/sci/ - Science & Math

First for my waifu.

What's your favorite pathological object / construction / counterexample, /mg/?
Mine would probably be the smooth bump function, infinitely differentiable but not analytic.

>>10900719
I don't consider that function pathological... It's the basis of like a LOT of differential geometry.

>>10900719
Mine would probably be the function from an interval to another interval such that it's a continuous bijection with nonzero derivative at a point, but its inverse isn't continuous at the image of that point, therefore also not differentiable. Something like zhw.'s answer in this: https://math.stackexchange.com/questions/2336977/differentiable-bijection-f-mathbbr-to-mathbbr-with-nonzero-derivative-w

The standard answers would probably be Hilbert's cube and the lake thingy. Also a non-contractible space with all of its (reduced) homology groups trivial. That kind of stuff.

Something like this:

>>10900719
nagata's ring (example of a Noetherian ring with infinite Krull dimension)

>>10900719
Infinite groups with two conjugacy classes

S6's exceptional automorphism group

Z[[x]] and Q are not free Z-modules, even though Q[[x]] and R are free Q-modules

C and C(x) are subfields of each other, but they're not isomorphic

The ring inclusion of Z into Q is a monomorphism and an epimorphism, but not an isomorphism

>>10900719
Conway's base thirteen function

okay /sci, could you explain to a fellow brainlet what the fuck is truly a matrix? I am doing a course on algebra and we learn the basic operations about matrices, as well as determinants and etc. But unlike in common algebra where one instictively understands numbers as entities, I don't really get matrices. Could someone explain this to me?

>>10900719
I really don't like pathological objects, they point out the weaknesses in our definitions.

>>10900571

Any good books to read?

>>10900571
i need some help with linear regressions

i have a bunch of data about firms, including research and development expense, gross margin, and new product sales over a n year average, where n is the earliest year data was available, up to 2010.

Im probably gonna use excel but can anyone explain how i should progress with this?

i want to show that the higher amount a firm invests into RnD, the better they are off financially in the past.

I just want a way to figure out the trend if any, and preferably to any degree of significance.

Strong emergence does not exist. All phenomena can be reduced to an explanation involving the direct result of simpler laws, even consciousness. Prove me wrong.

>>10901071
A matrix is a function which maps from one vector space to another. Think of it like f(x)=y is equivalent to Ax=y, where x and y are vectors instead of numbers. A square matrix maps to the same vector space, while a MxN matrix maps from R^n to R^m (correct me if I mixed that up). Image attached shows a matrix when multiplied against a vector x as above, y will be that vector but rotated theta degrees anti-clockwise

>>10900719
measure theory has the funniest examples
>there is a subset of [0,1] which has outer measure 1 and also its complement has outer measure 1
>there is a function [math] f : [0,1]^2 \rightarrow \{0,1\}[/math], such that [math]\int_0^1 f(x,y) dy = 0[/math] for any [math]x \in [0,1][/math],
but also [math]\int_0^1 f(x,y) dx = 1[/math] for any [math]y \in [0,1][/math]
>banach-tarski but its been memed too much

>>10901071
Depending on their application, you can think of matrices as simply 2D arrays of numbers, transformations from F^n to F^m where F is any module over a ring, such as R as >>10901116 said, or you can think of matrices as canonical representations of abstract elements of a linear map when a particular basis set is given. Square matrices with det != 0 have even more structure and can be described in even more ways.

I've discovered my first formula frens

>>10901118
What's the name of the first example?

>>10901221
I think that would be the existence of a non measurable set.

>>10901162
extremely elementary

>>10900990
>S_6

>>10900990
>S_6
Patrician taste

>>10901242
Technically, you can prove the existence of the example anon gave from the existence of non-measurable sets, but I`m not sure it`s the same.

>>10901275
Yeah, you do something like this https://math.stackexchange.com/a/157696
but you just do the induction by choosing two elements at each step of the induction. God I hate the axiom of choice.

Why does anyone give a shit about primes?

>>10900571
Asking the anons who have TA'd before: what is the highest fail rate you've seen in undergrad math courses. What is the fail rate in Real Analysis courses and have you ever seen a fail rate that you felt was unethically high?

