>The Bonnet problem asks when just a bit of information is enough to uniquely identify a whole surface.
>For the first time, mathematicians have found an example of a compact doughnutlike surface (as seen above) that shares its local geometric information with another surface, despite having a completely different global structure.
https://www.quantamagazine.org/two-twisty-shapes-resolve-a-centuries-old-topology-puzzle-20260120/

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>>16899933
>quanta magazine
gtfoh
t. actual math researcher

>>16899934
That article about quantum mechanics made "free" from complex numbers was really bad

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>Černý conjecture
Can /sci/ solve this open problem?
https://www.wikipedia.org/wiki/Synchronizing_word#Lenght

>>16899914
Ah, Hurewicz...his depths of knowledge indeed matched with his adventurous spirit

>>16900313
there's probably some kind of argument about how awareness of one's position and trajectory in space is a homotopic space and discrete stair steps are a homologic space, but that would be silly

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What's your favorite counter example

>>16899990
It'd be cool if /mg/ had some sort of running open problem that we could agree to casually collaborate on. Something with minimal technical overhead (i.e. probably combinatorics) and which isn't particularly well-known (unlike this one, apparently) so there's a modest chance of success. Like a random one of Erdős's conjectures or something.

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>>16900392

>>16900392
That’s not the definition of a square. A square is a parallelogram that’s equilateral and equilangular. This obviously doesn’t qualify.

>>16900392
You are missing this definition, a necessary condition for any polygon:
https://en.wikipedia.org/wiki/Polygonal_chain

>>16900392
Can you proof you can construct such a square?

>>16900512
>>16900618
Yeah that's the entire point.
It looks correct at first glance then boom a pathological shape/function that ruins your theory.
Another example is the weierstrass function, the dirichlet function, the vitali set, etc

>>16900721
The difference here is that there is no theory where that’s the definition of a square.

>>16900489
Art gallery?

>>16900721
>It looks correct at first glance then boom a pathological shape/function that ruins your theory.
>Another example is the weierstrass function
>the dirichlet function
>the vitali set
Those where great advances in their respective theories, if anything they helped rule out old theories or ways of thinking. Euclid's elements may have some faulty concepts, (i dont know which ones beyond missing axioms) but nowadays there is a level of precision almost similar to that of programming languages, which fail to compile due to even the slightest syntactic inaccuracy

>>16900745
>>16900798
Bro it's not that complicated. Forget about the square pic, it's just an example
I just wanted to learn about other examples of pathological mathematical objects which negate very intuitive and obvious implications/statements.
It's like the trolling version of mathematics. You have good theory of something? Here is the most contrived absurd object to prove you wrong.

>>16900809
have a look at arXiv:2502.06137

>>16900809
I accept your concession.

>>16900809
>I just wanted to learn about other examples of pathological mathematical objects which negate very intuitive and obvious implications/statements.
>It's like the trolling version of mathematics. You have good theory of something? Here is the most contrived absurd object to prove you wrong.
Counterexamples in Calculus - Sergiy Klymchuk
CounterExamples: From Elementary Calculus to the Beginnings of Analysis - Bourchtein, Andrei; Bourchtein, Ludmila
Counterexamples in Analysis - Bernard R. Gelbaum, John M. H. Olmsted
Counterexamples in Topology - Lynn Arthur Steen, J. Arthur Seebach Jr
Counterexamples in Probability and Real Analysis - Gary L. Wise, Eric B. Hall
Counterexamples in Measure and Integration - René Schilling, Franziska Kühn

>>16900835
That 100k wont last you. Fully over. Don't talk to me or about me or harass me or my family ever again.

>>16900952
**[spoiler]Boo![/spoiler]**

How can I prove that there exists a subspace of a infinite dimensional vector space which is not closed?
I tried taking two convergent sequences and proving that their sum does not converge within the subspace, but that ain't working

>>16901094
Take countable many linear independant vectors.
Scale the length of the k-th one to 1/k^2.
Now consider the subspace spanned by those vectors.
It doesn't contain the infinite sum of them.

>>16901109
>Take countable many linear independant vectors.

AOC?

>>16900837
>entire textbooks on counterexamples
what's the point tho? you would just become more insecure and unsure of everything

>>16901109
>It doesn't contain the infinite sum of them.
How so?

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In picrelated you have a quarter piece of a unit circle. Inside there is a square. One vertex of the square is on one side of the quarter piece, the other vertex on the other side and a third one on the arc.

What is the minimum length of the distance AB?

>>16899914
I think Wildberger is getting into my head. Lately I can't shake the feeling that nothing I do is "real" enough, whatever that means. It's making me question my decision to devote my life to math. Please, bros, tell me math is discovered not invented.

Combinatorics is not a unique subfield. It's enumeration theory on discrete varieties. CMV.

>>16901615
Essentially:
>Combinatorics is not a subfield but a collection of methods from algebraic geometry

>>16901615
All of mathematics is just a subfield of epistemology.

>>16901611
Wildberger shouldn't be the only "philosopher" whose views you consider, there are other finitists like Doron Zeilberger and of course dozens of platonists and dozens of nominalists/formalists and other positions, all of them with interesting written views. Even better, read some of the philosophical or autobiographical texts by Grothendieck (or at least a biographical work), some people find strenght on his quasi divine prose. Also, read the masters: Riemann, Cantor, Dedekind von Neumann, Gödel etc. may convince you in their own words about what they did and its importance

>>16901660
>strenght

>>16899914
Is it over I'd you're a ~30yo who dropped out of high school who wants to pursue mathematics? I'm doing khan academy rn and I'm only at arithmetic with fractions and can't help but feel perpetually retarded like I'll never catch up.

