11086157
Representations of the poincare group gives you the classification of quantum fields wrt to mass and spin. Its useful all the time when doing qm, so its nice for mathphys

[Anonymous No.11086800]

>>11085118
>unironically ever thinking you've discovered something new in mathematics
literally takes decades and decades of experience

[Anonymous No.11087156]

Is point set topology supposed to be easy/intuitive? I feel like I'm just picking up clever proof techniques with little intuition and it's destroying my self confidence de su

[Anonymous No.11087174]

>>11087156
have you taken real analysis and learned about metric spaces?
if so, yes
if not, no

[Anonymous No.11087190]

>>11087174
Took babby 1-variable analysis (Spivak, more or less) but metric spaces wasn't really covered. It just takes forever to sketch out proofs in my head.

[Anonymous No.11087204]

>>11087190
well i recommend reading chapter 2 of pugh's intro to real analysis (can probably find it online). shouldn't take long if you already have a spivak and it's very readable and fun. it will give you a wealth of examples and intuition on which to visualize topological notions - having a way to think about all of this and draw it is the right way to figure out point set proofs.

[Anonymous No.11087211]

>>11083761
based goy
probably could taken a few more before prison, weak.

[Anonymous No.11087215]

>24
>5 years since high school
>struggling with MAT151

Holy shit I actually have to put in work for an algebra course. It used to be so easy wtf bros

[Anonymous No.11087249]

>>11087156
Depends on what you're covering. Metric spaces were really intuitive for me, but I think that's how it is for most people. It helps to draw topological diagrams.

[Anonymous No.11087264]

I solved an exercise by using bruteforce.
But is there a mathematical reasoning behind pic related?

Basically I have 15 cards in a 4x5 big rectangle.
Collect each columns card from top to bottom, then place them by each line, from left to right.
How often do I need to do it (6 times as in pic related) to get back to the cards' original position?

[Anonymous No.11087292]

is there a mathematical proof of climate change?

[Anonymous No.11087333]

>>11087264
Okay.
If I color the rectangle like a checkerboard, it appears that the cards NEVER leave their original color, just positions.
Is there not a way to explain this?

[Anonymous No.11087336]

>>11087264
Better one.
BTW, how can I explain that the cards never land on another background color?

[Anonymous No.11087341]

>>11085494
trust me. every non-retard math person at semester 5+ knows this. moreover, they don't care because it's totally boring and irrelevant. we don't need to work with your kind.

[Anonymous No.11087510]

>>11087292
You can't prove a contradiction.

[Anonymous No.11087519]

Let M be a submodule of Q, prove that there is a maximal submodule E so that for any submodule N of E the intersection of N and M is nonempty.

My attempt: Consider the poset of all such E. For any chain the union of the chain is an upper bound. By Zorn we are done.

Sound good?

[Anonymous No.11087556]

>>11087519
no, absolutely braindead proof. Try proving that without using the axiom of choice

[Anonymous No.11087571]

>>11087519
If you want to use choice you should show that the result is equivalent to choice. Anything else is basically cheating.

[Anonymous No.11087609]

>>11087519
NOOOOO YOU CAN'T USE THE AXIOM OF CHOICE

[Anonymous No.11087626]

>that dilettante undergrad who is still working through basic calculus and linear algebra courses but still blabbers on and on about "alternative maths" topics like constructivism and category theoretic foundations

[Anonymous No.11087630]

How come 24 / 0.995 is different from 24 * 1.005 in my calculator?

the / 0.995 gives more decimal places

[Anonymous No.11087646]

>>11087626
>constructivism
Good.

>category theoretic foundations
Shit.

Constructive category theorists are classical mathematicians that got homesick.

[Anonymous No.11087653]

>>11087519
yes

[Anonymous No.11087657]

>>11087646
Dilettante undergrad detected. I suggest you focus on your basic studies and in receiving a breadth of mathematical understanding. It's too early to decide which topic of a subfield you're going to specialize in yet.

[Anonymous No.11087687]

>>11087626
Ah yes, the undergrad category theorist. We all know one.

[Anonymous No.11087695]

>>11087510
That's not more than rhetoric.
Of course you can prove a contradiction, all you need is any inconsistency.

[Anonymous No.11087725]

>>11087630
Because you calculate two entirely different things?

Why is 2*7 different from 4/10?

[Anonymous No.11087728]

>>11087630
> How come 24 / 0.995 is different from 24 * 1.005 in my calculator?
Because 1/(1-x) =/= 1+x.
1.005 * 0.995 = 0.999975

More generally, adding X% is a smaller change (in terms of ratio) than subtracting X%. E.g. "50% off" is a 2:1 ratio, "50% extra" is 2:3.

[Anonymous No.11087731]

>>11081780
Right now it's the set of 3x3 upper triangular matrices, with one on the diagonal over F3 with matrix multiplication

[Anonymous No.11087737]

>>11084212
there shouldn't be such p, because that would imply that every number has a multiplicative inverse in the ring where addizion and multipkication are taken mod p, but we know, that for example the divisors of of p aren't invertible.

