ahem...
eli5 this anon senseis >>16134585

>finished a degree in math with very good grades
>still know nothing about advanced math
Why is it like this? Was my university just shit tier or is there a huge gap between the contents of math lectures / textbooks and actual advanced math?

>>16135640
what's "advanced math" according to you?
undergrad math after the first year is advanced for non-math undergrads (and some postgrads)
by math standards, all the advanced specialized courses are post grad only.
i think it has something to do with the way faculties are organized rather than not wanting to give advanced math to undergrads.

>2 dimensions
Curvature of a wiggly line
>3 dimensions
Curvature and torsion of a wiggly line
.
>n dimensions
How many terms do you need to characterise the wiggliness?

Some guy made a video on youtube about 0^0 last month and there have been a few reply videos about it and all of them suck.
If you look at the riemann surface of z=x^y you can tell that z approaches zero as you get closer to the origin in every single direction.

>>16135652
Can somebody please name all books in this list?

>>16135736
>as you get closer to the origin in every single direction
it's not necessary to add this because in the complex plane all limits are taken considering all possible directions

0^0 is the empty product and therefore 1.
I will hear no further discussion on the matter. Cry more, analysts

>>16135652
>>16135771
http://imperium.lenin.ru/~verbit/MATH/programma.html
>I don't think all areas of mathematics are equally valuable; I am sure that mathematics in itself has no intrinsic value.
>Mathematics is only interesting insofar as it relates to string theory; this is a basic assumption that I don't want to discuss now. Relevance to physics is the only criterion we have left; and almost all the mathematics related to physics is related to string geometry.

>>16135799
List of useful books on mathematics

First course
Analysis" by Laurent Schwartz, "Analysis" by Zorich,
"Problems and theorems from functional analysis" Kirillov-Gvishiani
Differential topology (Milnor-Wallace),
Complex Analysis (Henri Cartan), Complex Analysis (Shabbat)
Second course
Lie groups and algebras (Serres)
Algebraic topology (Fuchs-Fomenko),
"Vector bundles and their applications" (Mishchenko)
"Characteristic Classes" (Milnor and Stasheff)
"Morse Theory" (Milnor),
"Einstein Manifolds" (Arthur Besse),
Commutative algebra (Atiyah-MacDonald),
Introduction to Algebraic Geometry (Mumford)
Algebraic Geometry (Griffiths and Harris),
Algebraic geometry (Hartshorn)
Algebraic geometry (Shafarevich)
Algebraic number theory (eds. Cassels and Fröhlich)
Number theory (Borevich-Shafarevich)
Galois cohomology (Serres)
"Invariants of classical groups" (Herman Weyl)
Third year
Infinite loop spaces (Adams)
K-theory (Atiyah)
Algebraic topology (Switzer)
Analysis (R. Wells)
Index formula (Atiya-Bott-Patodi, collection Mathematics)
Homological Algebra (Gelfand-Manin)
Cohomology groups (Brown, or something)
Cohomology of infinite-dimensional Lie algebras (Gelfand-Fuchs)
Kähler manifolds (Andre Weil)
Quasiconformal mappings (Ahlfors)
Fourth year in college
Geometric topology (Sullivan)
Etale cohomology (Milne)
Algebraic geometry - review by Danilov (Algebraic Geometry 2, VINITI)
Chevalley Groups (Steinberg)
Algebraic K-theory (Milnor)
Suslin's review of algebraic K-theory from the 25th volume of VINITI
Multivariate complex analysis (Goto-Grosshans)
The same from Demaia’s book (translation in preparation)
Fifth year
Gromov "Hyperbolic groups"
Gromov "The sign and geometric meaning of curvature"

>>16135736
y=x??

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What math do I need to know to understand Classical Mechanics?

>>16135851
symplectic geometry

>>16135851
just jump straight into Taylor

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>>16135585
champernowne constant
>https://mathworld.wolfram.com/ChampernowneConstant.html
I have a python script that finds the first occurance of any number in it.

>>16135851
Read Abraham Marsden's Foundations of Mechanics

>>16135640
Read https://sheafification.com/the-fast-track/

You need to work through textbooks written by the masters

>>16135640
See >>16135652
aim high, it doesn't just have to be a dumb desk engineering job either.
the effort doesn't just stop at reading hard books, if you just go for a programming job you're being lazy, and that's not why we're here on /mg/
it's not 'follow this hobby math curriculum and do a bunch of toy analysis proofs'
which is why our list doesn't look like a typical math or CS major, because it is not courses for courses' sake
it doesn't stop there, it continues in your work - if you know theoretical physics and a base of mathematics, you could be doing literally anything
it would be lazy and shameful to stop there
you should be working on something big and important, gather experience in a different field, transform it with mathematics, start your own company
anything less is a waste

what math do I need to understand the applications of derived algebraic geometry to the symmetry of PDE's?

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How do you solve this shit using Lhospital rule.

>>16136180
cringing at this website...

had a dream where I was a cancer doctor and I was cutting into someones legs where the cancer was and there I found the cohomology groups in the meat. the cancer was in the cohomology groups...

>>16136659
google

>>16136659
go to the emergency room and tell them you got assraped by Calculus AB

>>16136669
As someone who has only done analysis electives for fun (not a math grad, just like math and find it useful for my actual research area) I can't imagine spending more time than necessary studying algebraic topology. The little bit of topology I've been exposed to seems like there's always an army of schizos around every corner you turn.

Also, was the was all of the cancer in the same homotopy class?

rate my linear algebra book
https://archive.org/details/foundationsoflin0000malt/page/n7/mode/2up

>>16135640
>>16135684
I'd call advanced math something you might be proud of publishing in grad school. It's something that could be novel, but not groundbreaking or even field advancing. It shows that you're capable of connecting the dots between seemingly unrelated fields of math. It's something that might make a professor say, "neato," and be on with their day.

>>16135640
A lot of "advanced" math, especially the pedagogical parts, isn't actually written down in a publication anywhere. The only way to learn it is firsthand from people who know it
If you don't know anything cool after a math degree it's because you didn't socialize enough

>>16136669
holy shit you found a cohomological obstruction towards being cancer-free

>>16136180
buy an ad

>>16136718
>>16136767
Reddit is more helpful than a general dedicated to answering math questions. Do you know why? Because reddit isn't filled with miserable, jades fuckups that hate seeing people succeed. Your life is a fucking cope.

>>16136999
>people won't spoonfeed me homework answers so your life is a cope
go back

>>16136659
you want to get a 0/0 to use lhopitals. how do you do that to your equation? you might need to use trig identities

>>16136659
use lhospital to get lim 6x/10x
use it again to get lim 6/10
use it again lim 0/0
conclude 0/0 = anything = 3/5

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I used to be an analysis guy but now I have to admit that algebra Chads are indeed higher IQ. Every time I try to into algebra I get filtered hard.

>>16135640
Undergrad programs aren't designed to make you an expert in your field, especially in the U.S. where we still generally practice liberal arts education. The good thing about completing a math degree is that you should now have the 'mathematical maturity' to read and learn math independently. Ask yourself: what do you want to learn? What do you need to learn to get there?

why do so many, although ugly, women study maths? i thought they were supposed to be stupid compared to men.

>>16137488
In my experience math is 10% women, compsci is 1% and biology is 80%.

>>16137305
analysis gets you pussy, algebra gets you enlightenment

>>16137305
it's all about the morphisms nigga that's all

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Roll's Theorem!

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>>16138204
It's "Rolle's" for you!

>>16137600
in my uni, math is probably close to 40%. probably because it has an economy curriculum but still, how come so many of them are so good at math?
another thing i noticed is that there are very few black people here but most of them study math. fascinating.

Why does the only notion of higher categories index them by [math]\mathbb {Z}^{+}[/math]? I know they'll be useless but what's stopping us from creating 2.5-categories, 3.7-categories and [math]\sqrt{2}[/math] categories to fuck with grad students?