>>16901628
it is not

>>16901664
Thanks, but now that i think of it, the preposition "on" right after that typo is a bigger mistake. It should've been "in", it's a metaphorical use of "to find"

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>>16899914
H=i,j∑Pij(ci†cj+cj†ci)−σmap

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>>16901377
0 <= θ <= π/2

P = s*(Sin[θ], 0)
Q = s*(Cos[θ] + Sin[θ], Sin[θ])
R = s*(Cos[θ], Cos[θ] + Sin[θ])
S = s*(0, Cos[θ])

Cos[θ]^2 + (Cos[θ] + Sin[θ])^2 = (r/s)^2

(d/s)^2
= (Cos[θ] + Sin[θ] – r/s)^2 + Sin[θ]^2
= (Cos[θ] + Sin[θ] – Sqrt[Cos[θ]^2 + (Cos[θ] + Sin[θ])^2])^2 + Sin[θ]^2

θ ≈ 0.1154000604
r/s ≈ 1.4884550172
d/s ≈ 0.3970256621

θ = 0
r/s = √2
d/s = √2 – 1

>>16901377
Consider the three solutions to the cubic x^3 - 5x^2 - 5x -1 = 0. The two smallest are associated with local maximums, and the largest x = 5.87936
> = 1/3 (5 + 40/(251 + 3 i sqrt(111))^(1/3) + (251 + 3 i sqrt(111))^(1/3)) (wolfram alpha)
is associated with the minimum. Call 1+x = a

Let b = sqrt(1 / [x^2 + a^2] ). Then the minimum length is equal to b^2(1 + a^2) - 2ba + 1 = 0.0697166932857. The maximum formulas are a lil diff with negatives in diff areas

It's just algebra and calculus, no special tricks I saw

>>16903012

system:
r = 1
r/s ≈ 1.4884550172
d/s ≈ 0.3970256621

solution:
s ≈ 0.671838
d ≈ 0.266737

your answer is close to:
d^2 ≈ 0.071149

>>16903309
>d^2 ≈ 0.071149
correction:
d^2 ≈ 0.071148

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>>16901377

360 frames
15 seconds
24 frames per second

θ = k/180*Pi
s = 7
r = s*Sqrt[Cos[θ]^2 + (Cos[θ] + Sin[θ])^2]

line segment 1:
vertices: (–2*s, 0) and (2*s, 0)
edgecolor: (0, 1, 0, 1)

line segment 2:
vertices: (0, –2*s) and (0, 2*s)
edgecolor: (1, 0, 1, 1)

ellipse:
equation: s^2 – y^2 = (x – y)^2
edgecolor: (0, 1, 1, 1)

circle:
equation: x^2 + y^2 = r^2
edgecolor: (1, 0, 0, 1)

square:
vertices:
s*(Sin[θ], 0)
s*(Cos[θ] + Sin[θ], Sin[θ])
s*(Cos[θ], Cos[θ] + Sin[θ])
s*(0, Cos[θ])
edgecolor: (0, 0, 1, 1)

line segment 3:
vertices: s*(Cos[θ] + Sin[θ], Sin[θ]) and (r, 0)
edgecolor: (0, 0, 0, 1)

display:
|x| <= 1.01*(Sqrt[5] + 1)/2*s
ditto for y

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https://chan.alphakek.ai/sci/res/8.html

>>16903309
Yes yes, I didn't sqrt the guy, about .26 is the answer, accuracy depends on how far you want to go

>>16901377
>>16903825
according to wolfram alpha, the exact answer given my init instructions is the root of 5x^6 - 38x^4 + 60x^2 - 4 near x = 0.264039. Or the root of 5x^3 - 38x^4 + 60x - 4 then square your answer to a number near .26

>>16899982
Which one?

>>16903899
I should say that most of the problems with the Quanta article come from the arxiv preprint being retarded
https://www.quantamagazine.org/physicists-take-the-imaginary-numbers-out-of-quantum-mechanics-20251107/

https://arxiv.org/abs/2504.02808

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https://people.math.ethz.ch/~salamon/PREPRINTS/FERMAT.pdf

Can somebody explain to me the reasoning behind the u and v equations?
I can see they make perfect algebraic sense but why halve the addition and substraction of x and y?
How does one reach such equations?

Also, if there is a more detailed transcription of Eulers proof for n3 I'd appreciate it.

>>16904220
>they satisfy u+v=x, u-v=x
That's the key. The author doesn't say it, but
[math]u+v=x[/math]
[math]u-v=y[/math]
is a system of equations you can solve by taking x and y as constants, this what you start from to obtain eq. 18.

>>16904220
>why halve the addition and substraction of x and y?
You don't know what taking two numbers, adding them and dividing by two means?
You don't know what taking two numbers, subtracting them and dividing by two means?
Draw a number line. Choose two points x and y. What number represents u? What number represents v?

>the reasoning behind the u and v equations?
Well he uses the variable change for the rest of the proof, so the reason why he did the swap lies in there.
>Step 4. u is even and v is odd.
So, he proves in step 4 that u is even and v is odd. But this can't possibly be true, since (7+3)/2 = 5 and (7-3)/2 = 4, and this will be true for any chosen pair of odd numbers. So right off the bat we have a contradiction.