>>10901402
>unethically high
You can make calc 1 as easy as paper mario, and still 90% first year eng students will fail or change their mind about career choices before first semester is over (too much """math""" here) or give up so that they can try again to enter med school next year

>>10901221
many constructions are probably possible, look up bernstein set

>>10901402
I don't recall the exact failure rates but I'd estimate somewhere around 30% is the highest I've ever seen; this was for a Calc 2 course I TA'd in year 1 of grad school for a pajeet adjunct.
Failure rates in upper-division courses are generally much lower for many reasons; profs are less disinterested, smaller class sizes make feedback/tailoring easier, students are usually more motivated/less retarded than freshmen, etc. Realistically, mid-division courses like real analysis should not have more than 2-3 failures per class at the absolute max, and senior/grad courses should not have students failing them. If there are substantial failures in 300-400-level course, something went severely wrong. Of course if you count people who drop the class the rate would be substantially higher (a pretty big chunk of people drop real analysis) but I don't have numbers for that.

I would feel that (at least in the USA) a failure rate around or over 50% needs to be seriously looked at, although as a TA failure rates are not your problem and you absolutely should not try to tell the prof that he's doing a shit job of his course.

>>10901253
extremely homosexual

>>10901402
>>10901459
by "failing", do you mean failing the first exam, or failing all possible retakes and having to redo the course in future?

>>10901399
cause they're cool

The fact that they uniquely factor the integers apparently gives them tons of wonderful properties—properties which pose enormous challenges

how to prepare for calc 2 when it's been 14 years since I took calc 1?

I'm a little nervous

>>10901467
>14 years
What the fuck man.
Are you that one thirty year old cunt in the class?
Just fuck off dude.
Ask on /sqt/

>>10901162
good job, friend

Learn to enjoy the beauty of mathematical discovery <3

>>10901399
rsa encryption

>>10901467
If you still have your notes, run through the theorems that you did back then, and relearn all the rules and their proofs. Doing some prep learning on calc 2 can also help

>>10901467
Khan Academy is a great resource, especially for brushing up on calc and other high school stuff

>>10901472
Don't listen to that jerk, daring to try this is more than most people do

>>10901467
Retake calc 1

>>10901465
I'm referring to failing the course. Failing an individual exam (especially the first one) is very common. Most of these people either drop or get their shit together.

how to stay sharp after graduating?

>>10901492
get a job that isn't intended for braindead drones

>>10901490
Most people have the sense to take that W and bounce to an art school, seeing as the only way to avoid calculus in today's post-secondary education is to avoid a stem degree altogether.

>>10901492
Post on /mg/

>>10900719
Warsaw circle

>>10901419
I didn’t realize engies were that weak in calc, in my classes they tend to be the stronger non-math/physics students.
>>10901459
This makes sense to me but I’d heard that real analysis had particularly brutal fail rates from some anons here over the past few years. People saying they had professors failing 40-60% of the class and people dropping math and physics majors almost immediately after taking the course. That’s why I asked about that class specifically, it seems like its a strong weedout for people not prepared for proof based undergrad courses. I also hadn’t thought about taking W’s to avoid the embarrassment of failing but that should probably be factored in. >>10901467
Apostol, Courant, Spivak for textbooks; MIT OCW and Professor Leonard for YouTube university practice/techniques/cramming.

>>10901492
Answer stuff in /sqt/.

If I have an n-degree polynomial, should I use the fourier or taylor series to represent it as a sum of functions?

>>10901657
Nice joke mate

>>10901657

use the Parker transform:

[math]f \mapsto f + 0[/math]

>>10901657
>n-degree polynomial
>Taylor series
interesting proposition

>>10901674
>>10901743
Can't the taylor series be applied recursively?

>>10901686
You mean into Spiderman?

>>10901657
>Taylor expansion of an n-degree polynomial
That's a really advanced technique, are you sure you can handle it?

>>10901467
Well I suggest you watch sissy hypno porn and kill yourself you old fogey. Your gonna be the stupidest one in class and all the freshmen are going to be laughing at you when the teacher announces the lowest grade because they will know you got it

>>10901826
What is wrong with you? Are you replying to yourself? No one under 40 uses the word "fogey."

No seriously, imagine you have a really difficult function that you need to approximate as an electrical signal. So you first expand it into a sum of polynomials with the Taylor series and then you expand each polynomial as a sum of sin and cos, i.e. the Fourier series. Will this work? For example e^x? I mean you can probalby convert that one directly to the fourier series but I am sure you can find a a stubborn function that cannot be directly expanded into the fourier series.

