/mg/ - mathematics generalPreviously >>15420227Platonic solids edition.Talk math(s)!
Why?Why make a new thread when the old thread is on page 2?Why have mathematics in the title and math(s) in the body of the post?Why have an OP image that includes text describing the general, and why make that text italic computer modern instead of roman computer modern?
>>15433622Also, why is the icosahedron in the image non-regular when the other four solid figures are regular?
>>15433326I am a different Computer Program. >>15433341>Do you want a book on programming or abstract language theory?Yes. Both please.>Did you check the reference section on the Wikipedia pages?I almost never use Wikipedia. My understanding of that site is that it is primarily a smear laundering machine used by the media to SEO bomb things they don't like. I didn't know it had practical applications.
>>15433622The old thread hit the bump limit you moron
>>15433637>I almost never use Wikipedia. My understanding of that site is that it is primarily a smear laundering machine used by the media to SEO bomb things they don't like. I didn't know it had practical applications.wikipedia articles vary radically in quality, most are shit but a lot of them have really solid citations and references
>consistency of PA implies consistency of PA + ¬Con(PA)checkmate, Platonists
>>15433622Because the old thread HIT THE BUMP LIMIT!>Why have mathematics in the title and math(s) in the body of the post?to spite you!>Why have an OP image that includes text describing the general, and why make that text italic computer modern instead of roman computer modern?simply to annoy you! No other reason!>>15433628That too was to annoy you. Actually I'm not sure what you're talking about, it seems very regular to me.>>15433640Some people, I swear. He should be thanking me.
>>15433637>I almost never use Wikipedia. My understanding of that site is that it is primarily a smear laundering machine used by the media to SEO bomb things they don't like. I didn't know it had practical applications.Wikipedia is trash for anything that could be tainted by wokeness, but pure maths articles are usually legit
>ChatGPT and /g/ did not helpHow can I store any set out of all possible [sets with length 0-65536] that contain [unique integers from 0 to 65535 (16-bit unsigned)] in a single 64-bit unsigned integer while being able to lookup and add to the list in a few O(1) operations?Is this even possible? If not, how about with a lower maximum set length of 4096 or 1024?
>>15433859You can't because2^(2^(16)) > 2^(64)>how about with a lower maximum set length of 4096 or 1024Even those are impossible.
>>15433622some girl probably hijacked itcheck the platonic
>>15433859Pidgeonhole principle
>>15433637>My understanding of that site is that it is primarily a smear laundering machine used by the media to SEO bomb things they don't likeThis is the dumbest reprogramming I've heard so far. I can barely conceptualize the kind of bubble you have to live in where you didn't know Wikipedia was practical. Impressive, honestly.I bet you have plenty of non conventional ideas about math too. Are you working on a big idea, to prove the mainstream math establishment wrong by any chance?Definitely want to hear about those
>>15433859a set of n elements <-> an n digit binary integeryou need n=2^16 bits.maybe some hashing that works well enough for your application may exist
>>15433578Reposting this from the old thread because I am still curious about it. I may not be using all the correct terms, so if you don't get what I mean, just ask and I will do my best to elaborate.I have an array of k dimensions.The first dimension has length 2.The second dimension has length 3.The third dimension has length 5....The kth dimension has length p_k.I fill the array with values from 0 up to (but not including) the kth primorial. (The kth primorial is just the product of the first k primes.). Each value's position in the array is determined by its remainder when dividing by each of the primes. So, for example, the position of 3 in the array would be (1, 0, 3, 3, ..., 3). Because 3/2 has remainder 1, 3/3 has remainder 0, and 3/p for all other primes p has remainder 3.Do you have the array pictured in your mind?Now here is the game.You can pick k numbers.You take your first number mod 2, and its inverse in Z_2+, and remove all array values with those indexes in the first dimension.You take your second number mod 3, and its inverse in Z_3+, and remove all array values with those indexes in the second dimension.You take your third number mod 5, and its inverse in Z_5+, and remove all array values with those indexes in the third dimension....You take your kth number mod p_k, and its inverse in Z_(p_k)+, and remove all array values with those indexes in the kth dimension.The question is: how do you choose your k numbers so that the lowest value in the array is as high as possible? I am thinking of it like a game, like you make your choices, and I will find the lowest value in the array that I can, and that will be your score. How do you force me to give you a very high score?