>>16904293
=2*

mathlet here, is this impressive?
https://arxiv.org/pdf/2601.22401

>>16904323
When I saw this I groaned. Hard.
I fucking despise AI sloppers and their minions so fucking much.
>t. actual math researcher

>>16904325
explain to me like i'm a baby (i'm actually a baby), why i should not be impressed by this

>>16904220
>Can somebody explain to me the reasoning behind the u and v equations?
Because you can factor the LHS of [math] x^3+y^3=-z^3 [/math] into [math] (x+y)(x^2-xy+y^2) [/math], and the general idea is that [math] x+y [/math] and [math] x^2-xy+y^2 [/math] are coprime or near-coprime, so if their product is a cube then they must individually be cubes or close to cubes. So it's natural to make a change of variables to simplify these two factors. [math] u=\frac{x+y}{2} [/math] is just the first factor; division by 2 implicitly includes a parity condition, since you already know [math] x+y [/math] is even, so may as well.
As for [math] v=\frac{x-y}{2} [/math], that then just makes [math] x^2-xy+y^2=u^2+3v^2 [/math] work out nicely; in particular it decouples the variables so there's no cross-term. Again, division by 2 implicitly works in the parity condition.
This is a common substitution in general for these sorts of diophantine equations, particularly when you know two values have the same parity.

>>16904329
LLMs simply aren't at the point where they can make novel mathematical insights. This paper is basically just figuring out which conjectures are the lowest of low-hanging fruit, which evidently nobody cared enough about to even think about. The paper pretty much acknowledges this, too:
>One reason why it seems to be happening so frequently with AI-generated work on Erdős problems is that the solutions are so simple that they would not attract attention if they originated from humans. For instance, Erdős-1089 is answered by an offhand remark in a 1981 paper [BB81], where the authors seemed unaware that they had resolved an Erdős problem.
>In fact, for all of the AI-generated solutions which have not yet been located in the literature, we find it highly plausible that they were also discovered before by humans years ago (perhaps implicitly, as special cases of more general theorems), but were never published because they were not considered important enough.

>>16904347
that's a lot of words, but are you basically saying this is math for babies?

>>16904373
Yes, kind of.

That being said I am also a math researcher and I've been dealing with some existential dread as a result of AI.

Like most mathematicians, my skill at carefully cultivating arguments is something that I take deep, deep pride in. To see machines be able to replicate some of this ability is... disconcerting.

But I know that this is a deep, dense fog: the truth will eventually shine through. Someone will have an insight akin to Godel at the start of the 20th century, relating to the ontological and epstemic basis of mathematical truth.

>>16904373
Forgot to mention: I am >>16904456
and not >>16904347

>>16904456
What kind of math research do you do? And what are your thoughts on Galois theory?

>>16904459
I do combinatorics/probability... so the exact sort of stuff Erdos did.

I don't know much about Galois theory. My school has allowed me to get pretty close to doing a PhD without doing any courses in abstract algebra. I need to self-study it.

>>16904461
Nice. Do you have any favorite Erdos papers that you feel deserve more recognition?

>>16904456
Tb h I think this is going to be a very important year because reinforcement learning for math and programming will be scaled up significantly
We will see how far they can push those systems and there will be gains - be will also see if there's a ceiling and if they're missing something fundamental
Or they'll close the loop and it's over

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>>16900392
Your bad example aside, the concept of fractal geometry began as a perfomative contradiction to the assumption that all continuous curves are differentiable.
A fuckton of proofs had to be re-evaluated because of pic related.

>>16904456
>>16904470
I think we're just about at the plateau for LLMs' mathematical ability barring some really major breakthrough which who knows when it'll happen. There's really not that much good quality written material on most research-level topics to build semantic connections through unsupervised learning, and RL is a very limited way of inducing novel connections because you're ultimately still only training it on "solved" problems (it's not like, say, coding, where the emphasis is mostly on being a workhorse with a repertoire of standard tools rather than being genuinely creative). Maybe I'm wrong, time will tell, but I just don't see LLMs achieving even "good graduate student" levels without some fundamental new understanding in NLP representations.

>>16904293
>What number represents u?
A point in the middle of x and y ?
>What number represents v?
The distance to that middle point ?
Sorry, that still feels too far for me. How is choosing those two values specifically infered from a number reduction attempt?
>>16904333
>This is a common substitution in general for these sorts of diophantine equations, particularly when you know two values have the same parity.
I want to know more about these.
Any good book on Diophantine equations you would recommend?

>>16904466
Hahaha I haven't read any of his papers, interestingly enough. Most of my knowledge of what he did came through the seminal text by Alon and Spencer called The Probabilistic Method. I'd recommend reading it if you've got some familiarity with graph theory/combinatorics.

You can learn about his life through The Man who Knew Infinity.

>>16901628
All of mathematics is just the study of the morphisms of the free Boolean algebra with countably infinitely many generators

>>16904499
>Any good book on Diophantine equations you would recommend?
Unfortunately it's not really my field so I don't have a huge amount of experience with the textbooks. If you want elementary approaches to Diophantine equations, I'd imagine you're best off looking for either something from early last century or (probably more readable) something Olympiad-focused. A quick google gave me this:
https://mathematicalolympiads.wordpress.com/wp-content/uploads/2012/08/an_introduction_to_diophantine_equations__a_problem_based_approach.pdf
which seems pretty good.

>>16904514
>The Man who Knew Infinity.
You mean N is a Number, right?

>>16903827
>0.264039
Sqrt[(2/15) (19 – 4 Sqrt[34] Sin[ArcTan[(555 Sqrt[111])/2461]/3 + Pi/6])]

>>16901894

(d/r)^2
= ((Cos[θ] + Sin[θ] – Sqrt[Cos[θ]^2 + (Cos[θ] + Sin[θ])^2])^2 + Sin[θ]^2)/(Cos[θ]^2 + (Cos[θ] + Sin[θ])^2)

θ ≈ 0.1684742133
r/s ≈ 1.5173966608
d/s ≈ 0.4006521836
d/r ≈ 0.2640391889

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>>16903665

ellipse: r^2 – x^2 = (x – y)^2

circle: x^2 + y^2 = r^2

2.... plus 2.... equals.... 4!