[Anonymous No.11087741]

Anyone knows the prerequisite for the theory of rough path?

[Anonymous No.11087779]

>>11081694
Looking at two problems, which I am having trouble solving.

> Counting
Assumptions: Order matters, repetition allowed
University has 14-day exam period. You are writing math, physics and stat exams. How many choices of days are possible if maths exam must be held at least two days earlier than physics, and physics exam must be held at least two days earlier than stat.

Naive thought was 14x12x10, but the 'earlier' key word was the sticking point. I'm thinking math can be taken on days 1-10, physics on day 12 and stat only on day 14. Though, perhaps that is also a naive way of looking at it. Really not sure how to model the problem.

> Groups
How can you be sure that (Z_n3xZ_n3,+) has no more than 6 subgroups.
{(0,0)}
{(0,0),(1,1),(2,2)}
{(0,0),(0,1),(0,2)}
{(0,0),(1,0),(2,0)}
{(0,0),(1,2),(2,1)}
{(0,0),(1,1),(2,2),(0,1),(0,2),(1,0),(2,0),(1,2),(2,1)}

[Anonymous No.11087796]

>>11087779
>How can you be sure that (Z_n3xZ_n3,+) has no more than 6 subgroups.
Argue by order. If G is your group and H is a subgroup, then |H| divides |G|=9. It follows that |H| can only be 1, 3 or 9, the extreme cases of which are obvious. Now, how many combinations can you have for |H|=3?

[Anonymous No.11087803]

>>11087796
3!?

[Anonymous No.11087805]

>>11087803
You're a funny guy. That's why I will kill you last.

[Anonymous No.11087822]

>>11087805

[Anonymous No.11087826]

>>11087822
No but really. If you take x=(1, 1) and try generating a subgroup with x and (2, 2) in it, you get {(0, 0), (1, 1), (2, 2)}, but x and (1, 2) gives you (1, 1)+(1, 2) = (2, 0) and (2, 0)+(1, 1)=(0, 1), so you you already have more than 3 elements.

[Anonymous No.11087829]

>>11087826
I wasn't sure if it was readable lol.
I meant 3!=6.

[Anonymous No.11087831]

>>11087829
I know what 3! means.

[Anonymous No.11087837]

>>11087831

[Anonymous No.11087841]

>>11087837
You got your homework help. Now kys and never come back.

[Anonymous No.11087850]

>>11087841
But we didn't find a solution to the counting problem!

It's really clever though.

Maths and physics can be turned into 2 day objects, leaving 10 days. Minus stat, leaving 9 empty days. So you have three objects, 12 total, leaving 12C3=220.

[Anonymous No.11087861]

>>11087850
>But we didn't find a solution to the counting problem!
Use your fingers and toes.

[Anonymous No.11087907]

>>11087519
Based!

[Anonymous No.11088595]

Is the definition of [math] \otimes_R [/math] via

[math] \operatorname{Hom}_{\Z} (M \otimes_R N, G) \simeq \operatorname{Hom}_R(M, \operatorname{Hom}_{\Z} (N, G)) [/math]

good or are others more natural to gain an intuition and think of canonical examples of tensor product of modules?

[Anonymous No.11088628]

>>11087725
Then what's the inverse of / 0.995 ?

They give the same answer up to 2 decimal places

[Anonymous No.11088759]

>>11087695
>all you need is any inconsistency
Which does not exist in the Platonic realm. Do you have anything actually intelligent to add?

[Anonymous No.11088972]

>it's a "make a dumb transcription error and fuck everything up, having to start over from scratch" episode

[Anonymous No.11089028]

>>11087630
[math]
\frac{1}{0.995} = 1.0050251256281407035175879396985
[/math]

Not 1.005

[Anonymous No.11089059]

Is there an easier way to determine these without having to chart all the values for x and y?

[Anonymous No.11089079]

>>11088628
See >>11087728
> Then what's the inverse of / 0.995 ?
The inverse of /0.995 is *0.995. If you meant
> Then what's the reciprocal of 0.995
or
> Then what's 1/0.995
the answer is
1.005025125628140703517587939698492462311557788944723618090452261306532663316582914572864321608040201...
where the 99 digits after the point (including the leading 00) repeat ad infinitum. The reciprocal of a finite decimal isn't usually a finite decimal. A fraction only has a finite decimal representation if the denominator has no prime factors except 2 and 5. 0.995 is 995/1000=199/200; its reciprocal is 200/199 and 199 is prime.
> They give the same answer up to 2 decimal places
Well, given that your initial values have 3 decimal places, 2 decimal places isn't really enough.
(1+x)(1-x)=1-x^2. If x is small then x^2 is very small => 1-x^2 is very close to 1 => 1+x is very close to 1/(1-x). E.g. 0.005 is 1/200, (1+1/200)(1-1/200)=1-1/40000=0.999975.