>>10899745
I don't understand what you're asking exactly because you're using terms that don't mean what you think they mean. But I think it's rude to not at least answer you under a generous interpretation of what you might be asking.
At least in free theories, particles are sections of the associated vector bundle of a [math]\operatorname{Pin}^c_{1,n}[/math]-bundle. For [math]n\geq 3[/math], spin-statistics applies and bosons correspond to integer irreps and fermions are half-integer irreps. In other words, under the central extension [math]1\rightarrow U(1) \rightarrow \operatorname{Pin}^c_{1,n} \rightarrow O(n) \rightarrow 1[/math] bosons are irreps that pull back to [math]1 \in U(1)[/math] and fermions are those that pull back to [math]-1 \in U(1)[/math].
Physical theories require unitarity and the so-called "topological spin-statistics". The former states that the fields have non-negative norm (the associated vector bundle is actually a Hermitian bundle), while the latter states that Dehn twists of the underlying manifold by [math]\pi/2[/math] gets mapped to a "fermionic supercharge operator" [math]\operatorname{tr}(-1)^F[/math] that counts the number of fermions on the [math]\operatorname{Vect}[/math] side.
Aside from these algebraic/topological requirements, we may wish to equip the bundle with a [math]G[/math]-structure in order to gauge certain symmetries. This requires the underlying [math]G[/math]-manifold to be Hamiltonian and the existence of a moment map such that the [math]G[/math]-structure is compatible with the [math]\operatorname{Pin}^c[/math]-structure. This allows Atiyah-Bott localization.
In addition, you require either the Wightman axioms to be satisfied, or the Osterwalder-Schrarder (slightly weaker) axioms in addition to reflection positivity. Regularizability is given by the Strocchi-Swieca regularity conditions on the space of sections.

So please, study these (or at least read some Landau-Lifshitz) and ask coherent questions next time.

>>10900719

>>10900719
Construction of a 1-generic set using a Kleene-Post construction, which is a primitive version of Cohen forcing.

>>10901996
ive never seen Yukarifag mad at a brainlet before you must really be annoying lol

>>10902047
namefag isn't a brainlet, he's an underage who types incoherent nonsense pretending to be a schizo

>>10901473
Wholesome af, noice

>>10900719
Circumventing the Nielsen-Ninomiya theorem and the edge-bulk correspondence.
https://arxiv.org/abs/1903.08877

>>10901082
Moby Dick

What does this mean [eqn]\mathbb{C}\langle x_1,\cdots,x_n\rangle[/eqn]

>>10901493
Where?

>>10902521
gotta spell it out first

>>10902526
The set of complex numbers C is equipped with a sequence from x to x_n? I still forget what this means

>>10901082
Recolte et semaille

>>10901082
my diary, desu

>>10901836
Mate just cut your losses, there is no use fighting against your fate. You were destined to be a brainlet. It is obvious you were clamped and irradiated

>>10902530
Maybe complex coefficient generator of set numbers/polynomials? Pulling it out of my ass though.
https://en.wikipedia.org/wiki/Generating_set_of_a_group

>>10902637
don't wanna kill myself

>>10902521
C times the dot product of x1, dots, xn

>>10902521
https://www.encyclopediaofmath.org/index.php/Free_associative_algebra
It is the set of "polynomials" in the x_i where you do not assume that the variables commute

3+4i

how the fuck do i get this in polar form please help, me and my nigga been working on this we got two half CHIMP brains each

>>10903199
>Write number on a piece of paper
>Take piece of paper with you
>Go as far north as you can go

>>10903199
what have you tried ?

>>10901118
Analysis is a pathology itself.

>>10903199
draw a triangle my nigga

>>10901996
>>10902073
fuck off to discord rapcak or alternatively go shitpost in your physgen where all the lunatic congregate
>>10902047
look at this discordshill

wat goe in all field/?

orthogonal or eigen

>>10903763
perpendicular

>>10903763
Right-angled.

Simple algebraic field extensions make for finite dimensional vector spaces. But why is it I never see any matrix in field theory texts, then. Aren't there interesting operations within field extensions to be studied? I could guess that some Galois groups have matrix representations over the fields, but I never see them exlicitly, let alone used.