I am a bit confused about something in propositional logic. I managed to find an example that will best illustrate what's going on. Basically, this argument shows something false, but I don't really understand why it fails.

[math]\textbf{P1:}[/math] If [math]( A \rightarrow B )[/math] is a tautology, then it is false that for every formula [math]( B \rightarrow A ), ( B \rightarrow A )[/math] is a theorem of propositional logic.
Assume that [math]( A \rightarrow B )[/math] is a tautology, and that for every formula [math]( B \rightarrow A ), ( B \rightarrow A )[/math] is a theorem of propositional logic. So without loss of generality, we can take it that [math]A[/math] is logically inconsistent and [math] B [/math] is a tautology. Then [math]( A \rightarrow B )[/math] really is a tautology but [math]( B \rightarrow A )[/math] is logically inconsistent. In propositional logic, only tautologies are provable from the empty set, so [math]( B \rightarrow A )[/math] is not a theorem, contradicting our assumption.


[math]\textbf{P2:}[/math] If [math]( B \rightarrow A )[/math] is a tautology, then for every formula [math]( B \rightarrow A ), ( B \rightarrow A )[/math] is a theorem of propositional logic.
This follows from the completeness of propositional logic.

Putting [math]\textbf{P1}[/math] and [math]\textbf{P2}[/math] together:
If [math]( A \rightarrow B )[/math] is a tautology, then it is false that for every formula [math]( B \rightarrow A ), ( B \rightarrow A )[/math] is a theorem of propositional logic.
If [math]( B \rightarrow A )[/math] is a tautology, then for every formula [math]( B \rightarrow A ), ( B \rightarrow A )[/math] is a theorem of propositional logic.
Note that these two implications have contradictory consequents. So, if [math]( A \rightarrow B )[/math] is a tautology, then [math]( B \rightarrow A )[/math] is not a tautology.

What's wrong with this, exactly? I would very much like it to not repeat such mistakes again.

>>16139047
1. A->B is not a tautology, so if A->B is a tautology, then everything follows.
2. B->A is not a tautology either. Same as 1.

>>16137628
Actually anal. will give you asshole

>>16139056
I apologize anon, I don't quite understand what you mean. To clarify, A and B here stand for any formulas of PL.

>>16139061
A->B is not a tautology when A,B are free variables. For it to be a tautology would mean that for all choices of A,B in {True, False}, the evaluation of the formula is true.
If you mean A,B are some particular forrmula you're meant to substitue, things are a little bit different.
>then it is false that for every formula (BA),
This doesn't make sense. Because if you mean A, B are some particular chosen formulas, then B->A is also a single determinate formula, and it doesn't make sense to say "for every formula B->A".

To illustrate my point, it's analogous to saying
"If p is a prime number, then for every number p, p^2+1 is divisible by p".
You're mixing bound and free variables and that's not allowed.

>>16135585
Hello sci.
How do i increase my learning speed
How do i learn math faster
I consider myself an average person in terms of mathematical abilities but I have this autism of going deep into proofs and not moving on until i get a concept right

I even write down proofs myseld sometimes when reading a book
This is causing me to fall back in courses

How do i learn new math faster and do better in exams

>>16135800
>Algebraic topology (Fuchs-Fomenko),
Cant find online

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Anons I need you to help me with something.

During all our academic life(especially in chemistry, physics and mathematics subjects) all problems are "the same" in a way: look for a calculated result based on a series of data that we are given. This is done by doing a series of "actions" (passing a variable to the other side of the equation, integrating, deriving, substituting, etc).

Is there any branch of mathematics that formalizes and deepens all these "actions"? Is there any way to formalize that and discover its "patterns"? I have heard that the closest is category theory, homotopy type theory and categorical algebra but I have no FUCKING idea how to start, niggas I just want to solve my physics problems.

>>16137488
>>16138770
I think math culture is more approachable to fembrains than science culture is. It's quite teaching-focused so it attracts people who enjoy that, and it's a common degree choice for people who become teachers
It's also a degree where you can do almost literally 100% of the work chatting over a Starbucks in the math lounge if you want to

>>16139480
Perhaps it must be the textbook "Homological Topology" by them.

>https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.cimat.mx%2F~gil%2Fdocencia%2F2020%2Ftopologia_diferencial%2F%5BFomenko%2CFuchs%5DHomotopical_Topology%25282016%2529.pdf&psig=AOvVaw25T9XbDE0rhZHX_59UYh9i&ust=1713828174238000&source=images&cd=vfe&opi=89978449&ved=0CAgQr5oMahcKEwjgv9b9utSFAxUAAAAAHQAAAAAQBw

>>16139486
>Is there any branch of mathematics that formalizes and deepens all these "actions"? Is there any way to formalize that and discover its "patterns"?
depends on whether you consider programming to be math or not

>>16135640
I just finished my masters and I feel like I've only scratched the surface of 'advanced' math. There will always be stuff more advanced than you are doing and always more to learn than you ever can.

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>>16135597
Basically, everything is geometry. The process of squaring and rooting removes directionality from the system while also calculating the combined effect. The foundation of this is Pythagoras and its implications if you want to look into this part more.
The reason why it is a root^2 and not some other matching inverse exponent pair is because this distorts the topology of the domain.
You can see this by taking the equation for a circle and using higher exponents instead. The result is a square as the exponent approaches infinity.
One can eventually determine that the n=2 case is the only case where increasing one variable does not also influence other unrelated ones.
The 1/n factor is just to index the result to 1.

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What's your opinion of the Lean theorem prover?

So I was asked to find the premium p for an option of strike K that, at expiration, you pay p and gain [math] S_t - K [/math] if [math] S_t > K [/math], otherwise you pay nothing and get nothing.
I found that the payoff was [math] \mathbb{E}_Q( S_t-(K+p)) \mathbb{1}_{\{S_t>K\}} [/math]

I then attempted to solve (I think that's what I'm supposed to do but I'm really not sure)
[math] p= \mathbb{E}_Q(\mathbb{e}^{-rT} [S_t-(K+p)] \mathbb{1}_{\{S_t>K\}}) [/math]

and found
[math] p= \frac{ \mathbb{E}_Q(\mathbb{e}^{-rT} (S_t-K) \mathbb{1}_{\{S_t>K\}})}{1+\mathbb{e}^{-rT}P(S_t>K)}[/math]

I'm kinda confused since my results give a price lower than the price of a regular european call, which seems counterintuitive since the option I'm working on appears more "attractive", did I fuck up in the reasoning or the calculation ?

>>16140576
forgot to mention that the premium is calculated at t=0, and T is the expiration

>>16140324
Proof assistants are a great way to create an infinite stream of undergrad research projects without having to find things for young students to do but I'm not convinced there's any other point to them

>>16140736
What about teaching students how proofs work?

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>>16139486
>passing a variable to the other side of the equation, integrating, deriving, substituting, etc).
>Is there any branch of mathematics that formalizes and deepens
>passing a variable to the other side of the equation
Abstract algebra (passing a variable around is just adding the inverse of the passed element to both sides of the equation)
>integrating, deriving
Analysis, tells you what these are, proves why they do what they do, shows how they can work in higher dimensional spaces, also idk much about this but I think these can be defined much more generally something about derivating being an operator? No clues
>substituting
This also somewhat relates to the first point where you need to define what variables even are to you etc.
I think it's kind of a formal logic/model theory problem. Specifically substituting variables is touched on in universal algebra with algebras of terms (basically instead of numbers you just have the expressions with variables and operations on the expressions and see how those behave).

I know very little about category theory and fuck all about homotopy type theory and categorical algebra other than just some tiny tidbits I've heard/read but from those I think they are relevant but first you should study the basics.

For real analysis read Abbott - Understanding Analysis

For algebra there are tons of recommendations, try Aluffi - Algebra: Notes from the Underground.