>>15433637You’re right about Wikipedia. It’s not always objective with math either as midair authors will edit pages to insert their articles into the citations since no one would ever accidentally open them otherwise.Anyway, much better than wiki(pedo)ia is scholarpedia and Stanford encyclopedia of philosophy, which has extensive math articles as well
>>15433918>some girl probably hijacked itTeaching women to read has been a disaster for the human race
>>15434234https://mathworld.wolfram.com is good too, obviously it promotes the sponsor, but it's free from woke idiocy and midwittery. Britannica might be ok for some stuff too, back in the day they used to hire good mathematicians to write the articles, but the entire thing has become a woke mess run by libs in Chicago these days.
>>15433622>Why make a new thread when the old thread is on page 2?Go back to sfg.
Alright, we've talked enough about Analysis books, but what is the absolute BEST Calculus book/volume set for each of these two categories:SPEED - Learn single and multidimensional quickly so as to best enable a Physicist or Electrical engineer to conduct the rest of their applied studies. FOUNDATIONAL - Intended for the future math major, who will at least study analysis whatever else their speciality may become.I don't know what the answer is for the "Speed" category, but for "Foundation" I suppose that's Tommy I/II, Spivak, Piskunov, Zorich, Courant, or any the so called "Advanced calculus / analysis transition" books.
recreate this ploot and together we could make a lot of money. or dont i dont give a shit
how do i learn integrals?
>>15434748How do we make money from a thing like that? Anyway it looks kinda like x=cos(y)*cos(16y), with -pi/2 < y < pi/2
>>15435358Oh and I think the amplitude might be close to 2, so change it to x=2cos(y)cos(16y)
How to get tenure?
how do i learn to post better
>>15434474speed would be any calculus book that teaches you single and multivariable calculus in one booksince that doesn't exist, experience tells you that Stewart is the best book for speed, since it's what most non-math undergraduate students use to learn all of (applied) calculus in two "semesters" a
>>15434474Speed would be to skip Calculus and jump to Analysis. And the fastest analysis book is Baby Rudin, which you may want to follow up with Munkres' Analysis on Manifolds, which isn't very long either.
>>15433578Thinking of looking into some math textbooks and journals as a hobbyist, I've learned up to a bit past basic calculus and trigonometry in high school. Half the battle for me is looking into the fields of math and understanding how to negotiate it all (pic related is the closest I've seen someone address it in a way I understand). Do you learn the different parts of it as a gradated whole, or is it a bunch of separate tangents? What resources/books can I look at which can summarize all the different fields of math and how they might work together?
>>15435782Baby Rudin problems are too difficult for "Speed" and it teaches a formalism that just isn't useful to most applied users. Whatever the book is, it should barely require any proofs. Just geometric intuition.
>>15435544>StewartIt's surprisingly large though isn't it? It's both length and useless busy work for Amerifats, so I think the only people like like it are the publishers. Pic related by Kleppner exists for single (I haven't read it, but Kleppner might be known to some of you for his MIT mechanics book). I'm not sure what the "quick" multi variable book is.
>>15434090What year of uni would this be from?
>>15435953To my knowledge, It's not a uni question, it's just a question I made up. I can already solve it with a computer program which searches through every possible row combination of index deletions on the array. But this brute force method takes very long for high k. Once k gets to around 8, it already takes many hours. Beyond k=8 I have not solved at all, and I think the program would take days to solve. So I am looking for a more efficient strategy than just searching every possible combination.
Now say what you want, but TCS is a real interesting field to move into with a universal algebra, algebraic geom, or such degree in the pocket.
>>15434090Could you elaborate on what your filled data structure looks like.
>>15436513Yes. The k=1 array looks like[0 1]The k=2 array looks like[[0 4 2][3 1 5]]The k=3 array looks like[[[ 0 6 12 18 24] [10 16 22 28 4] [20 26 2 8 14]] [[15 21 27 3 9] [25 1 7 13 19] [ 5 11 17 23 29]]]So 0 is always in the (0,0,0,...,0) position of the array, and 1 is always in the (1,1,1,...,1) position in the array. In general, for any n lower than the kth primorial, the position of n in the array is given by (n mod 2, n mod 3, n mod 5, ..., n mod p_k).