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>>16901377

Input without braces and most commas:
(y – x)^2 + x^2 – r^2 = 0
(y – 0)/(x – r) = D[(y – x)^2 + x^2 – r^2, y]/D[(y – x)^2 + x^2 – r^2, x]
d = Sqrt[(x – r)^2 + (y – 0)^2]
r = 1

Solution 2 of 4 without commas:
d ≈ 0.2640391889203771424
r = 1
x ≈ 0.76019682008823566578
y ≈ 0.1105039736386732746

URL:
https://www.wolframalpha.com/input?i=%7B%28y+-+x%29%5E2+%2B+x%5E2+-+r%5E2+%3D%3D+0%2C+%28y+-+0%29%2F%28x+-+r%29+%3D%3D+D%5B%28y+-+x%29%5E2+%2B+x%5E2+-+r%5E2%2C+y%5D%2FD%5B%28y+-+x%29%5E2+%2B+x%5E2+-+r%5E2%2C+x%5D%2C+d+%3D%3D+Sqrt%5B%28x+-+r%29%5E2+%2B+%28y+-+0%29%5E2%5D%2C+r+%3D%3D+1%7D

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What is the maximum area of a rectangle which exists inside of a regular heptagon of unit area?

>>16905156
semi-half assed math gave me that given a unit side, the area would be (csc^2((3 π)/14) (4 + csc(π/14) (4 + csc((3 π)/14)))^2 tan(π/7))/(16 (8 + 8 cot(π/7) cot((3 π)/14))) = 2.28719.

So divide that by the polygon formula for 7

>>16898099
>>16900751
>>16901197
Are you still around? What do you think of this exposition? Maybe the references can help you
>Complex analysis: a brief tour into higher dimensions. R. Michael Range. Am. Math. Mon. 110, No. 2, 89-108 (2003).

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>>16905100
A = (x, y)
B = (r, 0)

> (y – x)^2 + x^2 – r^2 = 0
orbit of A

> (y – 0)/(x – r) = D[(y – x)^2 + x^2 – r^2, y]/D[(y – x)^2 + x^2 – r^2, x]
slope of line AB = slope of line perpendicular to orbit

> d = Sqrt[(x – r)^2 + (y – 0)^2]
distance between A and B

> r = 1
given radius

> d ≈ 0.2640391889203771424
15/2*d^2 = 19 – 4*√34*Cos[(a – π)/3]
a = ArcCos[2461/(1088*√34)]

>>16905156
you can get a first estimate by considering the circumcircle and incircle. doing some (not semi half-assed) math, you get the bound 1+cos(2π/7) = 1.62... ≤ A ≤ 2. here we've taken the circumradius R = 1 which is why i get a result different from the other answer.

>>16905290
i think setting a radius (circumcircle as i've done above or incircle) to 1 is probably a better idea as it avoids having to worry about the 7-gon area. you could also set the 7-gon area to 1 but that's hard to work with.

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>'First course in topology' notes
>section on homeomorphisms
>"these functions are the analogues of bijective linear transformations for vector spaces, (...) diffeomorphisms between manifolds, etc

>>16906421
sucks being a mathlet, huh hoss?

I'm trying to find the maximum of the Shannon entropy using a Lagrange multiplier, and I'm having trouble with this part of the proof

[eqn]\frac{\partial}{\partial p_i }(\lambda \sum_{p_i = 1}^k p_i - 1) = \lambda[/eqn].

Why isn't the answer [eqn]k\lambda[/eqn]

>>16906460
I know what a diffeomorphism is. But it's at the very least odd to include such a phrase in a textbook designed as a 'first-introduction' to topology.

>>16906474
There are k different pi (p1, ..., pk), and you're only differentiating with respect to one of them.

>>16906482
i was reading cutting edge group theory at 15 btw (i'm 32 now), you got filtered by a FUCKING INTRODUCTORY TEXTBOOK LMFAO

>>16906474
p_i = 1????

>>16906474
>>16906564
it's not \frac{\partial}{\partial p_i}, it's \frac{\partial}{\partial p_J}

>>16906566
What difference does it make? Is Pi supposed to strictly be a placeholder for values {1,...,k} whereas Pj denotes a specific value? Or is just sloppy notation?

>>16906573
No guy, it's a real difference. You're doing stuff like \int_0^t f(t) dt when it's supposed to be int_0^t f(t') dt'. The t' is supposed to disappear! t is not the same as t'

p_i is some number, but the number i is the index for what type of p you're referring to. Let k equal some constant, say 5. What would the expanded sum be?
It would be like p_1 + p_2 + p_3 + p_4 + p_5. The i is used with the \sum to contract those 5 objects and write it in a smaller and more readable way. If you expand the sum, then what the hell does \partial / \partial p_i mean? All the i's are gone when expanded! That's cause the thing is written wrong. It should be some partiial with respect to either p_1, OR p_2, OR p_3, or p_4, or p_5. Which one is the choice of the writer, so call it p_J instead.

And when you do the summation, what the hell does p_i = 1 mean? Does that mean p_1 + p_2 + p_3 + p_4 + p_5 where each term equals one? You aren't changing p_i, you're changing i!

If you struggle with more stuff like this, first make sure that you're copying what the book or vid or lecturer is writing exactly, and then expand it out fully, making sure it all makes sense each step of the way.

Was Paul R. Halmos the original >>>/sci/mg/ forerunner?
>>16906627
>>16906630

So pythagorean theorem length element dl, area element dA, and volume element dV all come from determinants of square matrices, involving the fact that they are rotationally independent, etc, essentially they respect the 2-norm. Is there any function similar to the determinant associated with any other type of norm? Sounds kinda useless but I wouldn't be surprised if some sort of taxicab volume exists.