I'm not sure about a logic book, A Friendly Introduction to Mathematical Logic is the only one I can think of that I could maybe recommend based off taking a peek inside twice or so. But don't overwhelm yourself. ACTUALLY try to read Serge Lang - Basic Mathematics first before jumping into the other fields. AND DO THE EXERCISES!!!

>>16140808
>(passing a variable around is just adding the inverse of the passed element to both sides of the equation)
Also I think I didn't give a fully satisfactory answer here as ideally you'd need to get to the root of what an equation and equality are, which is where I'd guess the model theory and category theory and foundations (one of which is homotopy type theory) come in but before concerning oneself with those and other "metamathematics" I think it's good to get the basics of more concrete mathematics down.

>>16139486
>>16140808
>ACTUALLY try to read Serge Lang - Basic Mathematics first
ok nvm probably you don't have to do this, if you could tell me what mathematics you've seen already I could maybe recommend some light discrete maths book first (have you seen sets, mathematical logic and proofs?) but in general I think you should be good going into Abbott's or Aluffi's book.

>niggas I just want to solve my physics problems.
What kind of physics problems?

>>16135585
so boring

>>16137488
i dont know maybe... maybe youre retarded or something...

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What's a good textbook on information theory? Is the subject interesting?

>>16142573
What's your level of math education and what are you trying to learn information theory for?

The general recommendation people give is cover and Thomas. It's super easy to find a PDF of (as well as solutions), doesn't require a ton of math experience beyond some basic multivariable calc based probability/Markov chains and some low level optimization, and covers at a decent level a lot of different topics.

MacKay's Information, Inference and Learning Algorithms is also pretty good but definitely less challenging.

If you're interested in a more rigorous and modern mathematics approach, Robert Gray's book on info theory is a great introduction to the measure theoretic approach to the topic. Doesn't cover as many topics as the other two books but is comprehensive on the topics it does cover.

Are there infinitely many odd primes [math] p [/math] such that [math] (p-1) = \mathrm{min}(\{ n \in \mathbb{N} : 2^n \equiv 1 \text{ mod } p \}) [/math] ?

In other words, are there infinitely many odd primes p such that the multiplicative order of 2 mod p is p-1 ?

>>16143230
Isn't that the same as asking if there are infinitely many primes of the form [math]2^n+1[/math]? Also, all primes except 2 are odd.

>>16143369
>Isn't that the same as asking if there are infinitely many primes of the form 2^n+1?
If p = 2^n + 1 , then 2^n = -1 mod p , which implies 2 has multiplicative order 2n mod p . And 2n ≠ p-1 . So I think what you're talking about is something else.

>Also, all primes except 2 are odd.
Yeah, so you can say either "odd prime p" or "prime p≥3".

>>16143380
>And 2n ≠ p-1
(Unless 2n = 2^n = p-1 , which only works for n=2 . But indeed p=5 is such that 2 mod 5 has multiplicative order 4 .)

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>>16135585
Just failed my calculus 2 exam (and the entire class), call me a retard please

>>16143852
Nah...I won't. Instead, I'll say fix yourselves.

wtf is k-theory on about? it reads like abstract nonsense for the sake of nonsense.

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What's the best curriculum for p-adic geometry? So far I've found intro books by Robert and by Koblitz, and a book about p-adic Lie groups by Schneider. Where would you go from there to get deeper into the subject?

>>16143230
For anyone wondering, this is a particular case of the Artin primitive root conjecture (https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots)
which is still unsolved in all its cases

(Got the answer from reddit)

>>16140324
In general based, but have cringe community, much like haskell

>>16136659
Multiply by 1, in this case $/frac{/frac{1}{x^2}}{/frac{1}{x^2}}$

I have to show that the root of [math]32x^6-48x^4+18x^2-1[/math] is not a constructible number. Can somebody tell me why this is true even though the polynomial is reducible?

>>16144508
Substitute in [math]u = x^2[/math] and see what happens.

Is there a single person on the entire planet who has actually worked through all of Kallenberg's Foundations of Modern Probability Theory? It's so damn terse.

What's the best way to learn math after calculus? Should I just pick a topic and dive into a textbook for it?

>>16144531
well it becomes [math](2u-1)(16u^2-16u+1)[/math] which to my eyes has constructible roots

>>16144541
Depends on what you want to do. Probably you should learn the rest of the "basic math" you'd need for most useful applications of math.

I'd say shoot for ODEs, linear algebra, some undergrad level probability, and then maybe some numerical analysis if you ever want to do math on a computer.

>>16144571
Yeah you're right, and the roots of your sextic polynomial are the square roots of those constructible roots so they are also constructible. I think you were right in the first place, in order for that sextic polynomial to have non-constructible roots its corresponding cubic must be irreducible over [math]\mathbb{Q}[/math].

>>16135726
I'm not sure of the answer, but here's my thinking:

- curvature of a curve: how much the curve fails to stay in a straight line
- torsion of a curve: how much the curve fails to stay in a flat plane

Continuing the pattern, we could define "n-curvature" of a curve to be "how much" the curve fails to lie in a flat n-plane.
Then usual curvature is 1-curvature, torsion is 2-curvature.

But I'm not sure how these higher curvatures would be actually defined, beyond just saying they measure "how much a curve fails" to be in a flat n-plane.

>>16135726
Not fully sure what you mean. Are you talking about the Frenet-Serret formulas for a 1-dimensional curve in n-dimensional space?

>>16143218
>What's your level of math education
I have a Master's.
>what are you trying to learn information theory for?
Just general curiosity. That's also why I'm asking how interesting the subject is overall.

Thanks for the advice.

i come up with this randomly during a "proof", is this true?
[math]
ab - \lfloor ab \rfloor = \frac{ab \mod n}{n}
[/math]

>>16139486
>category theory, homotopy type theory and categorical algebra
where did you hear that? a time traveler feom 2009?

>>16140324
Theorem provers (verifiers) are great, I hope LLMs can make good input for them soon. Because doing that shit by hand is many times more tedious than a normal proof, even if you are experienced.
>>16145299
What does ab mod n mean? Like the floor function but for multiples of n? No I don’t believe that equality is true. For example when ab=2 and n=3.

>>16145313
or ab=1 and n=2

>>16145313
Sorry im stupid, ab is the product of two numbers but i should've just put k when asking the question.
Anyway ab (or k) is a real number.

>>16145299
it would be [math]k-\left \lfloor{k}\right \rfloor =(k \mod n)-r[/math] where r is the remainder of [math]\left \lfloor{k} \rfloor \right/n[/math]

>>16145361
i also forgot to mention that i know k is equal to p/q.
let's say i'm given a number k and i have to find the fractional part of k, in another notation (k mod 1).
since i know p and q i can find out myself the fractional part of k.
[math]p = qx + r[/math] where x is [math] \left\lfloor\frac{p}{q}\right\rfloor [/math] and r is [math] p mod q [/math].
but then [math] k = \frac{p}{q} = x + \frac{x}{q} [/math] where x is an integer so x/q is the fractional part. but x/q = (p mod q)/q which is what i wanted to find.
what is wrong with this? sorry i forgot so many details.

>>16145511
i made a mistake in the final equation.
k = p/q = x + r/q and also in the conclusion i meant r/q.