>>15435945>It's surprisingly large though isn't it?yes but i guess you can skim through it pretty quickly. Other than Stewart you could use something like Strang, I guess, but that's only single-variable iirc>>15435782>analysis>applied
>>15436723Could you work k=2.
>>15437050Yes.The first dimension has length 2, and indices 0 and 1.The second dimension has length 3, and indices 0, 1, and 2.The rule says that for each dimension, you can delete one index and the index corresponding to its additive inverse mod p. So for the first dimension, you can delete index 0 (which is its own additive inverse mod 2) or index 1 (which is also its own additive inverse mod 2). For the second dimension, you can delete index 0 (which is its own additive inverse mod 3) or indices 1 and 2 (since 2 is the additive inverse of 1 mod 3).Of the choices available, the optimal option is for you to delete index 0 in the first dimension, and indices 1 and 2 in the second dimension. So your answer would be (0,1).I will now show the array that remains, with a # symbol representing deleted values.[[# # #][3 # #]]The lowest value remaining in the array is 3, so I am forced to give you a score of at least 3. This is the highest score you can force me to give you in the k=2 problem.Note that in this particular case, only one value remains- this will generally not be the case for k>2.
Did Hilbert have any good case again Brouwer of was he just a ree'ing old man, too lazy to find a clear proof of his claims?
>>15435358>>15435399obviously it uses sine and cosines most functions in fact all functions can and do use them look at fourier series for fucks sake. but you didnt even graph it so i dont give a shit still
>>15437687I mean I didn't graph it because I was too lazy to take a screenshot, I figured you could just graph it yourself once I gave you the function. But here it is. I still don't know what this has to do with money.
Is there such a thing as functional analysis, but for functions that only operate on integers or rationals? I mean obviously this is a subset of functional analysis with R/C, but I am interested if it's studied as a field in its own right. I.e. I am interested in how vector spaces, convolutions, etc. interact with such functions.I am doing CS bachelor in Switzerland, and CS here comes with an obligatory math minor. While I don't have a problem with the real analysis courses needed, I do find them a bit a baroque and extravagant addition for someone on a CS track. I believe that "up to philosophy", nature and computation only work on rationals. So I am more interested in this more focused subset.
>>15437822>that only operate on integersSeries analysis?I just made that up, but those two words put together sound about right?
>>15437822constructive analysis
why is /mg/ so slow? this thread is already 2 days old and only has 45 replies, less than 1 per hour. the current /sfg/ has 170 replies and its only 14 hours old, more than 12/hour and thats on a day when nothing is happening in spaceflight. its not like theres no math discussion on this board, its just this thread that gets ignored.why is that?
>>15437822>I believe that "up to philosophy", nature and computation only work on rationals.Does working with real numbers lead to any incorrect results on how just the rationals behave, though?
>>15437750>>15434748> I still don't know what this has to do with moneyThat graph could very well be part of a design thata company could use on their products and makea lot of money.Gina Ekiss, former designer of the SweetheartCup Co. (now absorbed into Dart Container Corp.),created the "Jazz" design for their products thatbecame part of the 1990s zeitgeist. However, itwas reported that she received no royalties fromthose sales. So, yeah...be your own company.
>>15438023I'm no business major, but it strikes me that logo design is a very small contributor to the overall profitability of a company. Knowledge of the math underlying an aesthetically pleasing logo is far from a guarantee that whatever company I start will be successful.
>>15438023that function fails the vertical line test, retard deserved to get no royalties
>>15437987 (me)to add, I think it's basically useless to learn all this complicated stuff just to get basically the same answers. unless you're very interested in armchair philosophy. like >>15437997
>>15438065That's not a philosophical question, it's a concrete one. Is there any one fact about rational numbers that traditional real analysis gives you which isn't actually correct?
>>15438491that's not his point, his point is that it is inefficient to learn analysis for all of the reals when you're only ever going to use the rationals in computing
>>15437050>>15437613To elaborate further, here's the reason (1,0) is the best.You only have four choices for the answer in the k=2 case. You can choose (0,0), (0,1), (1,0), and (1,1). All other pairs of numbers you could put in there are functionally equivalent to one of these, in the sense that they will result in the same indices for deleted values. Here are the arrays for each case:case: (0,0)Array:[[# # #][# 1 5]]I can give you 1.case: (0,1)Array:[[# # #][3 # #]]I can give you 3.case: (1,0)Array:[[# 4 2][# # #]]I can give you 2.case: (1,1)Array:[[0 # #][# # #]]I can give you 0.Since in the (0,1) case I give you the highest result, 3, the answer is (0,1).Let me know if that isn't clear enough, or if you would like to see the k=3 solution as well.