[Anonymous No.11089104]

>>11089059
You should know what y=1/x looks like. y=9/x is the same up to a uniform scale factor: y=9/x => (y/3)=3/x => (y/3)=1/(x/3). The "corner" is at (3,3).

[Anonymous No.11089119]

>>11088628
The inverse of dividing by 0.995 is multiplying by 0.995. Division is just multiplying one number by the reciprocal of the other.
[math]
y = \frac{x}{0.995} = \frac{x}{1} \cdot \frac{1}{0.995}

x = \frac{y}{1} \div \frac{1}{0.995} = \frac{y}{1} \cdot \frac{0.995}{1} = y \cdot 0.995
[/math]

[Anonymous No.11089123]

The inverse of dividing by 0.995 is multiplying by 0.995. Division is just multiplying one number by the reciprocal of the other.

[math]
y = \frac{x}{0.995} = \frac{x}{1} \cdot \frac{1}{0.995}\newline

x = \frac{y}{1} \div \frac{1}{0.995} = \frac{y}{1} \cdot \frac{0.995}{1} = y \cdot 0.995
[/math]

[Anonymous No.11089128]

The inverse of dividing by 0.995 is multiplying by 0.995. Division is just multiplying one number by the reciprocal of the other.
[math]
y = \frac{x}{0.995} = \frac{x}{1} \cdot \frac{1}{0.995}
[/math]

To reverse it:
[math]
x = \frac{y}{1} \div \frac{1}{0.995} = \frac{y}{1} \cdot \frac{0.995}{1} = y \cdot 0.995
[/math]

[Anonymous No.11089138]

>>11088595
>tensor product curries maps
is that not intuitive

[Anonymous No.11089149]

How do I find p so that the series converges? I'd assume for absolute convergence I can just remove the alternating part but what about regular convergence? I see an alternating harmonic series but then I don't know what to do with the 1/ln(n)^p part.

[Anonymous No.11089184]

>>11089028
>>11089079
I think I got it now

24 / 0.995 = X

24 - X = Y

Y is the difference I'm looking for I guess.

Doing 3D printing, so to fit a part I start with 100% of the hole then make it slightly smaller so when it's done printing it fits perfectly.

So I have the info I need now to make the "key" and fill it with whatever I want.

[Anonymous No.11089224]

>I took a control theory course and enjoyed it
Dammit, is it over?

[Anonymous No.11089229]

>>11089149
what is the alternating series test

[Anonymous No.11089233]

>>11089224
control theory is based, join the /dynamicalsystemssquad/

[Anonymous No.11089252]

>>11089229
thanks

[Anonymous No.11089265]

>>11089233
I'm already thinking about replacing a pure course with the operations research course next semester. Next thing you know I'll be employed.

[Anonymous No.11089469]

>English professor got mad at me because I agreed with him that 2 * 2 can sometimes be 10
how is it my fault that i know the basics of other bases

[Anonymous No.11089488]

Can y'all do my homework for me? Letting R be an infinite integral domain with finitely many units, how do you prove that R must have infinitely many max ideals?

[Anonymous No.11089502]

>>11089488
because it is

[Anonymous No.11089517]

>>11088595
I'd rather define maps [math]\varphi \colon A\otimes R \to A, a\otimes r \mapsto ar[/math] and [math]\psi \colon R\otimes B \to B, r\otimes b \mapsto rb[/math], and then let [math]A \otimes_R B[/math] be the (canonical) cokernel of [math]\varphi \otimes 1 - 1 \otimes \psi \colon A \otimes R \otimes B \to A \otimes B[/math].

[Anonymous No.11089612]

>>11089265
You could be one of the cool guys who studies exclusively the pure theory of something people normally consider to be applied math. There's nothing more satisfying than telling someone I do dynamical systems and stuff with hamiltonians (symplectic), hearing "oh, so you must do a lot of physics then" and responding "no, i don't know a thing about physics. is that relevant?"

[Anonymous No.11089616]

>>11089488
>y'all
why

[Anonymous No.11089657]

>>11089469
>bases
But then you are just changing the meaning.
Far more meaningful is that in certain number fields they are actually the same, eg. in the integers mod 2.

[Anonymous No.11089661]

>>11088628
>They give the same answer up to 2 decimal places
0.001 and 0.002 are the same number up to two decimal places too.
Does that mean that 1000*0.001=1000*0.002?

[Anonymous No.11089663]

>>11089661
It makes them the same for my purposes.

[Anonymous No.11089671]

>>11087630
idk anon what have you tried? those are two different values is why they're different, for example
1 / 0.5 = 2
1 * 1.5 = 1.5

[Anonymous No.11089675]

>>11089671
I guess, but by the third decimal place they produce the same result anyways

[Anonymous No.11089679]

>>11089616
Because "y’all" implies a certain quaint hospitality that I wanted to expend to y’all.