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I have a non autonomous ODE [math]\dot{\mathbf{x}} = A(t)\mathbf{x}[/math]
which has the following solution
[eqn]
\mathbf{x} = \left(I + \int_{t_0}^{t}\mathrm{d}\xi_1A(\xi_1) + \int_{t_0}^{t}\mathrm{d}\xi_1\int_{t_0}^{\xi_1}\mathrm{d}\xi_2A(\xi_1)A(\xi_2) + \cdots\right)\mathbf{x}(0)\quad.
[/eqn]
This series converges as long as
[eqn]
\int_{t_0}^{t}\mathrm{d}\xi \rvert\rvert A(\xi)\rvert\rvert < \infty\quad.
[/eqn]
After a certain time we have [math]\rvert\rvert A(t)\rvert\rvert = 1[/math], which means the series will not converge at infinity (I guess). But if I plot the phase space for the equation I get pic related. The different trajectories account for different initial conditions. The matrix [math]A(t)[/math] here is
[eqn]
A(t) =
\begin{pmatrix}
0 & 1\\
-e^{-t^2} & 0
\end{pmatrix}\quad.
[/eqn]
So, how can I interpret the phase space knowing that the series diverges at infinity? This is an non autonomous simple harmonic oscillator

>>16145511
this is different from your first formulation because it's p mod q instead of k mod n. For instance his counterexample of k=2 and n=3 does not work because it does not have this relationship. It would be 2-2=(6 mod 3)/3 which is correct. So it would be more apt to say [math]k-\left \lfloor{k}\right\rfloor=(p\mod q)/q[/math] when k=p/q

>>16145555
or if p=kq, then [math]k-\left\lfloor{k}\right\rfloor=(kq\mod q)/q[/math] where kq mod q is only nonzero whenever k is not an integer

>>16145521
>After a certain time we have ∣∣A(t)∣∣=1, which means the series will not converge at infinity (I guess)
uh, no? Guy, have you tried evaluating the integral? It's somewhat common. The determinant is \exp^{-t^2}. Set t_0 and t to -\infty and \infty, respectively. That's just the \sqrt{\pi}. The determinant is always positive, so ur integral is always less than \sqrt{\pi}. Looks like it should always converge.

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Retard here, please bear with me.

I was in a physics class and we were told
I = V/R intensity = voltage/resistance
P = I * V power = intensity * voltage
E = P * T energy = power * time

Then I remembered: momentum is the force integral with respect to time, and kinda saw something similar here. Then I wrote
I = V/R
P = ∫ I * dv
E = ∫ P * dt
E = ∫ ∫ I * dv * dt
She told me this was wrong and idk if I have to rethink my intuition.

>>16145611
I'm using the spectral norm, that means I need to get the max eigenvalue of [math]A^H A[/math] and take its square root.
The two eigenvalues are [math]e^{-2t^2}[/math] and [math]1[/math]

>>16145750
The issue is [math]\int IdV=\int(V/R)dV=V^2/(2R)[/math] because V and I have a strict relationship (the relationship should be V^2/R anyways, so the integral would need a factor of 2). I would describe it as this: force and time and momentum are very fundamental quantities, but voltage and current are highly derivative quantities, based off of charge and electric field and time and other more basic quantities.

>>16145076
Information theory has essentially three different uses at this point (as far as I'm aware):
1) Classical information theory work for channel capacities and encoding. This is especially relevant in multi-user information theory where many schemes have no known closed form solutions for maximum channel capacity.
2) Information theoretic statistics and machine learning. Fisher information provides your bounds on optimal estimation for a particular likelihood formulation, and the eigenvalues of your fisher information matrix give you observability conditions for your estimate (i.e., if your FIM is singular, you aren't guaranteed to get a convergent estimate because your observations don't contain enough information about the parameter you want to estimate). KL divergence provides an optimal error exponent for classifiers and hypothesis tests.
3) Optimization and optimal control theory/RL. KL divergence provides a very common metric of "proximity" for descent directions in your cost function. A common way of ensuring stability for your optimization method is making sure that your expected cost distribution between any two steps are "close to each other" which is usually done in an information theoretic way (either by setting a bound for the minimum joint mutual information between policy/state updates or setting a bound for the maximum KL divergence on the same quantity).

There's also a lot of applications of information theory to optimal portfolio theory and multi-criteria decision making but I'm not really very familiar with those.

>>16136999
>>16136659
You are like a baby. Feel free to quit and film yourself at mcdonalds.

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I failed my Module and Category Theory class, pretty hard.
This makes the 3rd time I failed a class.
I don't want to give up, but man I feel like no one will take me for a masters, I tried so hard too.
I was stupid.
Please me tell me it's going to be alright, /mg/.

>>16145955
Just do more exercises

>>16145955
Do you know why you failed? If it's correctable then you'll be fine.

>>16145521
well, idk bout this method's convergence, but
>"[b]y the same reasoning as for the scalar case in Example 9.6, we see that eigenvalues of A with positive real parts yield exponentially growing solution components, eigenvalues with negative real parts yield exponentially decaying solution components, and eigenvalues with zero real parts yield oscillating solution components."
>"Thus, the solutions of this ODE are stable if Re(λ_i) ≤ 0 for every eigenvalue, and asymptotically stable if Re(λ_i) < 0 for every eigenvalue, but unstable if there is any eigenvalue such that Re(λ_i) > 0."
Eigenvalues of A are always imaginary for all time, t, so it looks like your solution is oscillating between a finite number of possible values.

>>16145772
For finite time t, the integral is still finite?

>>16145750
>>16145787
1/2 * V^2/R = 1/2 * VI
E = 1/2 \int VI dt = 1/2 \int V dq
Looks pretty right to me. What did she say was wrong with it?

>>16145955
What's a question you struggled with from an exam? I found that abstract nonsense tends to get easier for you to do just with time. If you zoom in too close on it, it's very easy to get lost in definitions. In the case of category theory, it's much easier to first try and realize exactly what you're trying to create with the language, and then realizing why the complex diagrams and definitions HAVE to be what they are.

>>16146217 (You)
So i was rethinking this, and actually >>16145750 is originally correct, and >>16145787 is misleading.

If you're talking about a wire, don't think of your first eq. as V = IR; think of it as V_diff = IR. >>16145787 replaces I with V/R, which is wrong - it should be the constant V_diff, not the variable V.
For your second eq, P = V_diff * I since I is constant.
Your third eq is fine, maybe say E_loss to be more specific and clear.
For your fourth eq then, V_diff is constant, I is constant, so E_loss = V_diff * I * t_diff, which is correct.

>>16143852
>he didn't cheat on his math exam
ngmi

>>16145521
Write x(t+dt) = x(t) + (dt)*x'(t) = x(t) + (dt)*A(t)*x(t)
At t=0, A (the "pull") is just 90 degree rotation clockwise relative to the direction x is pointing.
This is just circular motion with constant angular velocity.
As t goes to infinity, A will simply "pull" x along the the first component at a velocity equal to its second component.

Now if you decompose A into the 90 degree rotation part + remainder, you can understand the transition between the 2 situations.
The remainder term tells you x will be "spiraling" away from zero in quadrants 1 and 3 and "spiraling" toward zero in quadrants 2 and 4.

Since the equation is homogeneous, the path for a given x0 can just be re-scaled by k to get the path for k*x0.

>>16146369
I'll add that there should be a special angle for x0 that converges.

>>16145820
>1) Classical information theory work for channel capacities and encoding. This is especially relevant in multi-user information theory where many schemes have no known closed form solutions for maximum channel capacity.
This one sounds interesting. I think I'll look into this in the following year.

>>16146204
Where did you get this piece of text from?
I'm interested in reading more
>>16146208
It is
>>16146369
>>16146372
This is a very good explanation, thank you. Just did the decomposition, studied it a bit, then ran a mathematica code of the vector field associated with the ODE, and things make more sense now.

>>16146007
That's the thing, I did all the exercises, but the professor asked things directly from the theory...
>>16146104
Well, see above. I guess if I just study the theory harder it'll be OK. Professor kind of lied though, he said the exams would be like the exercises but they were straight up taken from the theory.
>>16146272
One I really struggled with was showing the morphism that is part of the definition of the cokernel is an epimorphism. It's exactly what you said I think, I just got lost with the diagram.

>>16145560
>>16145555
thanks. im happy i was able to see this during my other proof. it took me awfully long though

How do you isolate x in the equation (x +y)x(x+3)=0 It feels like something I should be able to solve but I have no clue.