>>15437997mathematicians act like the answer is negative but truth is they have no idea
>>15438569Meant to say at the beginning that (0,1) is the best.
>>15437613Sorry I meant how you got the values in the first place.
>>15438699Oh, sure. We are filling the array with all values less than 6 (since 6 is the second primorial). The array has dimensions 2x3.We place the value 0 at location (0,0), because 0 mod 2 is 0, and 0 mod 3 is 0.We place the value 1 at location (1,1), because 1 mod 2 is 1, and 1 mod 3 is 1.We place the value 2 at location (0,2), because 2 mod 2 is 0, and 2 mod 3 is 2.We place the value 3 at location (1,0), because 3 mod 2 is 1, and 3 mod 3 is 0.We place the value 4 at location (0,1), because 4 mod 2 is 0, and 4 mod 3 is 1.We place the value 5 at location (1,2), because 5 mod 2 is 1, and 5 mod 3 is 2.And that's the full array, leaving us with[[0 4 2][3 1 5]]A simple way to think about it is that as you step through the values, you increment all the indices by 1, but apply the mod rule so that you pop out on the other side when you go over the length of the dimension.
>>15432158>Good luck on your finalI got an A in Abstract algebra (post-curve)We did it bros....
>>15439014it's over, we're back
*blocks your path*
>>15438491>>15438567I wrote it poorly. "complicated stuff" means constructive analysis etc. I was trying to agree with the linked post but rereading it, it seemed like I was disagreeing.
>>15439014Nice, I'm glad it worked out.
>>15438051>>15438023I agree completely. But every successful companywould need an aesthetically pleasing logo ordesign to work with. And that's a job/businessthat one can provide to these companies, mathknowledge varying.Famous graphic designer Saul Bass createdlogos of his own style for some good money atthe time. One example is the Bell System logo(lasting 1969-1984).https://www.youtube.com/watch?v=xKu2de0yCJI>>15438060Damn, bruh, that's a rekt...
Let A be the algebra of continuous real-valued functions on some compact Hausdorff topological space X. Then is every prime ideal of A also a max ideal of A?Note: a standard exercise shows every max ideal of A is given by evaluation at a point of X. However I’m not sure what can be said about the prime ideals.
>>15439136I think that even if my math skills suffice, my artistic and design skills are not great enough to support such a career. That said, if you or someone else wants to pay me for finding the function, I won't object twice.
>>15439160>>15439136>That said, if you or someone else wants to pay me for finding the function, I won't object twice.There you go...that's a niche little thing you canpotentially do. Then you can expand on it downthe line.
>>15437995>the current /sfg/ has 170 repliesProbably there's a discussion about some stupid topic there
>>15437995/sfg/ is really really low quality, even below typical /sci/ standards. Space travel and exploration is a media sexy topic that attracts low quality people that like to talk a lot even if they have nothing to say.
>>15439143>every max ideal of A is given by evaluation at a point of Xshould say, every max ideal of A is *the kernel of* an evaluation homomorphism at some point of X
>>15439014> I got an A in Abstract algebra (post-curve)Congrats! I took four classes this semester including a weed-out class which had a really shitty final everyone hated and the TA wasn't good at preparing us for. I was sweating bullets for weeks thinking I'd have a C or worse in that class, and with the weight it turned out I got all As. Not that it matters as much for a grad degree, but this is the first year I've done any of this shit in college, so I feel like I'm on schedule. WAGMI anon
Has anyone read this?I kinda want to buy a physical copy but I don't know if it's worth fifty bucks.
>>15439866http://libgen.is/book/index.php?md5=22A5EBCFABF203F38C2038F68FC27293
>>15439888ah yes, the sekrit website that no one has ever heard ofso what?
>>15437822>but for functions that only operate on integers or rationalsWell, the integers don't even form a field, that's quite a problem. Integer problems get insanely difficult. The rationals at least form a field and yes, if you think of it, many standard theorems in functional analysis work for spaces over Q, but what's the point exactly?