>>10901162
Nice penmanship friend. Who else here like using fountain pens? Or just pens in general? If I have to use a pen, it's the 0.7mm pilot g2. The 1mm makes nice, thicker lines, but it bleeds through too much unfortunately.
>>10901492
Recreational math! On the AMS website they have featured tons of books with fun problems. Further, there are monthly journals with challenging problems with reader submitted solutions. Fun, and you can get your name out there.
>>10904079
Eh, could be the fact that matrices in finite dimensional vector spaces serve the role of transforming one vector into another, but with fields, the elements are already sufficient to do that, no need to specify a new object to do that.

>>10901162
[math] {d^n \over dx^n} f(g(x))
=\sum \frac{n!}{m_1!\,m_2!\,\cdots\,m_n!}\cdot
f^{(m_1+\cdots+m_n)}(g(x))\cdot
\prod_{j=1}^n\left(\frac{g^{(j)}(x)}{j!}\right)^{m_j} [/math]

>>10904088
Is it clear that all linear maps of a field extension (say in Q(2^(1/3))) can be represented by field element multiplication in the algebra?

>>10901162
basically copy pasted from binomial theorem

>>10904100
>Is it clear that all linear maps of a field extension (say in Q(2^(1/3))) can be represented by field element multiplication in the algebra?
If you're viewing the field just as a (n-dimensional) vector space over Q, then the answer is no, since the space of all linear maps has dimension n^2, and the space of multiplications has dimension n.

>>10904116
smart

I've been toying with pic related, thoughts?

>>10904185
You're just looking at functions that generate a finite monoid and ask for the order of the group element

I think you're just asking for monoid representation theory of finite groups where the mutliplication is given by function concatentation.

For the invertible functions (groups), see the list of options here
https://en.wikipedia.org/wiki/List_of_small_groups
In particular, for cyclic grups, in C, for f(x):=x*exp(2pi/n), you get f^(n)(x)=x.

Your example are probably finite subgroups of
https://en.wikipedia.org/wiki/Modular_group

>>10904231
Just to be clear, the notation I was using wasn't exponentiation.
[math]
[f]^3(x)=f(f(f(x)))
[/math]
I'm a complete amateur mathematician but I'll try to read the links that you posted

>>10904116
If you're gonna take that perspective, then any linear map you could think of could be represented as a tensor and be defined by the field elements and the field multiplication.

>>10904185
Hit me up with theorems about cyclic Darboux functions.

>>10904240
I know, I understood you correctly.
E.g. for n=4, choose
f(x):=i*x
f(f(x)) = i*(i*x) = -x
f(f(f(f(x)))) = -(-x) = x
You already do the same trick when you write +- in the last line.

Btw. a more explicit article article than the modular group one is
https://en.wikipedia.org/wiki/M%C3%B6bius_transformation

Also your 4th one is
1/(1-1/x) = x / (x-1)
and it's clear that you'll not come back to x as soon as you e.g. use x^2 somewhere.

>>10904268
Okay I see what you mean, in your initial response you had
[math]x*exp({2\pi \over n}[/math]
instead of
[math]x*exp({2\pi * i\over n}[/math]

>>10904280
Oh yes, the i was in pi and it needed a second one.

Btw. note that once you found a solution, you can generate new ones by group element conjugation f -> h f h^-1

For example, since
f(x) := 1/(1-x)
has
f(f(f(x))) = x,
so does
g(x) = sin^{-1}(1/(1-sin(x)))
where sin(x)(sin^{-1}(y))=y

>>10904280
Oh yes, the i was in pi and it needed a second one.

Btw. note that once you found a solution, you can generate new ones by group element conjugation f -> h f h^-1
https://en.wikipedia.org/wiki/Conjugacy_class

For example, since
f(x) := 1/(1-x)
has
f(f(f(x))) = x,
so does
g(x) = sin^{-1}( 1/(1-sin(x)) )
where sin(sin^{-1}(y)) = y

>>10904300
a clearer example might be
[math] k(x) := \sqrt{ \frac {1} {1-x^2} } [/math]

>>10904307
>>10904300
I guess what I'm trying to figure out, (as delusionally ambitious as this may be), is if I can find all functions that are n-idempotent. (please don't give me the full answer I want to figure this out on my own I just need a little help)

>>10904311
I don't think I'd have much more to say anyway.
One thing that might make it more reasonable would be to define conditions on the functions.