>>16906671
>Sounds kinda useless but I wouldn't be surprised if some sort of taxicab volume exists.
No, the standard volume measure on [math] \mathbb{R}^d [/math] is unique up to scaling once you assume some basic desirable properties (Borel, translation invariant, finite on compact subsets, etc). This is uniqueness of the Haar measure; in fact, this uniqueness then proves that it's also rotationally invariant, since composing the Haar measure with a rotation gives another Haar measure.
The connection between volume measure -> rotational symmetry -> L2 norm says more about the latter (it has an inner product, and hence a larger isometry group) than the former.

>>16906628
I see what you mean. Thanks for the explanation.
>And when you do the summation, what the hell does p_i = 1 mean?
Yeah I originally meant to write i = 1. That was a careless mistake.
>If you struggle with more stuff like this, first make sure that you're copying what the book or vid or lecturer is writing exactly, and then expand it out fully, making sure it all makes sense each step of the way.
The textbook that I'm working through completely glosses over computing the partial derivative of the summation as I've written above, so I wanted to work it out myself. Having not used what I learned form calculus in years, I'd say I'm pretty rusty.

>>16899914
Is the yoneda lemma basically saying that however a morphism actually works is irrelevant, the only thing that's relevant is how other objects map into an object?
An object is fully determined by the way other objects map into it.
For example, if you have an inital object 0 and an object 1, and there's a morphism from 0 to 1, then the 1 object is fully determined by that morphism. So you don't really even care about the object, you just care about how other objects map into it.
Furthermore, if there are multiple morphisms into an object, then those morphisms have some (unique?) shared structure, which is 'encoded' by the morphisms between them.

>>16906715
Thanks.

If A is false, and B is false.
Then is A <=> B true, because they are both equivalent, or false because they are both false and equivalent?

>>16907849
Are you asking if "A if and only if B" holds for some arbitrary A and B because they happen to share the same value?

>>16907865
Now that you say that I see the problem with my question.
No what I mean is, let's say I want to show NOT(A and B) <=> NOT A or NOT B.
I create a whole truth table, I see ok the truth/false values for each case are the same for both expression, therefore they are equivalent, and I showed that NOT(A and B) <=> NOT A or NOT B.

So my question is for the specific case where [NOT(A and B)] is false, and [NOT A or NOT B] is false. Is the equivalence also false, or is it true because both are false and thus the same?

>multiple mandatory courses on just generalizing integrals
>learn nothing about derivatives besides 1st year calculus.
why?

>>16907943
do you not consider gradient/curl/divergence to be generalisations of derivatives? or is vector calculus first-year for you

>>16907869
Why would the equivalence be false? The equivalence is false if and only if A != B.
In both of your cases, true = A = B.

>>16907944
>or is vector calculus first-year for you
yes
>do you not consider gradient/curl/divergence to be generalisations of derivatives?
No. it's not as rigorous or general.
how do you take a derivative in a function space?

>>16908036
It's called the Fréchet derivative and it's very important in nonlinear Functional Analysis which itselft is important to study any nonlinear PDEs.

>>16908036
>>16908042
I think by "[integral] in a function space" he's thinking of Riemann–Stieltjes integrals, in which case the closest thing to a derivative analog would be the Radon-Nikodym derivative. But I don't think I've ever seen Riemann-Stieltjes integrals done seriously in anything that would be a mandatory undergrad course, so I'm not sure. I guess it's possible.
Also,
>gradient/curl/divergence isn't rigorous
what

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>>16906493
If that's true then why are you on a Mongolian human sacrifice studies forum?

>>16908055
>ut I don't think I've ever seen Riemann-Stieltjes integrals done seriously in anything that would be a mandatory undergrad course, so I'm not sure. I guess it's possible.
ok. lebesgue integrals are standard curriculum taught to every math majors.
and if you are learning lebesgue measure then you are also about measure spaces in general and integrals on these spaces wrt measure.
integrals, integrals, integrals. nothing on derivatives.
>>gradient/curl/divergence isn't rigorous
>what
Not general or rigorous enoug, at least the way i learned it.
there was some shit about open sets, delta balls and jordan something but that's it

>>16908071
If your program is actually rigorously constructing sigma algebras and measures and Lebesgue integration from first principles in a required undergrad course then I don't know, that's not usual, that stuff is usually reserved for first year graduate analysis.
But if you've seen partial derivatives defined as limits (which I'd hope you have) then you've seen gradients, etc. in pretty much full rigor, I don't know what you would find non-rigorous about it.

>>16908086
Idk wym by "first principles" but in probability theory you spend a lot of time only doing set stuff. Granted that's not a required course.
what I'm trying to say is that i have an intuition of what integration is in any space or domain.
For derivatives my intuition and understanding has made no progress beyond the fresher's vector calculus.
It's odd that integration appears so often in math but derivatives are far rarer

What is the font used in Douglas Bridges lecture notes? Looks beautiful.

"In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, with a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere. (In fact, there is so much extra space that any two of the 12 outer spheres can exchange places through a continuous movement without any of the outer spheres losing contact with the center one.) This was the subject of a famous disagreement between mathematicians Isaac Newton and David Gregory. Newton correctly thought that the limit was 12; Gregory thought that a 13th could fit."

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>>16909381
https://www.wolframalpha.com/input?i=1%2F%282+%283+ArcSec%5B3%5D+-+Pi%29%29+%3C+k%2F%284+Pi%29+%3C+1%2F%282+Pi+%281+-+Cos%5BArcCsc%5B2%5D%5D%29%29

>>16909140
> For derivatives my intuition and understanding has made no progress beyond the fresher's vector calculus.

That's because derivatives are a much simpler construct. There are tons of functions which can be integrated that are not meaningfully differentiable.