>>15439892Don't spend money on a book that you can get for free. The publishing industry with their paywalls will never die if people like you keep buying books.
>>15439898So writers for the AMS don't get paid by copies sold?How much they earn with articles/books published?
>>15439905Journal articles don't pay anything. The publisher takes 100% of the profits.
>>15439908>The publisher takes 100% of the profits.Get out of here. There's no way this true.
>>15439919it is
>>15439898paying for books is not a matter of being for or against the publishing industry it's simply about having morals
>>15439925morality gets out of the window when the authors don't earn any profit from itnow, if I want a physical copy I would buy a used copy instead
>>15439919Maybe you will believe stackexchangehttps://academia.stackexchange.com/questions/162966/how-much-percentage-royalty-do-i-get-from-springer-as-the-papers-author-and-h
>>15439942Holy lmao, every day I understand Ted Kaczynski more and more.
>>15439042>are elements of [math] [1,\inf] [/math]So infinity is inclusive now?
>>15440022[math]L^\infty(X,\mu)[/math] is the space of essentially bounded functions. That is those functions [math]f[/math] for which there is a number [math]K[/math] such that the measure of all points [math] x \in X[/math] with [math]\|f(x)\| > K[/math] is zero.
>>15439940Springer's prices aren't even that high compared to some like Pearson and McGraw-hill.
>>15440022Look up the extended real line
>>15439908>>15439919>>15439940>>15439942Huh, didn't know that. Welp, thank God for scihub then, I've also gotten hundreds of books from libgen. Is this the same for books, do contributors get any royalties or does it all go to publishers?If I ever publish an article or book in my field, I'd pretty much expect scihub and libgen to provide access to them and ask their teams for the download analytics.
>>15440086Put it this way, if you write a book, and you actually do make money, you'll be earning less than minimum wage. The only way to be successful is to write a 1st year textbook that almost every student needs. Something like Stewart Calculus made that gay Canadian a multimillionaire, and it's crappier than some old soviet publication you can get from Mir for free or nearly so for the printed version.
is there a fundamental theorem of tensor calculus?
am I just dumb? it wouldn't take 0 either
>>154406240 and -1 it is
>>15440624… What's hiding under that Foxit Alert?
>>15440653if there's no solution, I should put in "empty"
>>15440655Empty is always the correct answer.
>>15437750it not like im just gonna spill a bunch of spagetti and post any proofs ive developed. you got the basic function down but i think missed a portion of the essence of ploot. anyway if you want to understand where the money part comes in look at digital filter designs and what a wavelet like that is designed to do. or dont cause thats just another bull shit primrose path
what's /mg/'s opinion on nonstandard analysis and modern "infinitesimal" calculus? i'm learning "standard" calculus through apostol right now and although i don't have problems understanding it, the concept of infinitesimals is just so much cooler
what're some good resources for learning stochastic calc? I'm rusty with math in general but trying to convince myself i'm smart enough to do real math before even thinking about going back to school
What do you enjoy about algebra? Maybe i'm not smart enough to see the beauty in it, but it is the only area of math i've ever been genuinely bored by. I don't find any of the results pretty, it's just a means to understanding certain parts of topology for me. So, educate me. What the heck am I missing?
>>15440978https://quantnet.com/threads/which-book-is-best-for-my-self-study-of-stochastic-calculus.49808/
>>15433578Is it still worth even studying mathematics when you can just use WolframAlpha to calculate everything, and have ChatGPT explain anything you find complicated? I literally do not see why there is any point in "proofs" and exercises in frustration like symbol manipulation? Like why do this in 2023, because it looks cool? If I'm building a bridge, I can just look up whatever I need from ChatGPT which has knowledge of engineering tables, and any math that I need to perform for cost calculations can just be done in wolfram alpha. It just makes sense to use the tools we have available instead of everything trying to reinvent the wheel learning it from scratch by themselves. Use your time more efficiently.