[Anonymous No.11089683]

>>11089663
No. Just because you round every number to two digits after the comma does not mean you get a result accurate to two digits after the comma.

Your entire issue is that you are calculating entirely different things, which have no relation to each other.

[Anonymous No.11089685]

>>11089679
No, it implies that you are a good damn fucking nigger who speaks an abomination of the English language.

[Anonymous No.11089708]

>>11089679
Nah sorry just implies you're stupid!
>>11089685
Watch the racial slurs.

[Anonymous No.11089711]

>>11089708
>Watch the racial slurs.
Why the niggerphilia?

[Anonymous No.11089919]

I don't understand numbers and how they are even possible.

How can pic related be possible? How is it that, three times 3,33..... can be 10? What is the magical property at work here that can transform 3 into a different number.

How can 1 ever be anything else (for say, example 2) when you can just slam numbers behind it? Where does it cross the line into two? Why can I even add numbers behind? Like 1 and then 1,01, and then 1,000011235....

Sorry am low IQ but I have had nightmares of the above for the past two weeks. Am not trolling just genuinely fucking stupid hick retard trying to understand how is 1+1=2 even possible

[Anonymous No.11089928]

>>11089919
1/3 = .33333....
2/3 = .66666.....
1/3 + 2/3 = 1

https://www.youtube.com/watch?v=G_gUE74YVos

[Anonymous No.11089950]

>>11089928
So numbers are transcendent? 1 is both 1 and 2? So how can you know 10+10 is 20 and not 19 or 21?

[Anonymous No.11089957]

>>11089950
stop trolling and just watch this channel

https://www.youtube.com/watch?v=Mfk_L4Nx2ZI

[Anonymous No.11089961]

>>11089957
I'm not trolling. How can you add numbers together and say they're something like 10 when they're composed of numbers which can be between 0 and 1 at same time. So is it just probability? That 5+5 is 10? It's confusing me the fuck out.

I'll watch the video, alright.

[Anonymous No.11089970]

>>11089961
Just look up what a whole number is, compared to complex or other types.

You could also just do things physically for real world math vs conceptual math.

[Anonymous No.11089971]

>>11089149
It converges better than the log series expansion, so you're done

[Anonymous No.11090675]

Anyone here attending Broto's birthday?

[Anonymous No.11090816]

>>11089961
If you really want to know what's going on here, it's a long and complicated story. It's fraught with people going insane, starving themselves, and realizing their decades of work are worthless. Over and over and over. It's the story of foundations - when people in the late 19th century to the early 20th century wanted to answer the questions you're asking (well, they asked questions which were a bit more knowledgeable, but with the same spirit) completely logically and rigorously, with no room for error.
Here's what might help you: we need to define a system in which we can work. We want this system to make sense with what we see in real life. So Peano came up with some symbols (0 and Successor) and some axioms (rules for how the symbols work) which together define something called "Peano arithmetic". The rules are really simple, the main one is that you can write Successor in front of any existing object in Peano arithmetic.
So we know 0 is in Peano arithmetic, and we can write Successor in front of 0. Let's give Successor(0) a nickname cause it's long, 1. By the rules, we can write Successor in front of this new thing to get Successor(Successor(0)). Let's call this one 2, it's really long.
Now Peano wants to define +. Well, one way to do that is just to take the Successors on one thing and slap them on to the other. So 2 + 1 which is a nickname for Successor(Successor(0)) + Successor(0), we slap the successors before the plus onto the thing after the plus and get Successor(Successor(Successor(0))) which we could nickname 3.
So Peano does this with his basic symbols to define more, like ×, but this system turns out to kind of suck and not get you very much more than basic arithmetic. That's when other people come in to introduce new symbols and axioms. This is the field of "logic."
What do we use today that gets us all these fancy things like 3.3333... and irrational numbers? We use a set of symbols and axioms called "ZFC."
cont

[Anonymous No.11090824]

Please help with this question.
Using the knowledge of x^2n is congruent to 0 or 1 modulo 4, I need to show that -1 + 4x +x^2 +8x^3 +x^4 = 0 has no integer solutions.

I sent 1 to the RHS and then moduloed (?) 4x and 8x^3 which is 0 mod 4 and the other 2 is 1 mod 4. Its most likely wrong and I am just confused on how I should proceed.

[Anonymous No.11090834]

>>11090824
mod 4:
dude just plug in 4 numbers to check. It's not that hard dude

[Anonymous No.11090838]

>>11090834
I have to prove it mathematically so I can't just plug in numbers.

