Hurewicz space editionITT: discuss mathematicsPrevious: >>16876118
>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/
>>16899933>quanta magazinegtfoht. actual math researcher
>>16899934That article about quantum mechanics made "free" from complex numbers was really bad
>Černý conjectureCan /sci/ solve this open problem?https://www.wikipedia.org/wiki/Synchronizing_word#Lenght
>>16899914Ah, Hurewicz...his depths of knowledge indeed matched with his adventurous spirit
>>16900313there'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
What's your favorite counter example
>>16899990It'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.
>>16900392
>>16900392That’s not the definition of a square. A square is a parallelogram that’s equilateral and equilangular. This obviously doesn’t qualify.
>>16900392You are missing this definition, a necessary condition for any polygon:https://en.wikipedia.org/wiki/Polygonal_chain
>>16900392Can you proof you can construct such a square?
>>16900512>>16900618Yeah 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
>>16900721The difference here is that there is no theory where that’s the definition of a square.
>>16900489Art 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 setThose 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>>16900798Bro 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.
>>16900809have a look at arXiv:2502.06137
>>16900809I 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 KlymchukCounterExamples: From Elementary Calculus to the Beginnings of Analysis - Bourchtein, Andrei; Bourchtein, LudmilaCounterexamples in Analysis - Bernard R. Gelbaum, John M. H. OlmstedCounterexamples in Topology - Lynn Arthur Steen, J. Arthur Seebach JrCounterexamples in Probability and Real Analysis - Gary L. Wise, Eric B. HallCounterexamples in Measure and Integration - René Schilling, Franziska Kühn
>>16900835That 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
>>16901094Take 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 counterexampleswhat'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?
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?
>>16899914I 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.
>>16901615Essentially:>Combinatorics is not a subfield but a collection of methods from algebraic geometry
>>16901615All of mathematics is just a subfield of epistemology.
>>16901611Wildberger 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
>>16899914Is 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.
>>16901628it is not
>>16901664Thanks, 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"
>>16899914H=i,j∑Pij(ci†cj+cj†ci)−σmap
>>169013770 <= θ <= π/2P = 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]^2θ ≈ 0.1154000604r/s ≈ 1.4884550172d/s ≈ 0.3970256621θ = 0r/s = √2d/s = √2 – 1
>>16901377Consider 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 = aLet 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 areasIt's just algebra and calculus, no special tricks I saw
>>16903012system:r = 1r/s ≈ 1.4884550172d/s ≈ 0.3970256621solution:s ≈ 0.671838d ≈ 0.266737your answer is close to:d^2 ≈ 0.071149
>>16903309>d^2 ≈ 0.071149correction:d^2 ≈ 0.071148
>>16901377360 frames15 seconds24 frames per secondθ = k/180*Pis = 7r = 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)^2edgecolor: (0, 1, 1, 1)circle:equation: x^2 + y^2 = r^2edgecolor: (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*sditto for y
https://chan.alphakek.ai/sci/res/8.html
>>16903309Yes yes, I didn't sqrt the guy, about .26 is the answer, accuracy depends on how far you want to go
>>16901377>>16903825according 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
>>16899982Which one?
>>16903899I should say that most of the problems with the Quanta article come from the arxiv preprint being retardedhttps://www.quantamagazine.org/physicists-take-the-imaginary-numbers-out-of-quantum-mechanics-20251107/https://arxiv.org/abs/2504.02808
https://people.math.ethz.ch/~salamon/PREPRINTS/FERMAT.pdfCan 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=xThat'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
>>16904323When I saw this I groaned. Hard.I fucking despise AI sloppers and their minions so fucking much.>t. actual math researcher
>>16904325explain 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.
>>16904329LLMs 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.
>>16904347that's a lot of words, but are you basically saying this is math for babies?
>>16904373Yes, 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.
>>16904373Forgot to mention: I am >>16904456and not >>16904347
>>16904456What kind of math research do you do? And what are your thoughts on Galois theory?
>>16904459I 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.
>>16904461Nice. Do you have any favorite Erdos papers that you feel deserve more recognition?
>>16904456Tb h I think this is going to be a very important year because reinforcement learning for math and programming will be scaled up significantlyWe 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 fundamentalOr they'll close the loop and it's over
>>16900392Your 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>>16904470I 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?
>>16904466Hahaha 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.
>>16901628All 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.pdfwhich seems pretty good.
>>16904514>The Man who Knew Infinity.You mean N is a Number, right?
>>16903827>0.264039Sqrt[(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 ≈ 0.1684742133r/s ≈ 1.5173966608d/s ≈ 0.4006521836d/r ≈ 0.2640391889
>>16903665ellipse: r^2 – x^2 = (x – y)^2circle: x^2 + y^2 = r^2
2.... plus 2.... equals.... 4!
>>16901377Input 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 = 1Solution 2 of 4 without commas:d ≈ 0.2640391889203771424r = 1x ≈ 0.76019682008823566578y ≈ 0.1105039736386732746URL: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
What is the maximum area of a rectangle which exists inside of a regular heptagon of unit area?
>>16905156semi-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>>16901197Are 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).
>>16905100A = (x, y)B = (r, 0)> (y – x)^2 + x^2 – r^2 = 0orbit 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 = 1given radius> d ≈ 0.264039188920377142415/2*d^2 = 19 – 4*√34*Cos[(a – π)/3]a = ArcCos[2461/(1088*√34)]