>>15441071Well the WolframAlpha is a bit silly and old - math is usually not about reducing the most formulaic expressing, integrals, PDE's, etc. You learn theories and attack research problems - the famous ones being intractable for 200 years now. And you must enjoy it a bit, to do it professionally.You can't do it for fame. That said, I wanna take your point seriously. There's a 1 in a 100k of getting famous outside of math, with math. Like of 100k people who study till second semester, only one will make it to more than being mentioned in Quanta magazine once. Also, personal AI's will (incresingly) be society altering, and most people will have AI's (vocally) explain most things to them, for better or worse. Now from a research perspective this is also just a tool - it might be a while before the computer does more than brute force and if the AI gives you an idea, you'll still work enough on it to call it your own.But we can look at your question form the other angle? What do you want to do with your life and how is there an angle to it without being proficient in math? Are you satisfied with engineer level statistics?If you ask me, I would say study physics instead - but I'd already have said that 10 years ago, not because of AI. In math you'll find the formulaic problem solvers, the ones who want the more clear cut presented problems and certainty that putting work into attacking it will bear some fruit, at least to you. The smart people, who are comfortable to solve problems outsize the ZFC formalism, and the people who are best at mathematics, were always physicists. So that's my recommendation.gl
Will you anons help me with statistics homework? My degree depends on it. Need it by May 17 3:00pm Eastern time. If you accept, I can post some challenging statistical problems. They're really not that hard. I'm just bad.
>hand math professor a donut>pour boiling hot coffee into the donut>coffee goes through hole in donut>professor gets 3rd degree burns all over their arm>professor is hospitalized>professor tries to sue me>I bring topology specialist to court for expert testimony>judge says I shouldn't have poured boiling coffee into the donut>topology specialist explains how donut is a coffee cup>case dismissed
>>15441468Are you implying you pour your coffee through the fucking handle of the mug?
>>15441071>I can just look up whatever I need from ChatGPTyou can't effectively use chatgpt unless you already know what it's talking to you aboutif you try to build a bridge with it you're going to kill 5000 people because the model spat out gobbledigook and you used it anyway because you were too ignorant to recognize bad information
>>15441480GPT is dumb and gets problems wrong, I'll prove this later today
>>15441472Where else is the coffee going to go if you pour it all over a donut
>>15433578man, this threads went to shit
>>15441071>ineering tables, and any math that I need to perform for cost calculations can just be done in wolfram alpha.wolfram is almost completely useless for mathematical optimization unless you are already very well-versed in the subject, and if you are already well-versed in the subject then you'd be using anything but wolfram alpha
>>15441233>The smart people, who are comfortable to solve problems outsize the ZFC formalism, and the people who are best at mathematics, were always physicists.hahahahahahahahaha
>>15433622critical miss
>>15441480Oh fuck lol bridges are gonna start collapsing because idiots who cheated their way through college and lied through their teeth for job interviews are gonna use AI to completely design bridges without checking any of the machine's work.
anyone else think the planet is much larger than it seems to be and we're actually in the middle of an epic?
>>15438699>>15438821Do you have any further questions about this?
>>15439014I remember in the previous thread you were working on a solution to this problem >>15434090Are you still interested, now that your final is finished?
are there any good texts on rewrite systems? stuff like confluence, normalization and so on
Textbooks by committee and webassign math homework is astronomically more organized and superior to any faggot with a huge ego that thinks only he can single handedly understand the best way to teach a course. If your intent is to make it more difficult to learn to waste student's time and filter out the lazy, sure, use your own curriculum. But if your intent is for people to actually learn concepts and receive constant immediate constructive feedback, you aren't going to beat webassign and books like Stewart.
What hole? >:(
Springer's "Texts in applied mathematics" series is probably the best there is. Maybe a good runner up from Mcgraw Hill's international series in pure and applied mathematics, but that has gone to shit in print quality and selection for a while now. Early on I really wanted to collect Bourbaki, but to be perfectly honest most of it doesn't interest me nor is it in any way pleasant to read.
>>15441387fuck you useless bastards, I figured it all out myself
>>15439363mind the glass house friend
>>15442518>telling people how they should teach coursesNobody cares, and you're not good at math. Nobody here is good at math besides me. You're all lazy, dumb, gay pony nerds. Cloppers. Absolutely incapable of doing anything useful, if your life depended on it. You cloppers couldn't prove the Pythagorean theorem if I pointed a gun at your heads. You should all be very ashamed of this fact. Utter failures. Go have sex. Oh wait, you can't. Your autism is so advanced that it causes physical deformities. Your skin is pale and curling from lack of sunlight. Your hair is thinning or bald. Your pubes are overgrown, and your teeth are yellow. You are a disgrace of a human being.