E.g. if you use the full axiomatic machinery of "conventional math" then you end up with properties that "existing" exotic functions are exceptions for, see e.g.

https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation

>>10904185
Consider f(x)=(ax+b)/(cx+d). For any n, f^n(x)-x=0 gives you PQ/R=0 where P=cx^2-(a-d)x-b and Q is a polynomial in a,b,c,d (not involving x). Finding values of a,b,c,d for which Q=0 gives you a function satisfying f^n(x)=x. E.g. for n=2, Q is a+d, so any function of the form f(x)=(ax+b)/(cx-a) satisfies f^2(x)=x. For n=3, you have ad+bc+a^2+d^2=0, for which one solution is f(x)=(x-3)/(x+1).

Note that any "continued fraction" expression, i.e. starting with x and repeatedly replacing x with a+b/x for some a,b can be reduced to an expression of the form (a+bx)/(c+dx). Such expressions correspond to projective transformations on the real line.

In short, you're looking for a 2x2 matrix M=[[a,b];[c,d]] where M^n=kI, i.e. M is any nth root of the identity matrix, multiplied by an arbitrary scalar (which will be present in both numerator and denominator, and so cancels).

>>10904079
matrices are big gay, the seething regarding their lack of use in abstract lin alg texts is very stupid

>>10903572
don’t ever reply to me again sperg
>>10903083
you should all stop asking questions in /mg/

>>10904369
>gay
Why the homophobia?

>>10904369
I cope

>reals aren't computable
>not only that, the reals that are computable have measure 0
this fact makes me existentially uncomfortable

>>10904437
epic
>>10904378
gays are predisposed towards pedophilia, promiscuity, and psychosis

>>10904437
Pretty sure there's a good sense in which a subset of the reals having non-zero measure is quite a special occasion. So them having measure zero shouldn't Rob you sleep.

There's no strong understanding what subsets of infinite sets A are, as evidenced by the fact that you can force models where 2^A are.
E.g. it's consistent that there's intermediate sets |w0| < |w1| < 2^|w0| rejecting the continoum hypothesis, and then in L you have |w0| < |w1| = 2^|w0| but You CA. Also have |w0| < |w1| < |w2| = 2^|w0|.
Collections of subsets are up for grabs and we can't control them, so fuck it

>>10904617

what is a functor

>>10904734
morphism between categories

>>10904734
A category is a way of talking about relationships between objects. The morhpisms of a category are the relationships between objects. Commonly these morphisms are structure preserving functions (homomorphisms, continuous maps, order preserving maps, etc) but they don't have to be.
For example the category of groups consists of all groups and all group homomorphisms.

But we want to go one step further and talk about the relationship between categories. A functor is then a certain type of structure preserving map between categories. For each object in the source category it assigns an object in the destination category and for each morphism in the source category it assigns a morphism in the target category. These assignments are structure preserving in that they respect identity morphisms and composition of morphisms.

As an example, in topology we can assign every space a fundamental group and every continuous map of topological spaces induces a group homomorphism between their respective fundamental groups.

>>10900772
Sure, it's not particularly pathological in the sense that it is used everywhere. but it's fucking weird that it works on an analysis level, and it fucks with my intuition for the proof of taylor's theorem.

>>10904100
what is the sum over? it's some weird shit if i recall correctly, like m_j such that m_1 + 2m_2 + ... + n*m_n = n.

>>10904361
What is "R"

find the smallest positive integer n for which [math]\left\lfloor\log_2 1\right\rfloor+\left\lfloor\log_2 2\right\rfloor+...+\left\lfloor\log_2 n\right\rfloor=1994[/math]
do you solve this using the Stirling's approximation?

What's the most autistic math you have studied?

this variety of math has lost it's usefulness
math is useful when you are calculating things, but this not so much

>>10900571
>math general

yes, but how does it apply to reality?

>>10905058
a better question would be, who cares?

thoughts on differential topology and guillemin and pollack's book? should i bother learning this at an undergraduate / early graduate level if i already have some classical differential geometry, a good amount of topology and measure theory, and i plan on taking more advanced graduate courses in smooth manifolds? i'm not sure if it's a waste of time or not.

>>10901162
you have very elegant handwriting, good for you. I can't stand people that think its alright to scribble everything.

>>10905134
i can't stand you, but you won't stop posting, will you?
enjoy this picture as a symbol of my distaste for you

>>10905136
lol