>>16146840
You can't, you could isolate y though.

>>16146840
leave the x(x+3) alone, call it A. So we have A(x+y) = 0. Expand this, then isolate the y, but do not cancel out the A's! Notice you just get y = -x A/A. This is usually easily simplifyable, but this isn't true when A/A = 0/0. So if you were to graph y(x), you would get some holes in your function, even tho the limit exists.

From here, it's obv to isolate x, but just remember to consider the potential 0/0.

>>16146217
>>16146307
She didn't elaborate, she said: these are not integrals.
I'm not familiar with these quantities but she teaches math as well, that's why I was puzzled, at the very least
E = ∫ P * dt
should make sense to anyone I think?
Thanks for clearing it up.

>>16139042
Higher categories have (n+1)-morphisms going between n-morphisms, idk what a 1/2-morphism would be let alone an irrational one.

>>16139486
Look into computability theory (Turing machines etc.). Sipser is a good introduction.

>>16147314
That means she has issues with your P = \int I dV.
If we have finite charges q, then W = \int P dt = \int F dx = \int \int E*dq/dt dx dt, which means P = \int dq/dt *E dx. Idk why she would be against E dx = dV.

>>16139486
Not sure if this is what you are asking about, but if you just want to know if these actions are valid then you can treat them as steps in a proof.
For example, passing the variable a to the other side would be applying the function f(x) = x-a.
Because f is a function you get
L = R => f(L) = f(R)
and, because f(x) is injective you get
f(L) = f(R) => L = R
so
L = R <=> L - a = R - a.
Or maybe you are looking for abstract algebra, computability theory.

>>16135652
Honestly not that much of a meme post. Maybe just push most of it back a year and rearrange some topics, and you get pretty close to what a gifted student does here at my uni.

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>>16136180
Why is it only mathematics with a geometric / topological flavour that is promoted, also in the other well-known reading list on sci? Finite / discrete group theory and combinatorics are still active and important areas of research.

>>16135912
Do you also take part in Project Euler? It is one of the problems there.

>>16149162
Cuz it's made for physics/scientists, not mathematicians

I'm sick of googling and this is real stupid, can someone give me the latex (so i can look at it on my end) for the vector and matrix/tensor form for the following QFT operators:

d_{nu}(psi) d^{nu} (psi)

where psi is a scalar field
and

d_{nu} A_{mu}

where A is a 4-vector field (in both cases d is to be interpreted as partial)

>>16149162
because this is /mg/ and not r/math
trve mathematics is founded in geometry
you can discuss combinatorics and other allegedly valid fields of mathematics in r/math if you must

Combinatorics is the queen of modern mathematics.

>>16149162
ALL math is geometry

you better start finding some important results now, because once AI can do maths properly, then things will spiral out of reach.

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Is pic rel a good book? I'm looking for a short but broad overview of anal number theory.

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>>16149711
That book you posted is 50 years old and it's author died like a decade ago. In a fast moving area like mathematics you always will want to get recent books like pic related.

>>16149711
Yes, it's fine for a standard introduction.

>>16146840
x=0 or x=-3 or x=-y

If you want it in one formula, start with (1)^(1/3) then mobius transform the 3 cube roots of unity to be the roots of your equation.

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Can any complex analysis aficionados explain the intuitive meaning and significance of this theorem to me? It's used in the book twice, to prove that the complement of a simply connected domain in the extended plane is connected and in the course of the proof of Runge's theorem

>>16149667
Wrong. All math is logic.

What’s the quickest way to show if [math] a_1,a_2,\ldots,a_N [/math] are N points in [math] \mathbb{R}^n [/math] , then the function [math] f : \mathbb{R}^n\rightarrow \mathbb{R} [/math] given by [math] f(x) = \sum_{i=1}^{N} ||x-a_i|| [/math] attains its minimum at the “average” [math] x = \bar{x} := \frac{1}{N}\sum_{i=1}^{N} a_i [/math] ?

I tried differentiating and found [math] \nabla f (x) = \sum_{i=1}^{N} \frac{x - a_i}{||x-a_i||} [/math] , but I’m not sure how to show this is 0 only if, or even if, [math] x = \bar{x} [/math] .

>>16150246
It's only needed for Runge's theorem. Intuitively, you just want Gamma to contain K and then Cauchy's integral formula takes care of the rest. It's easy once you show K has an epsilon-neighborhood contained in U.
>>16151076
If this is Euclidean distance, it's wrong unless you square the lengths before summing.

New wildberger: Pure maths has painted itself into a corner
https://www.youtube.com/watch?v=YhN4X56E7iM

what field of pure math is good for a real schizophrenic (me)

>>16151076
>If this is Euclidean distance, it's wrong unless you square the lengths before summing.
Thanks for pointing this out, I see you're right in many examples. Also yes, ||•|| is Euclidean distance.
However, if the [math] a_i [/math] are e.g. the 3rd or 4th roots of unity in the plane, then it seems the minimum of [math] f(x) = \sum_{ i=1}^{N} ||x-a_i|| [/math] does occur at [math] \bar{x} := \frac{1}{N} \sum_{i = 1}^{N} a_i [/math] .

Is there a way to figure out whether the minimum of [math] f(x) = \sum_{ i=1}^{N} ||x-a_i|| [/math] will be at [math] \bar{x} := \frac{1}{N} \sum_{i = 1}^{N} a_i [/math] , or at one of the [math] a_i [/math] ’s (these being the only two possible cases) , just from knowing the [math] a_i [/math] ’s ?

>>16150263
kek

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So I've started reading picrel. And first exercise has stunted me.

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>>16151855
>Prove that every Boolean Algebra is isomorphic to power set [math]P(A)[/math] with the three set operations [math]\cap , \cup , \sim [/math] as the AND , OR and NOT operations, respectively.
Bros , what do I do here?

>>16151869
define a suitable isomorphism of boolean algebras (hint: think)

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Hi anons, I pretty much dropped math after high school.
Right now I have very vague memories of HS/early college math (introductory calculus, linear algebra and real analysis, some notion of what differentials are, basic statistics, shit like that) but it's all quite dusty and chaotic in my head

I'd like to update my knowledge and advance to higher levels of mathematics through self-study, especially since it'd help me in my career (I'm a software dev). But I'm also just interested in it for its own sake and I've always wanted to git gud so that I could understand things like topology, high level geometry and whatnot.
What are some good resources to get started that aren't too dry or a slog to get through? Self-study is tough so I don't really care about the "best" resource as long as I'm getting something engaging enough that I'll actually be motivated to stick to it.

>>16152037
By the way I speak french fluently and I know france is renowned for math, so books in french are fine too

>>16152040
Godement algebra

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>Legendre symbol

>>16151869
It's false. Counterexample: the free boolean algebra on a countably infinite set of generators. It's countably infinite but all powersets are either finite or uncountably infinite.

>>16135800
This is shit, here's the real /mg/ list:

"Linear Algebra and Its Applications" by Peter Lax
"Principles of Real Analysis" Walter Rudin, "Real Analysis with an Introduction to Proof" by Steven R Lay to help you
"Real Analysis" by Royden to learn Measure Theory and Integration
"Real and Complex Analysis" by Rudin for a terse Measure theory reference
"A First Course in Sobolev Spaces" by Giovanni Leoni for Sobolev Spaces before Functional Analysis
"Partial Differential Equations" by Lawrence C Evans to learn PDE and Sobolev Spaces
"Real Analysis: Modern Techniques and Their Applications" by Folland for Point-Set Topology and Intro to Functional Analysis and all around reference for common theorems
"Stochastic Calculus" by Karatzas and Shreve
"Functional Analysis" by Nagy
"Functional Analysis" by Peter Lax
"Riemannian Geometry" by do Carmo
"Topics in Optimal Transportation" by Villani
"Gradient Flows in Metric Spaces and in the Space of Probability Measures" by Ambrosio, Gigli, and Savaré

Don't you even DARE call yourself an analysist until you've memorized ALL of these books.