>>15442572Hope you find some sort of contentment in your life, anon. Rooting for you
>>15439143No. Let X be a continuous compact manifold and consider the map from global continuous functions (aka A) to the stalk of the structure sheaf at some point p. Take the kernel.An explicit example: continuous functions on S^1 (parametrized by a variable in (-pi,pi]). Say f and g are equivalent if there is a neighborhood of 0 where they are the same function. Define addition and multiplication of equivalence classes in the obvious way. This gives a ring S and we have a natural surjective map from A to S. Also, check S is an integral domain and that it has a nonzero noninvertible element. Isomorphism theorem shows the kernel of the map is prime but not maximal.
In pic related there is an exercise which asks to prove that f(m,n) = 2^m(2n+1) - 1 is bijective (for the purpose of showing that N^2 is equinumerous to N). I have proven that it's injective, but am struggling to show that it's onto. My approach was to break it into even and odd cases. When even, finding the ordered pair that maps is straight forward but I can't figure out how to map odd numbers. any advice?
>>15443116[math]0\in\mathbb N[/math], right?Given x in N. Look at x+1 and divide by 2 until you can't (maybe 0 times). That puts it in the form [math]x+1=2^m\cdot\text{odd}[/math] where m is the number of times and the odd part is positive. A positive odd number can be written as 2n+1 where n>=0. So [math]x=2^m(2n+1)-1[/math], hence surjective.
>>15443116Letting t=2^m(2n+1)-1, observe that this implies that t+1=2^m(2n+1).Now consider breaking t+1 down into its prime factorisation.
>>15443130>>15443133Thanks and checked. Obvious in hindsight but I guess that’s always the case.
Is [math](f \circ g)^3 = I[/math] possible to prove?[math](f \circ g)^2 = I[/math] is trivial, so [math](f \circ g)^3 = I[/math] reduces to proving [math]f \circ g = I[/math] or otherwise that [math]f = g[/math]. I feel like I'm missing a "trick" of some kind, please give me only a hint if possible.
>>15443834Consider[math]S = \mathbb{Z} \\f(x) = -x \\g(x) = x[/math]
>>15443878Yeah, after I posted I started to consider concrete cases and looked at the wikipedia page for involutions, and now I'm thinking that if [math]f \neq g[/math] then you can't prove [math](f \circ g)^3 = I[/math]and it's not specified that f and g are equal, so there's no proving
I am so burnt out on studying and anything math and uni relatedHow do I keep going brosIs it over for me
>>15443907You can at least prove that[eqn](f \circ g)^n = \begin{cases} I & \text{for n even} \\f \circ g & \text{for n odd} \end{cases}[/eqn]That is because the group generated by f,g is a finite generated abelian group so you know there are only very few possibilities. Namely it's isomorphic to either [math]\{0\}, \mathbb{Z}_2, \mathbb{Z}_2^2[/math]. In either of the possible cases every element has order 1 or 2 so even powers vanish.
>>15443917Yeah I wrote down the same thing breaking it down into even and odd cases, except I don't have the algebraic(?) vocabulary and knowledge you do so I kind of just left it at that.Thanks for the help, I'll google all the unknown words when I get home later.
ask me anything about the platonic solids
>>15443936how can I cure my schizophrenia?
>>15443942have you tried prayer, fasting or exorcism or spending a prolonged time away from the computer?
to summarise the algebraspeakA group is just a set (in this case, f, g, and I) with an operation (function composition) that's associative (function composition is by definition), has an identity (I), and where every element has an inverse (operates on itself to get I; every element here is self-inverse).It's Abelian if it commutes (f(g)=g(f)).It's finitely-generated if it can be produced from repeated operation of a finite number of elements (in this case, every element can be produced from some combination of f and g).Technically this wouldn't account for everything, because the integers with addition are also a finitely-generated Abelian group and there certainly aren't "very few" of them, so we want to throw in "free" as a qualifier, too (i.e. we can produce any element in the group in one unique way, up to cancellation, from a set of generators, here f and g, in finitely many operations)
>>15443931>>15443973Forgot to include a point on the first go, forgot to actually reply on the second.Fuck this, I need to get to sleep already.