[Anonymous No.11090887]

>>11090838
Why not? An entirely reasonable proof would be the following: let [math]f(x)=-1 + 4x +x^2 +8x^3 +x^4[/math]. Then if there exists [math]x\in\mathbb Z[/math] such that [math]f(x)=0[/math], in particular we'd have [math]f(x) \equiv 0 \pmod 4[/math]. But [math]f(0) \equiv -1 \equiv 3 \pmod 4[/math], [math]f(1) \equiv 1 \pmod 4[/math], [math]f(2) \equiv -1 \equiv 3 \pmod 4[/math] and [math]f(3) \equiv 1 \pmod 4[/math]. Since all integers are [math]0,1,2[/math] or [math]3 \pmod 4[/math], then there are no integer solutions.

[Anonymous No.11090891]

>>11090887
Thank you so much bro I hope you have an excellent day and god bless you.

[Anonymous No.11090932]

hurr durr more than half questions in this thread could have been easily avoided if only the asker learnt a little bit of logic and proof theory

[Anonymous No.11090944]

>>11090816
ok nevermind i cant finish this for a while
just look up ZFC if you want, but its complicated

[Anonymous No.11090951]

>>11090932
Apparently I need to use AGM (arithmetic geometric mean inequality) here to solve the problem but I don't see how I am supposed to apply it. Any tips?

[Anonymous No.11090958]

>>11090951
Sorry for retarded rotation perhaps this is better

[Anonymous No.11091000]

>ywn solve an open problem
why live?

[Anonymous No.11091003]

>>11091000
Git Gud

[Anonymous No.11091006]

>>11091000
Forget about open questions, answer this one instead. Also nice trips.
>>11090951

[Anonymous No.11091054]

>>11090951
im a brainlet so i just subtracted 1/2 from the sum and got a fraction that is always positive for positive a and b ergo the original fraction is always greater than 1/2

[Anonymous No.11091073]

>>11090951
Hey man, nice problem you got there. Multiply out the denominators and fully multiply all terms until you are left with a^2 + b^2 >= 3/2ab. We know by the AM GM inequality that a^2 + b^2 >= 2ab >= 3/2ab for all a,b >= 0. QED

[Anonymous No.11091093]

>>11091073
My bad I made an error, I think it reduces down to a^2 + b^2 >= ab/4 which yields the same result. Sorry for the mistake. Studying has burned out my brain.

[Anonymous No.11091112]

>>11091093
In fact, there is a simple algebraic proof to show that the lower bound is actually 2/3 and not 1/2. This was probably done in order to obscure the fact that AM-GM needed to be applied as a last step as it would have had the identical form had it been 2/3. Anyways, hope this helped you.

[Anonymous No.11091138]

>>11090824
> Using the knowledge of x^2n is congruent to 0 or 1 modulo 4, I need to show that -1 + 4x +x^2 +8x^3 +x^4 = 0 has no integer solutions.
-1 + 4x +x^2 +8x^3 + x^4 = 0
=> 4x + 8x^3 = 1 - x^2 - x^4
=> 4(x + 2x^3) = 1 - x^2(1 + x^2)

Clearly 4(x + 2x^3) ≡ 0 (mod 4)
x^2≡0 => 1 - x^2(1 + x^2) ≡ 1 (mod 4)
x^2≡1 => 1 - x^2(1 + x^2) ≡ 1 - 1(1+1) ≡ 1-2 ≡ 3 (mod 4)
So the LHS is congruent to 0 while the RHS is congruent to 1 or 3; thus the two can never be equal.

[Anonymous No.11091155]

>>11091054
>>11091073
>>11091093
>>11091112
Thanks a lot guys. Helped me a great ton.

[Anonymous No.11091422]

>>11089961
Found a better one maybe
https://www.youtube.com/watch?v=1EGDCh75SpQ

[Anonymous No.11091558]

>>11089657
Most normies aren't going to know that shit and it's harder to explain to other normies.

[Anonymous No.11091648]

>going through multivariate calc easily
>have found the vast majority of math boring until and including this point
Should I just do something else or is there something I'm missing?

[Anonymous No.11091681]

>>11091648
you realize you haven't scratched the surface of math yet, right? unless you consider being a mathematica monkey the 'peak' of math.

[Anonymous No.11091685]

>>11091681
I get that. I guess I'm mainly wondering if it gets more interesting/fulfilling. If I'm not enjoying these basic elements then am I just going to burn out on later things?

[Anonymous No.11091706]

>>11091685
abstraction is only fulfilling once you've seen the practical side of things.

[Anonymous No.11091819]

>>11091685
>If I'm not enjoying these basic elements then am I just going to burn out on later things?
Probably. It's true that calculus isn't a totally accurate representation of what later, theory-focused math courses are like, but it's also true that calculus is still math, and it does indicate interest/talent for more advanced stuff somewhat accurately.
You should take a linear algebra course (a proper one about vector spaces, not the hurf durf I row-reduce matrix one that has 300 engineers in it). Abstract linear algebra is (in my opinion) the best general litmus for whether somebody likes "real" math at all. If you're bored in linear algebra, you're very likely just not mathematically inclined in your interests.