>>16152177
do carmo doesn't belong there

>>16152512
Do Carmo is the only reference I’ve seen that treats Riemannian Geometry as NOT some abstract algebra bullshit and also NOT as some general relativity bullshit.

>>16152174
He difinitely means finite Boolean Algebra

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>>16152177
https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

>>16152177
Don't worry, I'll never call myself an analyst.

>>16135640
You need to read a textbook, completely.

>>16135851
Single variable calculus and ordinary differential equations

>>16152062
Thanks. How broad is it, will it act as a good refresher on everything I learned or does it omit certain subjects (not counting probability and statistics, I know these require a separate book)

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>>16152838
Here are the contents (book is also available in French). Basically it does basic set theory, algebra (groups, rings, fields, linear algebra). For basic analysis read baby Rudin and Ahlfors' complex analysis.

>>16151611
bump

>>16152717
That's an incredible list with really wonderful commentary. Have standards really fallen or is UC Berkeley just a really brutal undergrad program? I have never once heard of papa Rudin being taught at an undergraduate level (every program I've ever seen has it as a first year grad course) and they place it in intermediate for undergrad.

Either way, thanks for posting the list. It's really interesting.

>>16152717
>https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
This list is abstract algebraist propaganda. Use >>16152177 to become an analysist.

>>16152788
"Analyst" is something different. An "Analysist" is someone expert in Analysis.

>>16153108
>>16153109
What the fuck are you babbling about?

>>16153128
Are you ESL? Which words are you confused by?

>>16152037
Éléments de mathématique is in french

wildberger has ascended
https://www.youtube.com/watch?v=9IlLbS8700U

Let n≥2 be an integer, let w be a primitive complex nth root of unity. Let [math]U = \mathbb{C} \setminus \{1,w,w^2,\ldots,w^{n-1}\} [/math] be the set of complex numbers which are not an nth root of unity.
Define [math]f : U \rightarrow \mathbb{C} [/math] by [math] f(z) := \sum_{k=0}^{n-1} (z - w^k) / |z - w^k| [/math].
- Since the sum of the nth roots of unity is 0, it follows that f(0)=0 .
- We note f is smooth, though not holomorphic, on U .

My question is, does f(z) have any zeros other than at z=0 ? How do we show this?

>>16153219
I thought it was going to be the standard Wildberger argument "infinite number of twin primes don't be real cus infinity don't be real" but it was something even more retarded.

Is it a better idea to take abstract algebra or linear optimization?

The abstract algebra class focuses on group theory.

I'm double majoring in math and humanities.

>>16153258
f(z) = 0 is the same as |f(z)| = 0. I would rotate the complex plane so that z is aligned to the positive x axis, call it z'. If z' > 1, then the real components of each term in the sum is greater than 0, so the magnitude of the sum is > 0. If z' = 1, then only one term can possibly have a real component of 0 or less (cant be less), while the rest of the terms have reals > 0. If z' is > 0 but < 1, it's always uneven.

Idk, im just drawing it in my head. There def no other zero other than at z=0, tho

>>16153522
Is algebra not a requirement for your math degree? I would recommend both if you have any interest in applied math, but algebra is far more central so I have to recommend it if you need to pick one.

>>16153660
An "introductory algebra" class that covered polynomial rings and similar things is mandatory.

The "proper" algebra class in group theory isn't mandatory for people who double major.

But I do enjoy applied math too and would love to see real world applications of linear algebra in linear optimzation.

So maybe I should drop my plans to take measure theory so I can take both these classes.

>Retaking trigonometry... Probably going to fail cuz I just don't understand identity's at all.

>Knowing my luck I'll fail precalc as well.

>It took me 5 years to get an associate degree in cyber....I'm trying for my bachelor's but I fucking hate how much math I need to take. I hate the gen ed classes and how basically nothing transfered.

>I am so pissed at my family who act like its all OK and I just need to try harder and study more.

>I'm slowly burning out again, and I'm starting to feel like the true retard that I am.

>TLDR: retard that might become homeless

>>16153698
Oh, I see. OK, my recommendation in that case is linear optimization and measure theory. Extra semesters of algebra are not required for undergrad in my opinion, especially if you tend more towards applied.

>>16153764
Measure theory is one of those subjects that can be taught to such absurdly varying levels of difficulty (depending on the professor). If they just keep things to measures on the real line like Royden it's not so bad but things can get pretty messy when it gets more general and when you start dealing with normed functional spaces.

Why is it so difficult to induce collaborations between mathematicians and physicists?
I (math) had a discussion about that topic with a colleague (phys) some time ago, and we came to the conclusion that it fails because both sides stick to their respective dogma and refuse to learn notations/conventions of the other side, thus making it impossible for them to follow their talks and seminars. In particular the students, both undergrad and grad, never really learn to deal with for example coordinate-free formulations (phys) or how to actually calculate stuff in neatly chosen coordinates (math).

>>16154136
Only the first part of Royden keeps measures on the real line. The third part of Royden is extrapolating everything from the first part to arbitrary measure spaces.

>>16154440
For what it's worth almost every field that takes things from the work of mathematicians just renames everything and invents new notation because if the math isn't used in their field already then they can claim they invented it.

Also, physicists disgust me.

>>16153698
I agree with >>16153764
Measure theory is invaluable and should be studied as soon as possible after you finish real analysis (or honors analysis or whatever your school calls it)

I finally realized why Spivak keeps getting shilled on 4chan in lists like >>16135652
Turns out Spivak tries to shoehorn “category theory” into every topic despite no real mathematicians ever using it for any application whatsoever. Category theory, like the Rust programming language, is only popular with transsexuals, and similar to Rust there are no jobs that actually use it. Recommending Spivak is part of yet again another long con on 4chan to get you to send femboy pics of yourself to sadists on discord. Don’t fall for it! Follow a REAL Mathematics reading list like >>16152177 instead!

>>16154689
Where is Spivak in >>16135652?

>>16154757
Sorry I just assumed he was there because he keeps getting shilled on /sci/

>>16154773
Verbitsky read Spivak in kindergarten while he was taking a shit and then used it to wipe his ass

>>16154773
Where is the category theory in Spivak? Maybe in his differential geometry books?

>>16154790
He/she is probably just a brainlet confusing Michael Spivak and David Spivak

>>16154136
The measure theory class at my uni is notorious for being the hardest class, so I think it focuses on the wacky stuff.
>>16153764
The weird thing is I'm in my MA.
You can get an MA in my country that's essentially just 2 BAs if you double major.
I'll probably self-study some graduate-level math though.

>>16154581
Royden doesn't even develop the Lebesgue measure from the outer measure until the end of the book. That was like week 2 of the course when I took it. Our course didn't really follow any particular textbook (it was all from the professors written lecture notes and exercises) but probably the closest to our course structure would be Bass's Real Analysis for Graduate students or Cohn's Measure Theory (which is pretty great, if a little handholdy, and doesn't get recommended enough in my opinion).

>>16154835
Well, the hard and wacky stuff ends up being the most important for a lot of real uses of measure theory and theory of integration (functional analysis, probability theory with random mappings to more general spaces, Fourier analysis and PDE's, etc.).

If you have the opportunity and the sweat you should give it a shot. You'll learn a ton. It'll probably be hellish but you'll learn a ton.

>>16152062
>>16152857
>For basic analysis read baby Rudin and Ahlfors' complex analysis
kek don't do this
read Understanding Analysis by Abbott first

>>16154854
> Royden doesn't even develop the Lebesgue measure from the outer measure until the end of the book.
Wrong, Royden’s book has three parts and it sounds like you’ve only seen the first part.