If you want more interesting notions of derivatives, you really want differential forms from diff geo/smooth manifolds. Then you can get into more interesting ideas like exterior and covariant derivatives.

>>16909369
You can extract this data with pdfxmeta
https://github.com/Krasjet/pdf.tocgen

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>>16908002
got it, thanks

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>>16906493
>t.
Have fun being an unsufferable autist, I guess.

>>16909140
>It's odd that integration appears so often in math but derivatives are far rarer

Derivatives can very interesting in highly rigorous and sophisticated contexts. See:
>Fréchet derivative =/= Gateaux derivative.
Two common ways to understand derivatives not always agree in the context of normed spaces (mostly used for function spaces).
>Lebesgue differentiation theorem
"Almost-everywhere" analogue of the fundamental theorem of calculus for Lebesgue integrable functions.
>Differential geometry
Diffeomorphisms (which are differentiable functions with differentiable inverse) allows for a deep study of the geometrical properties of very intricate spaces that would be otherwise impossible to study. Diffeomorphism groups also lead to interesting connections to algebra.

The 'multivariable' inverse function theorem is a key result for modern geometry, but for most (unmotivated) undergrads it will comes off as a curiosity, or simply as a tool to prove the (equivalent) implicit function theorem which is then used for calculation purposes only.

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>>16909564
I see no pattern.
Does logic have 'order'?

>>16904456
I'm not a math researcher I'm just a GS-1520 loser and still consider myself lucky for it, but I only see AI as a big google search at best, its pretty worthless and makes itself evident when I want to go home or have a deadline and am feeling lazy but have to actually think, and then I get hungry and ruin my cut. Not sure there has ever been a serious math guy with abs.

Anyways came in here to say, I'm doing the Hopkins ACM masters for 7 courses now and am disillusioned, all humans are idiots and there is no hyper math daimyo, we are all petty ronin.

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Two equilateral triangles are inside a regular heptagon so that all the vertices of the triangles are on the perimeter. The heptagon has unit sides.

One triangle has one side which is parallel to one side of the heptagon. The other triangle is the same except rotated 90 degrees counter-clockwise.

What is the exact surface area of the blue shaded region?

>>16912274
Your setup is inconsistent, there don't exist two triangles in a heptagon like you describe.

>>16899914
Hey bros, anyone have a good recommendation for a problem solving book from algebra to differential equations. Or just your favorite problem book in general (with answers).

>>16912761
>from algebra to differential equations
So you already have the theory under your belt?

>>16912771
yeah I do. I studied applied math in uni but I was naturally very good at programming so the role I always performed in group projects etc was always the programming. I knew enough theory to pass the tests but I always felt unsatisfied with my maths after uni.

>>16912761
Zeitz's "The Art and Craft of Problem Solving" is a classic.

>>16912761
>>16912778
Try Vladimir Arnold's Trivium
>[...] we must specify not a list of theorems, but a collection of problems which students should be able to solve. [...] The compilation of model problems is a laborious job, but I think it must be done. As an attempt I give below a list of one hundred problems forming a mathematical minimum for a physics/applied math student.
https://www.physics.montana.edu/avorontsov/teaching/problemoftheweek/documents/Arnold-Trivium-1991.pdf

>>16913493
Thanks anon, I'm struggling with some of these so it does go to show that I'm below the 'mathematical minimum'. And what makes this great is I always have a reference test for myself now.

>>16913012
thanks anon, downloaded and looking at it.

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>>16912274
m = Pi/7
r = Csc[m]/2
P(v) = –r*(Sin[v*m], Cos[v*m])

red circle: x^2 + y^2 = (1/100)^2
red polygon: P(0), P(2), P(4), P(6), P(8), P(10), P(12)

t = Sqrt[3] + 3*Tan[m/2]
z = –r*Cos[m]*(Sqrt[3] – 2*Sec[m/2] + Tan[m/2])/t

n = Pi/3
s = r^2*Csc[3*m/2]*Sqrt[39 – 48*Cos[m] – 18*Sin[m/2] + 42*Sin[3*m/2]]/2/Sqrt[3]/t
Q(v) = –s*(Sin[v*n], z/s – Cos[v*n])

blue circle: x^2 + (y + z)^2 = (1/100)^2
blue polygon: Q(0), Q(2), Q(4)

x + y = z

let y be equal to x

z = 4

Solve for x

>>16915185
I wonder if this is a trap...

Math is so back. I got a job as an AI Engineer at a Fortune 500 company with an applied mathematics degree as a fresh graduate. I'm so excited for the future.

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>>16915386
shit man I hope you got the 300k starting to keep the meme alive

I was just thinking that if you just add infinity to both sides of the equation then the other terms are so small compared to infinity that they become negligible so everything just becomes infinity=infinity and I think that could be really useful for simplifying calculations

[math]V^{*\otimes k} \cong \operatorname{Hom}(V,\cdots,V;\mathbb{K})[/math] by an isomorphism [math]\psi^k[/math]. How does [math]\bigoplus_{k=0}^{\infty}\psi^k[/math] respects the tensor algebra multiplication and is an algebra isomorphism?

>>16915479
Nah, I'm in Mexico, they're probably near shoring to keep expenses low, but still, it's a really nice job compared to pretty much anything I could get here.

>>16911641
>Not sure there has ever been a serious math guy with abs
All you need is a clean diet to have visible abs.

number theorists can’t seem to fathom bases outside of base 10

>>16916932
don't they use mod arithmetic all the time?

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Pretty sure my professor thinks I'm retarded because I do cross products by calculating them as quaternion multiplication. I just find those things easier to work with than vectors. Is there anything I can do to fix this or am I doomed? Master's student btw

>>16916932
Sounds like you have an insightful research direction. I look forward to your preprint.