[Anonymous No.11091886]

>>11091648
>>11091685
multivariate calculus is literally just "do it 3 times"
don't let it change your perceptions of math
start checking out linear algebra, totally different flavor

[Anonymous No.11092109]

>>11082112
The only field where that's universally true is theory of PDEs

[Anonymous No.11092141]

>>11087336
You're moving a card from index k to index 3k. (I'm using 0-based indexing as it's simpler in this case.) The indexes are in $\mathcal{Z}_{15}$. I'm sure you can take it from here.

[Anonymous No.11092158]

>>11083630
>Rees algebra
REEEEEEEEEEEEEEEEEEEE

[Anonymous No.11092249]

[Anonymous No.11092280]

>>11092109
you can't solve any problems with intuition in PDEs

[Anonymous No.11092431]

>>11092109
>universally true
That is a meaningless statement in 2019.

[Anonymous No.11092437]

>>11092431
Universally false post

[Anonymous No.11092443]

>>11092437
universally stupid post

[Anonymous No.11092481]

>>11092280
But something like maximum principles are very intuitive once you understood the physical interpretation of e.g. the Laplace equation.

Although it obviously Intuition gets more and more useless the more abstract you get, I doubt that a physical interpretation for more general maximum principles, e.g. on linear second order PDOs has any physical meaning.

[Anonymous No.11092488]

>>11089919
>just genuinely fucking stupid hick retard trying to understand how is 1+1=2 even possible
One stone and another stone make two stone. Grug sure.

[Anonymous No.11092491]

>>11092488
>One stone and another stone make two stone
I'm inquiring about the nature of '2' and 'make' here since for any of this to make sense, 1 has to be 1, it can't transcend itself into '2' or then you couldn't be sure that 1+1 = 2 because it could be actually 4.

But that's what you get if 0.333 can be 1

It's insane voodoo magic

[Anonymous No.11092496]

>>11092491
What? Stop being braindead, the definition of the word "2" both in common use and in mathematics is 1 and 1 put together.

There is no "transformation" or "transcendence" 2 is just the shorthand we use for 1+1.

[Anonymous No.11092510]

>>11092496
What does "1 and 1 put together" mean this is so insane, it could be anything from 0 to 2 because for some reason 0.9 is actually 1 and not 0 so when you are putting '1 and 1' together it could be 0 if the probability aligns right and when the 0.9 is actually 0 instead of being 1 so how do you know you have two instead of 1 and when you 1+1???

im sorry

[Anonymous No.11092519]

>>11092481
i didn't realize we were talking about fucking maximum principles, now, did i?

[Anonymous No.11092521]

>>11092510
>What does "1 and 1 put together" mean
What does "mean" mean. This is a stupid game.

Nothing you write makes any sense it is impossible to communicate with someone for whom words mean something different and you simply are incapable of formulating a sentence I could interpret.

>it could be anything from 0 to 2
No, by definition it is 2.

>because for some reason 0.9 is actually 1
No.

>so when you are putting '1 and 1' together it could be 0 if the probability aligns right
No.

>so how do you know you have two instead of 1 and when you 1+1???
Because that is the definition of 2.

[Anonymous No.11092523]

>>11092519
>i didn't realize we were talking about fucking maximum principles, now, did i?
I didn't talk about anything before that, I just wanted to comment that indeed there are situations in which physics intuition can be used to make sense of PDE results.

[Anonymous No.11092524]

>>11092521
>>>because for some reason 0.9 is actually 1
>No.
but the entire thing of 1/3 + 2/3 which is just 0.333 puts together thrice means that 0 is 1. so if you have 1 it can also be a 0 because 0 can be 1. so how do you know that when you put 1 and 1 together its not 0 and 0? or 0 and 1?

[Anonymous No.11092527]

>>11092510
I already explained the logic thing to you before, dude. Everyone in this thread is just going to be mean to you.
You're looking at this all wrong - every day I wake up, and when I look at my hands, I put one finger up, and there's one finger. And then I put one more finger up. And there are two fingers up now.
Mathematicians DEFINED (this is non-arguable, not ambiguous at all) an operation to handle this common and seemingly unbending phenomenon, and gave it the name "addition" and made a + symbol for it.
You're asking why 1 + 1 is 2, but it just is, because that's the way the + symbol is defined. It is a function which takes two numbers and always always gives you the same result, a very specific result which is very well defined, no matter what you do.
This is not about counting things on fingers anymore, this is just a definition. Just like any dictionary definition.
In a similar way to how you can't just use "of" anywhere you like in a sentence, you can't just write 1 + 1 = 1.4. There's a definition, and you'd be ignoring it, so the thing you'd be saying wouldn't make sense.

[Anonymous No.11092530]

>>11092524
>but the entire thing of 1/3 + 2/3 which is just 0.333
No.

>means that 0 is 1
No.

> so if you have 1 it can also be a 0 because 0 can be 1. so how do you know that when you put 1 and 1 together its not 0 and 0? or 0 and 1?
No.