>>16154790
Also in his Algebra books and he literally wrote a paper called “Backpropagations as Functors” in order to make his shitty category theory relevant to more hyped up topics

>>16154915
two completely different spivaks lol

>>16143369
>>16143380
Why don't mathematicians just come up with a word for primes that are at least 3, like 'trime' or something?

Question about monoid algebras:

Let (M,•) be a monoid, let F be a field. Let F[M] denote the "monoid algebra" generated by M with coefficients F; i.e., F[M] is the free (noncommutative) associative unital F-algebra generated by the elements of M, quotiented by the relations ab=c whenever a•b=c in M. Let [math] \iota : M \hookrightarrow F[M] [/math] be the canonical inclusion.

Then is [math] \mathrm{image}(\iota) [/math] a linearly independent set over F ?

>>16154910
No, I have a copy right in front of me. Royden doesn't develop the Lebesgue measure from outer measure until chapter 12 out of 15. If that's not near the end of the book, what is?

Compare that to Cohn's measure theory, Bass's Real Analysis for Graduate students, Axler, Folland etc. where it's literally the way they construct the notion of measure to begin with. This is such a weird thing to fight about because it's so clear that Royden's book is far less focused on general measure theory and leaves it to the end.

>>16155046
What have you tried?

>>16155046
>the free (noncommutative) associative unital F-algebra generated by the elements of M
Slight correction: technically we need the free associative noncommutative nonunital algebra. However, after we quotient by the ideal written in that post, F[M] is indeed unital with identity the identity element of M .

>>16155054
My guess is no; aiming for a contradiction, suppose some nontrivial F-linear combination of elements of [math] \mathrm{image}(\iota) [/math] gives 0 in F[M] ; then some nontrivial F-linear combination of elements of M in the free noncomm assoc algebra [math] F \langle M \rangle [/math] lives in A, the ideal generated by elements of the form xy-z for x,y,z in M with x•y = z in M .
Can we show any nonzero element of A has to have a nonzero term of "degree 2", i.e. A cannot just be a linear combination of elements of M ?

>>16155063
>My guess is no
Meant to say yes, i.e. [math] \mathrm{image}(\iota) [/math] is linearly independent over F

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>>16135585
I havent done math since high school and I want to be >le smart.
If I finish picrel will I be ready for a calc textbook? I made it to AP Calc AB but I don't know what I remember from that course or earlier years. This was 3 years ago.

>>16155094
calculus is close to bare minimum in math, the equal of "can you read chapter books?" Any calc book will be fine for you at an intro level. Get like 2 or 3 calc books, and compare their chapters, and skip the overlaps. You'll be fine

>>16155094
Yeah the. I’ve thing is that call is popular enough that there’s a million books to choose from. There’s even some manga books that teach calculus

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https://www.youtube.com/watch?v=fWRm9dISpNk&list=PLSx1kJDjrLRSS4Aio9WfQlNnH6X6pC_8t&index=20, great video from arthur parzygnat on youtube, explaining affine supspaces and transformations

>>16155046
>>16155063
Isn't this just the definition? A group/monoid algebra literally is just the set of formal linear combinations of group/monoid elements

>>16155164
So why are they linearly independent from the definition alone?

I'm a 25 year old codemonkey who got his BA in math from a top 5 uni. I really want to spend more time exclusively with math and have better degrees. Does it make any sense to go back to uni? How easy is it to get a scholarship to do a masters + phd for someone in my situation?

>>16155220
If your grades are good enough and you have profs from undergrad that can speak to your potential research performance apply directly to a Ph.D otherwise you're probably going to have to either pay for a master's or write a paper on your own and publish it. Ph.D programs care more about your ability to do research than anything

Why would someone imply 'contradiction' isn't used in math, but rather contraposition?

The context was a (real analysis) assignment. It needed to prove that there is no function f: [0,1] -> (0,1) which is both continuous and surjective.
I started with "Proof via contradiction: [etc. the rest is unimportant]"
But the corrector wrote literally this remark next to it:
>Contraposition :) Contradiction you'd need in Logic
(yes, it was a woman)
If it makes a difference, it was in Kraut where we actually use two different terms (Widerspruch & Kontradiktion), but they mean the same thing. The word Kontradiktion is preferred by Philosophical Logic, but I still don't get why all of this was pertinent to this person.

>>16155251
you derive the contradiction via the contrapositive
I agree it is a retarded nitpick

>>16155251
I think this is just a pet peeve people have. It annoys me when I grade papers
Many people write fake proofs by "contradiction" where they don't actually need the contradicting assumption anywhere. i.e. they might write something like
>assume there exists some B which breaks these assumptions
>now I will prove that any continuous function on [0,1] cannot be surjective
>therefore my assumption was false, and B cannot exist
this is a technically valid proof, but it's shit proof writing, because the first line is useless but looks important

Not all examples are this bad but a LOT of contradiction proofs can have the contradiction pretty easily removed

>>16155249
I got a first. IDK if I could get a good recommendations letter. I was pretty lazy and didn't do my homework, instead preferring to play video games. I also wouldn't like to be bothering them all this time later. None of them probably even remember me.

>calculator screen broke
>there isn't a SINGLE fucking online calculator that allows for the usage of °, ', ''

What the FUCK am i supposed to do? I still need to do this shit. Why is a literal mechanic calculator necessary in this day and age?

>>16155308
retvrn

File: square spiral.png (64 KB, 897x897)

64 KB PNG

Powers of two as a "square spiral". This autistic diagram is meant to show that powers of two-and their respective proper factors-may be represented pictorially as a "square spiral", with each doubling creating either a whole square, or a 2x1 rectangle.

Since the only proper factors of a power of two are the lesser powers of two, these factors themselves can be represented as "blocks", as in the picture, which almost-but-not-quite-add up to the whole number. There is always a unit left over. This is why we say that the powers of two are "almost perfect" numbers.

To generate the proper factors of a power of two, we have a multiset with one element-which is two-such that combinations of prime factors can only be generated by picking 2, eh, let's call it "k" times, depending on the case. just counting from 0 up to n-1. To get perfect numbers, of course once we introduce a convenient prime, then it becomes the only other item which may be chosen, and the whole business works out (algebraically).

This is a very limited (2D) and autistic way of thinking about perfect numbers, but could a "pictorial" representation in several dimensions be used to rule out the existence of odd perfect numbers (which is an open problem), or (I wouldn't expect this) confirm the existence of one? When I think of 28 for example, I currently think of a 2x2x7 3D "block", using that inner 2x2 sector, stacked seven high. But "coloring" these blocks in the number's various factors then becomes problematic in a 3D model.

The probability of a piece of mathematical writing being schizo gibberish increases by roughly 20% for every random "word" which is put "inside" quotes for no particular "reason"

>>16149162
>Finite / discrete group theory and combinatorics are still active and important areas of research.
Along with algebra these will take over in the age of computers.

Can someone tell me an application of category theory to Calculus of Variations? All the applications I can find seem to be in Algebra and I’m trying to understand if category theory is worth learning

>>16155251
Im not a math guy (specially not real analysis). I can def believe it that no function exists, but how does one prove it? Like, what's the gist of the proof?

Also, what's a good book for stuff like this?

>>16135652
Kek. Meanwhile in reality the first course in algebraic topology at MIT is a graduate course.

>>16153525
If z'=1, actually it would be 0/0 as the one term

>>16155251
She is probably complaining about an unnecessary use of contradiction. You can prove a statement P => Q directly, by contradiction (show P and ~Q => contradiction), or by contrapositive (show ~Q => ~P). Really contrapositive is just a special case of contradiction, but it is considered better form not to use contradiction proper unless you really need it (ie, you actually use consequences of both P and ~Q), because it makes arguments less clear to read. Many undergrads perform a lot of unnecessary contradictions, and professors will often be very aggressive in trying to break them out of the habit.