>>16916932
I read this and was confused for a moment. Then I remember there exist retards like you who think "number theory" doesn't mean sheaves, Picard curves and L-functions but instead means "HURR HURR NUMBERS LOL".

>>16917010
You should be familiar with about 2 or 3 ways of doing something, even to the extent that you can demonstrate that this yields the same result. Cross-compatibility, as I would say.

I'm curious. How do you do cross products with quarternions?

>>16917121
Take vectors <a,b,c> and <x,y,z> and represent them as the quaternions 0+ai+bj+ck and 0+xi+yj+zk
The quaternion you get by multiplying these two together has their dot product as its real part (albeit negative) and their cross product as its nonreal part

>>16917010
>>16917200
>multiplying 0+ai+bj+ck and 0+xi+yj+zk
>professor thinks I'm retarded
No he doesn't. It's the same math in the same amount of time. Who cares how you visualize it when they're basically the same. It's only dumb if you also calculate the real part just to toss it after

>>16899914
guys , I don't want to add problems here but can you please tell me your idea on this video.

I really need it , its really important.
https://www.youtube.com/watch?v=JWaukcx-upg

>>16917410
My suggestion without opening the link is concerning the use of british "maths". Why isn't "Matter and Mathematics" a title that appeals to a broader audience?

[eqn]\boxed{ ~\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mathbb{WARNING} \\\ ~ \\\ ~~~~~~~\mathfrak{This \ \ \ meme \ \ \ contains \ \ \ content \\\ ~~~~~~~~~~that \ \ \ may \ \ \ trigger \ \ \ autists. \\\ ~ \\\ } ~~~~\mathcal{MATURE \ \ AUDIENCES \ \ ONLY ~~~~~ \\\ } }[/eqn]

>learn arithmetic
>learn algebra
>everything else in math is just referencing formula and tables while applying arithmetic and algebra
it's that simple but mathematicians would have you believe that they're doing wizardry

>>16917539
That is quite possibly the most impressively retarded take on mathematics I have ever heard. This sounds like what someone who has never actually applied themselves and tried to do mathematics thinks is involved.

>>16917541
this is a mathematician trying to fool you into thinking math is hard so you'll overpay him

>>16917539
>>16917410
Its this referinching the video anon.

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>>16917541
>>16917543
What did you say?

>>16917546
>look I hid the math behind a bunch of greek letters it's really complex you know? give me $200k/year and I'll solve it for you

>>16917547
You are so fucking ignorant about modern mathematics it's unreal. Mathematics is not about numbers, arithmetic or equations anymore, that was like 4 centuries ago. Nowadays we have the Langlands program, infinity categories, sheaf cohomology, Hopf algebras, etc. all of which far transcends your barely-there understanding of the subject.

>>16917543
I'm not a mathematician. I'm an electrical engineer. I don't need to fool you to earn a better salary for my work. Nobody needs to be fooled into believing "math is hard." You are simply ignorant of what mathematics actually has to offer.

As an example, real analysis very often turns into finding ways to approximate quite complicated functions or relationships by infinite series. The "challenging" part of the problem is not carrying out the arithmetic operations in a sum. It is producing an algebraically tractable representation of whatever function/mapping/relationship is of interest.

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>>16917546
>Tate
He is truly the philosophy king-warrior of our era.

>>16917574
John Tate, you fucking retard. Tate is a relatively common surname.
https://forebears.io/surnames/tate

https://en.wikipedia.org/wiki/John_Tate_(mathematician)

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Does the partition principle imply excluded middle?

When I throw a ball into the air it doesn't blink out of existence for an instant after flying precisely 4ft.

>>16917740
depending on how short "instant" is, you'd never know

What is the most historically significant incorrect math proof?

>>16917574
Don't let them take this from you. People hate to see a Romanian living that LA life regardless of what it is that you accomplish.

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>>16917554
>An actually good post.
Kinda jumpscared me, ngl.

>>16917541
>>16917547
>>16917574
>>16917740
>>16917941
This is why brainlets should not be allowed anywhere near mathematics.

>>16917944
I meant to quote >>16917539 for the first post instead it was >>16917541.

mathlet here, is this man a mathjester or is he arithmaxxing:
https://www.daniellitt.com/blog/2026/2/20/mathematics-in-the-library-of-babel

>>16917959
>MUH AI
kys
kys
kys
KYSKYSKYS

>>16917962
i see your cortisol levels are spiking

>>16917551
Ok but this isn't really math. Real math is adding numbers, solving equations and finding absolute truths. Practical shit. Calculus, differential equations, lin alg. But as soon as you introduce shit like fields and rings and algebraic anything it becomes nonsense.
>heres a set of all polynomial functions now lets perform some esoteric analysis on.

>>16917970
You are incredibly retarded.
>t. mathematician with published papers

>>16917971
>mathematician with published papers
You learned to babble schizo bullshit with other schizo. Good for you, I guess.

>>16917970
You just can't resist seething at my intellect and my much greater understanding of the subject, can you, boy? Be an adult. Read Baby Serre. Then come back.

AFCIA is free as a PDF online. If you read it then I guess that's one step towards understanding mathematics as well as I do. But until then, you don't want serious discussion. You're just a novice who wants praise. Now I have a paper to attend to.