I won't solve another Captcha for you. Your words do not have the same meaning they have for other people.

[Anonymous No.11092531]

>>11092524
you've made a mistake somehow, nothing about 1/3 + 2/3 = 1 implies that 0 is 1.
but it is true that if 0 is 1, then you can derive any result you like no matter how contradictory.
how exactly are you getting from 1/3 + 2/3 = 1 to 0 is 1 again?

[Anonymous No.11092535]

>>11092523
No worries, I was just being smug and pretentious for fun. Now I'm going to do it again.
>maximum principles
>physics "intuition"
>PDE "results"

[Anonymous No.11092539]

>>11092531
>nothing about 1/3 + 2/3 = 1 implies that 0 is 1.
It literally adds zeroes together because 1/3 is 0.333 and I've been thought that zero times three is still zero but suddenly not, its actually 1.

[Anonymous No.11092554]

>>11092539
see this is the part that doesn't make any sense to me
1/3 + 1/3 = 0.3333... + 0.3333... = 0.6666...
1/3 + 1/3 + 1/3 = 0.3333.... + 0.6666... = 0.9999....
Are you just upset that 0.9999.... = 1? Yeah, that's a residue of a poorly chosen decimal representation system for numbers. But if you think about it, it makes sense: 1 - 0.9999... should be 0.0000.....001 but if you have infinitely many zeros how the fuck are you supposed to have a 1 after them? this is just 0.
there are no problems here

[Anonymous No.11092570]

>>11092554
>Yeah, that's a residue of a poorly chosen decimal representation system for numbers. But if you think about it, it makes sense: 1 - 0.9999... should be 0.0000.....001 but if you have infinitely many zeros how the fuck are you supposed to have a 1 after them?
You understand why my brain freezes thank you.

[Anonymous No.11092599]

Feeling adventurous, might well-order an arbitrary set later

[Anonymous No.11092600]

If [math] R [/math] is a commutative ring with equal left and right action on modules [math] M [/math] and [math] N [/math] over an extension of [math] R [/math], does this imply that

[math] M \otimes_R N [/math]

is just the tensor product?
If yes, can I loosen my assumptions on the three entities?
If not, what do I have to tweak to make this true?

[Anonymous No.11092637]

>>11092600
>is just the tensor product?
You mean [math]M\otimes_RN = M\otimes_\mathbb{Z}N[/math]?

[Anonymous No.11092640]

>>11092637
I think yes. In this notation, are M and N already defined to be right and left modules?

[Anonymous No.11092648]

>>11092640
Let's check that shit out. Let [math]r\in R, m\in M, n\in N[/math]. Now we have [math]r(m\otimes n) = (rm)\otimes n = (mr) \otimes n = m \otimes (rn) = m \otimes (nr) = (m \otimes n)r[/math], so it does indeed act like the tensor product over the integers.

[Anonymous No.11092655]

>>11092648
I mostly try to understand the very definition and look at
>>11089517
or
>>11088595

Actually I find it strange that the integers come into play here. What I know is that Z is initial, so integers will be everywhere when you talk about rings - but why do the above two definitions relate to the integers?

My thinking was that the module product ([math] \otimes_Z [/math]) is all about those elements that come from identifying where you can at least move [math] r[/math] between the left and right side (between M and N from the right resp. left) - so I guessed that if everything commutes, then you just have a normal tensor product.

[Anonymous No.11092660]

>>11092655
If we try this >>11089517, then we have [math](\varphi\otimes 1 - 1 \otimes \psi)(m \otimes r \otimes n) = (mr) \otimes n - m \otimes (rn)[/math]. The two tensor products are the same iff this difference is 0 for any triple, as then the cokernel is trivial.

[Anonymous No.11092685]

>>11092570
Think of this: Whats the number between 0.999... and 1? If there isnt one, theyre the same number. Infinity is a powerful thing.

[Anonymous No.11092735]

>that retard in my class who mumbles to himself, says how easy things are, shouts out the answers constantly despite being wrong, and doesn't let anybody else answer

[Anonymous No.11092762]

Saw this in a thread yesterday and was curious. What is [math]\displaystyle \text{Card}\left(\bigcup_{k=1}^{\infty} \mathcal{P}^{k}\left(\mathbb{R}\right)\right)[/math] ? Seems like it ought to be pretty big.

[Anonymous No.11092768]

>>11092762
should start at [math]k=0[/math] with the convention that [math]\mathcal{P}^{0}(A)=A[/math].