>>16155251
>If it makes a difference, it was in Kraut where we actually use two different terms (Widerspruch & Kontradiktion), but they mean the same thing.
Literally nobody says "Kontradiktion" in German. It's an extremely rare and antiquated word. You outed yourself as either a fedora wearing autist or a migrant. Proper math terminology is "Beweis durch Widerspruch".

>tfw interested in applied math but everyone in applied classes is retarded and lazy

>>16156080
If you're like me and you're interested in the theoretical aspects of applied math, then you will likely want to look into the subfield of "Applied Analysis". It's what my PhD specialization was.

>rationals are closed under addition
>do addition but just keep doing it forever
>suddenly it's not closed anymore

>>16156296
>finite sets closed under union
>do union but just keep doing it forever
>suddenly it's not closed anymore
maths is a scam

>>16156233
I took a look at the "applied analysis" book by Bruno. Is that similar to your field? If so, it seems pretty similar to what a lot of us in the "systems" side of EE end up doing for systems modeling, stability analysis and applied probability theory.

>>16156344
Yeah I took a look and I'd say that book is a pretty accurate intro. My dissertation was taking a lot of concepts in the last chapter of his book and extending their concepts in novel state spaces in order to make useful tools in stability analysis for people working with nonlocal aggregation models. You can see my dissertation here to get an idea https://kilthub.cmu.edu/articles/thesis/Properties_of_Minimizers_of_Nonlocal_Interaction_Energy/6721172

>>16156377
I'll take a look! Stability analysis is a super huge part of my field (optimal state estimation and sequential signal detection). I've always found myself drawn towards the more analysis side of things (e.g., measure theoretic approaches to probability theory/statistics and modern Fourier analysis) but in many cases it doesn't end up buying you a lot in terms of practical research benefits.

>>16156383
Yeah in my experience measure theory shines in domains where you have state spaces where states can be discrete, continuous, or both at the same time. However in most practical cases you're only going to be in some discrete domain *or* continuous domain and not have to consider both possibilities at the same time.

>>16156383
Also sorry I killed your backlink, I wanted to rephrase how I characterized my dissertation

>>16156390
>>16156389
You're good, I downloaded your dissertation. CMU is a great school so I'm sure your dissertation is no exception.

Surprisingly enough, one of the biggest challenges within signal detection are these mixtures. Look up probabilistic data association or multi-hypothesis tracking. It's all about trying to associate discretized realizations of continuous Markov chains to a discrete number of possible hypotheses. As a result you get these weird hypothesis distributions where your likelihood functions have compact continuous regions separated by discrete probability masses to account for the discrete nature of the track hypotheses. These kinds of continuous-discrete mixtures are exactly where the difficulty lies in data association, and knowing some measure has been very handy for that. The problems come that really none of the existing literature in the tracking world uses measure theory as its language do you have to be willing to play translator and do the work yourself of converting everything to that worldview to make use of it.

>>16156397
That sounds really cool and a great use-case for measure theoretic statistics!

>>16136988
this is only because of heavy protectionism/ mafia style information concealing in math in order to protect jobs. After a while you realize that people obfuscate A LOT.

What's the best way to build automaticity for math?
I'm 20 fucking years old and I still don't know my times tables, and I still feel the need to count on fingers some times.
Some days I really think I'm retarded.

>>16156525
Time tables should be learned by memorization. Make some flashcards and keep treating yourself with them

>>16156525
This >>16156553 is true. If you can learn the alphabet, phone numbers, words, you can memorize times tables. I was forced to memorize them in 1st grade.

Building intuition in math requires practice, which means you need to do homework (too late now?) or do practice problems from books or online resources.

>>16136659
First step let's put csc(4*x) as 1/(sin(4*x))
Second stet let's put -cot (4*x) as -cos(4*x)/sin(4*x)
Third step let's add them together and we'll have:

(1-cos(4*x) ) /(sin(4*x)) +cos(4*x)

Fourth step let's add the cos(4x) to what we have, we'll do this by multiplying and dividing cos (4x) by sin(4x), well have as a result: cos (4x)*sin(4x)/sin(4x).

We now add this:
(1-cos(4x) +cos(4x)*sin(4x))/sin(4x)

let's express cos(4x)*sin(4x) as 0.5*sin(8x) we can do this because cos(x) * sin (x) = 1/2 * sin(2x)

Now we know that cos(0+) =1 and sin (0+) =0 so our expression is:

(1-cos(0+)+0.5*sin(0+))/(sin(0+))
(1-1+0)/(0)
this is a 0/0 situation so we can use Lhopitals rule

once we differentiate the upper and lower part we have:

(4sin(4x)+4cos(4x))/4cos(4x)
when substituting the 0+ we have
(0+4)/(4) which is 1

What are the units of dcos(θ)?

cos(θ) is dimensionless and unitless, but is dcos(θ)? We have:

dΩ = dcos(θ)*dϕ
dcos(θ) = sin(θ)*dθ

dθ has units of [rad], so does that mean dcos(θ) has units of [rad]? This would make sense to give units of [sr] for dΩ.

I have people telling me dcos(θ) is unitless, but everything seems inconsistent if it is. But if it's not unitless, then why is cos(θ) unitless but dcos(θ), an infinitesimally small step in cos(θ), isn't?

>>16157261
So, this might sound kind of strange but the output of trig functions are in general unitless quantities. dcos(theta) is unitless, but dcos(theta)/dtheta has the units of "per radian."

It's the same way that Hz for frequencies operates. When you have a function which gives you F(t) = # of periods elapsed at time t, and you differentiate with respect to time you get frequency, # per second.

It's the same thing with trig functions when you have radial frequency (per rad instead of per second).

>>16157261
>>16157304
I forgot to add, radians are themselves unitless in a certain sense.

Think about the arclength/sector formula.

L (m) = r (m) theta (rad).

If radians were themselves possessing a unit in terms of dimensional analysis, this wouldn't work. Radians are themselves unitless quantities. This is also why conversion from rad/s to Hz doesn't violate any dimensional analysis rules. Just like "per second" has nothing in the numerator, "rad/second" is dimensionless in the numerator.

>>16157312
So that means

dcos(θ) = sin(θ)*dθ
[unitless] = [unitless]*[rad]

is still correct and valid? And the same for this?

dcos(θ)/dθ = sin(θ)
[rad^-1] = [unitless]

That's a good example with arclength, I didn't consider that. I understand that radians are dimensionless, but I guess I still expected [rad] to propagate consistently through equations, but I guess it doesn't have to. Thanks.

>>16157336
Yes, as counter intuitive as it might seem, "per rad" is actually not a proper "unit" in the way meters etc. are.

There's actually a reason this has to be the case that shows up in analysis (angle space is non-metric) but for now I would just try to come to terms with [rad] being effectively not a proper unit in the way something like meters, Newtons, volts, seconds etc. are.

>>16157340
>angle space is non-metric
Interesting. I never took analysis because my degrees were in science, not math. I've heard it's insanely difficult but it sounds interesting and I'd love to know more about why math is the way it is.

>>16157347
Well, I personal find analysis very interesting and compelling and my degrees are in EE. If you want to try to learn, it's never too late. Just give yourself patience. Pushing yourself to learn challenging material at too fast of a pace can just cause you to give up too early or cram a bunch of concepts that you don't properly understand together.

Basically the problem with angle space is that the set of all points which are zero "distance" from any particular point is non-empty. If you pick a point in the angle space, you will arrive at the exact same point by adding 2 pi radius any integer number of times. As a result the "ball" of zero radius surrounding any particular point in the angle/phase space has a countably infinite number of members. This even though it is of Lebesgue measure zero (meaning it is a countable set of isolated points with no "length" between them) it is non-empty despite having radius 0. Thus, it can't have a proper distance metric applied.

>>16149724

>fast moving

the last interesting math breakthrough happened 10 years ago

>>16149724

math is the slowest of all fields. You can expect something interesting to happen like once every 10 years