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>>16917970
>as soon as you introduce... algebraic anything it becomes nonsense
>lin alg is cool though

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>>16917959
>Frontier models are not yet performing these tasks at a level comparable to their ability to answer well-posed questions that are accessible to known techniques. I think there is no indication that their doing so is impossible, and some evidence that it is possible.
it's possible via the same mechanism that getting the zeros of an algebraic function from rolling dice is possible, it's just better than completely random guesses.
which is to say, it's completely useless at a research level unless a human is the one checking the output, and the human in that system is the only part of it doing anything "intelligent."
still useful as a semi-random search through existing ideas, and a way to come up with potential starting points for real research, but the further you stray from the published data (i.e. the entire point of research), the closer to random output you get.

it's been really disappointing to see tech companies satisfied with scaling this architecture when these billions could have been spent on research into, y'know... actually emulating intelligence instead of statistically approximating its output. it's become abundantly clear in everything but marketing and hype that the two are not even remotely equivalent - even the most genius biological mathematician needs less information than LLMs by dozens of orders of magnitude to come up with new ideas in mathematics (and at a fraction of the thermodynamic cost).

i want to see scalable intelligence emulation architecture that can do at least nematode-level real-time learning in my lifetime, dammit. we had perceptrons in the fucking 1950s.

>>16917908
There are incorrect proofs of Euclid's fifth postulate, the fundamental theorem of algebra, the four color theorem, and Jordan curve theorem. Some of these are of historical interest. Maybe the Jacobian conjecture is interesting because it is a fairly elementary problem and still unsolved

if you're going 60mph and you add 60mph you're going 120mph
if you're going 1hpm and you add 1hpm you're not going 2hpm

math is broken

>>16918108
Seems to be working fine? If it's taking you 1 hour to go a mile and then you add another 1 hour to that then it takes you 2 hours to go one mile.

>>16918108
Common core has destroyed a generation.

>>16917959
He seems pretty sensible so I'm a bit confused as to where his sudden optimism comes from. The First Proof results are basically in line with the higher end of his prior expectations but with a couple major asterisks, and even in the most generous reading is more a demonstration of consistency rather than ability to push on.

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I had push it to the limit on repeat the entire time I learned limits

>>16918222
>muh ai
kys

>>16918885
AI is the future of mathematics. No single human can comprehend all fields of mathematics at the same time, let alone recall every formula and table in existence from memory. For AI? Not a problem. They can also work 24/7 without ever taking a break or getting distracted, at speeds orders of magnitudes faster than the most autistic savant to ever exist.

>>16918893
I'm an arithmetic geometer, IQ tested somewhere north of 140. I can tell you AI will never be able to put a dent in my fucking work let alone my job, boy. Keep rotting your brain with sycophantic chatbots while I write papers. Better yet, be an adult and realize AI is a steaming pile of shit.

>>16918894
You know what doesn't have to cling to the false hope that it'll still be relevant in the future? AI.

>>16918896
Cool. You might want to read up on AI psychosis, AI hallucination, Moore's Law, model collapse, water debt, etc. before posting another reply though. Fucking infantile know-nothing corpo fanboy.

remember to include trigger warnings with your posts for the one guy who has anti-llm psychosis

>>16918896
Guess what likes to make noise about a subject it knows absolutely nothing about? A corpo fanboy like you. Nigger.
>>16918898
Anti-LLM psychosis? Yeah I'm the "muh ai kys" serialposter but that doesn't mean your made up neologism is accurate. I'm perfectly sound of mind unlike the niggerdroidballlickers like you who spam this board with their fucking bottom of the barrel garbage.

>>16918899
I'm fucking sick of it. You retarded tourists will leave when this craze blows over and the money incinerators like Anthropic and OAI are finally sent packing.
Mark my words.
You were never citizens of my country regardless. You just wanna be called smart while doing nothing.

having an absolute 140iq melty

>>16918898
>the one guy
Imagine believing this. You are hated almost universally.

>>16911576
Who can say

Im working on some crazy shit with ChatGPT. Trying to create a new branch of mathematics
Paper will be on viXra and then will be submitted to Annals or smth

The number line eventually loops back on itself. Prove me wrong.

>>16919068
Nice, going to tell ChatGPT to add spherical numberlines to my new branch of mathematics
Numberspheres if you will

This is what happens when you morons reply to every bit of obvious bait put in front of you. The thread turns into a circus of (you) farmers, and not even good ones.

>>16919073
use case?

>Euclid is prounounced you-clid
>Euler is pronounced oiler
Sorry, but no. It's you-ler.

oy-klid

People like Euclid and Euler are only famous because they were lucky enough to be born at a time when there were still basic problems left to be solved. They spent their entire lives trying to grasp concepts that a modern 2nd grader is expected to know. To get famous nowadays you have to solve problems that probably don't even have solutions in the first place.

>>16919514
Yeah, their brilliance is greatly overstated. Most modern maths professors wouldn't have struggled solving any of the problems Euler faced in the 1700s. Though back then, maths was seen as a fruitless hobby with no academic future reserved for only the strangest of people. Doesn't help that much of the population were devout Christians either.

>>16919515
>>>/r/eddit

>>16919516
I'm not an atheist

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>heh, you solved that problem? well, I just memorized your solution so that makes me just as smart as you

i have to learn group theory in 3 weeks

>>16919453
It better models how numbers work in reality

>after years of telling you not to trust graphs they throw in a question that has to be solved visually from the graph

I just realized that a polynomial is basically just the sum of its terms which could each be treated as an independent function ugyuu~

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https://80.lv/articles/see-how-you-can-animate-basic-run-cycle gimme maths that can either do cycles like footage one or whole sequence like footage two or even idles or linking between any to another. Also making a characterization like limps, happy,etc
Like, I feel I can, but, I also know, I won't be correct so...why not have some math, from, anyone better?

Basically a simple equation to rely on rather than, shooting out of false confidences

>>16919703
Wait until you realize this is generalizable to all functions and even across classes of functions.
Try plotting "sin(x) + x" into your graphing calculator some time

it's... it's literally the definition of +...