[Anonymous No.11092777]

>>11092599
mad lad

[Anonymous No.11092928]

>>11092762
Should be that of V_{w+w}.
That's a model of Z (not to be confused with ZF), so it's pretty big.

https://en.m.wikipedia.org/wiki/Von_Neumann_universe

Basically, for a setty foundation you need to be able to map sets X and Y to the function space X->Y a finite number of times. E.g. indeterminant integrals map (R to R) to R and Fourier transform adds one more set slot and so on and so forth. So 30 iterations of the powerset on N is what you actually need
Unless you want to express things in terms of universal properties, for which you need functors, which map between large objects (e.g. the class of groups is as big as the class of sets, so for functors you can't just go with power sets and instead need Grothendieck universes, which mimic classes but inside a set)

[Anonymous No.11092965]

>>11092762
>>11092928
R at the stage of V_{w+1} btw.
If your theory is string enough you can of course take that limit object you have defined (at the ordinal w+w, in your case) and carry on

[Anonymous No.11093031]

does the choice of p-norm affect the convergence rate of Newton's Method for nonlinear systems of equations?

[Anonymous No.11093036]

>>11092660
Thank you

>>11093031
I'd assume so - I mean it will depend on the class of problems too.
But I couldn't find a point blank statement about it when quickly googing just now.

[Anonymous No.11093087]

>>11092570
>>11092685
Yeah, a good way to think of this is as a limit. When we write "a number repeating" like this, what we really mean by that is that the number is the limit of all the little finite length parts (this is a definition! It's one way we can define the real numbers!) So when we write 1/3 = 0.3333... we mean to say 1/3 is the limit of the sequence 0.3, 0.33, 0.333, 0.3333, and so on.
Now just check for yourself what the limit of the sequence 0.9, 0.99, 0.999, 0.9999, etc. converges to, you should be able to convince yourself it converges to 1. You can even use the definition of convergence of a sequence here.

[Anonymous No.11093149]

>>11083245

Going through that book as well.

Is Aluffi's Chapter 0 a good introduction to Algebra?

[Anonymous No.11093299]

>>11093149
>Is Aluffi's Chapter 0 a good introduction to Algebra?
Why don't you read it and find out?

[Anonymous No.11093307]

>>11093149
>>11093299
to be fair, it's probably a strange introduction to algebra

[Anonymous No.11093317]

>>11093149
Shafarevich is the only introduction to algebra.

[Anonymous No.11093342]

>>11093307
this very much

[Anonymous No.11093612]

>>11092762
>Axiom of replacement
Enjoy your inconsistent foundations.

[Anonymous No.11093657]

>>11093612
lel

[Anonymous No.11094091]

>>11093149

[Anonymous No.11094143]

>>11092762
it's a big set

[Anonymous No.11094149]

>>11093612
huh?

[Anonymous No.11094153]

>>11094143
for you

[Anonymous No.11094215]

>>11092928
>>11093612
>>11094153
Numberphile just posted a video using (large countable) ordinals

https://youtu.be/0X9DYRLmTNY

[Anonymous No.11094340]

>>11082690
God damn you fucking csniggers really can't comprehend the world not being like your gay little vidya consoles can you?
Isn't it enough that you ruined neuroscience by putting the meme that brains are like computers into everyone's head? Now you have to do it to the rest of the physical world apparently
Go spank it to Stallman's greatest hits or something

[Anonymous No.11094466]

>>11094340
>csniggers
Why the racism?

[Anonymous No.11094471]

[Anonymous No.11094476]

>>11093149
Absolutely not. Find an algebra text meant for undergrads.

[Anonymous No.11094509]

>>11094476
>Find an algebra text meant for undergrads.
Nothing in Aluffi is too advanced for undergrads.

[Anonymous No.11094545]

/mg/pill me on stacks

[Anonymous No.11094554]

dead general

[Anonymous No.11094579]

>>11087731
based

[Anonymous No.11094580]

>>11094554
Agree, let's just give up. We're not going to make it anyway.

[Anonymous No.11094601]

>>11092109
>pdes
>universally make use of physical intuition to understand results
Why yes, whenever I use Riesz to prove there's a weak solution I interpret it in terms of physics, how did you know?
>>11094545
A sheaf, but instead of groups or sets it's categories.
>>11094554
Nah.

[Anonymous No.11094631]

New thread at >>11094565
Good night, lads.

[Anonymous No.11095325]

>>11094601
>Why yes, whenever I use Riesz to prove there's a weak solution I interpret it in terms of physics, how did you know?

>She can't imagine L^p in terms of geometry like R^3
Lol

[Anonymous No.11095929]

>>11094631
faggot

[Anonymous No.11095933]

bump 1

[Anonymous No.11095938]

bump 2

[Anonymous No.11095941]

bump 3
dumb idiot retard

[Anonymous No.11096293]

>>11083626
>>11083598
this exercise is extremely easy, just saying

[Anonymous No.11096436]

What is the fastest way to introduce myself in cutting edge modern topology and homotopy theory?

[Anonymous No.11096468]

>>11096436
may

[Anonymous No.11096576]

>>11096468
Are you talking about concise course? I've read that, but that's just basics. What do you read next?

[Anonymous No.11097881]

>>11095325
>she
No.