Retard here, i'm getting into math.In a videogame you the way you gain energy is Energy gain = (max energy - current energy) * 0.1If I started from 0, how would I find the number of iterations required to cap energy?
this will blow your mind-1 * -1 = 1
>>15444182letting EG, ME, and CE stand for the obvious1: EG = ME * 0.1, new energy = 0.1ME2: EG = (ME - ME * 0.1) * 0.1 = 0.9ME * 0.1 = 0.09 ME, new energy = 0.19ME3. EG = (ME - ME * 0.19) * 0.1 = 0.81 ME * 0.1 = 0.081 ME, new energy = 0.271 MEetc.in general the pattern we see is that at iteration x, your current energy is 1-(0.9^x) times your maximum energyStrictly speaking, the relation as defined means that you'll never actually see equality, but presumably there's a smallest unit of energy that counts. In that case, you'd just keep going until 0.9^x times your maximum energy fell below that floor
>>15444198Thank you, that helped.
How many math courses did you take during your undergrad? I'm almost done and realized I only took 16 math courses....
>>15444190Check this out[math]\Pi_{i=1}^{2n}(-x)^{2m+1}=x^{(2m+1)\,\cdot\,2n}[/math]
>>15443973>>15443977Thanks. It's all clear now. I look forward to learning more about algebra in the future so I can solve some exercises and get a better feel for all the definitions
>>15443914take a break for a day or two; it's not the end of the world
Do non-math specific study guides help with learning math? I'm referring to stuff like "How to read a book" and "Learning how to learn".
>>15440624american math graduation exam
Should I take a PhD position I got offered? The university is world top 50 in the meme rankings and the research group is good.I just can't see the point of doing research if I'm not the new von Neumann, Grothendieck or Lovász.Fuck, I wasn't even that interested in math before uni. Producing garbage papers for the following 3 or 4 years just sounds like mugging myself. Not that I'm too keen to enter my wagie-cage, but it sounds like a more mature decision.
>>15444820If you're not sure, the answer is probably no.Would you spend your free time right now to learn as much as you can about whatever that phd is about?
>>15444852>Would you spend your free time right now to learn as much as you can about whatever that phd is about?Yeah, I really like the area. Teaching undergrads is also fun. It's just that I'm not into producing papers for the sake of producing them (which is what PhD positions are about unfortunately).
Suppose I take an ideal I (of a commutative ring R) that has no zero-divisors, and then I create a new set by replacing 0 with 1.That's multiplicatively-closed, right?
>>15444820>I just can't see the point of doing research if I'm not the new von Neumann, Grothendieck or Lovász.I cannot think of a worse way to live your life than this.
>>15444820>>15445254The way that I see it, I figure that even if I don't become a luminary in my field of research, I can at least do my part for a larger group project beyond myself which benefits knowledge, extend study into a pet theory, or maybe have a little rule or hunch which gets some consideration in the literature. Maybe it's a sucker's thinking, but I have money at the moment, prospects for if I need to quit, and I'm not getting any younger.
>>15444315what does it mean???
>Do a calculation>Feel like a genius >Read a mathematical proof>Want to kmsAny cure?
>>15445725
>>15445741I will give it a look
>>15445054yes, the only thing to check is 1*ideal in ideal and no product ab of either 1 or ideal element is 0
>>15435796Some are connected, some not at all.I recommend you find a general field of mathematics that interests you, and then build your knowledge base around it.If you can tell me what your main field of interest in mathematics is (in detail or very broadly), I can provide you a study plan that will give you good guidelines.If you're not sure and you'd like to have a taste of everything before you decide, that's absolutely fine too and this is what is generally done in a typical math major in the first 3 years, to this end I'll recommend you a few books.Algebra: Michael Artin's Algebra is an excellent book covering many topics of undergraduate algebra, which also gives you insights into more advanced developments. In the topic of Linear Algebra, you may want to also check Friedberg's book, even though Artin's book covers most of what you'll need.Analysis: I think Michael Spivak's Calculus book is pretty good as an introductory text, also has lots of intuitions developed, not just the definitions, lemmas, theorems. You'll have decent foundations with these two, and I think you'd be ready to basically take into detail any kind of specialized topic you want from there. I didn't recommend 1 million books, but keep in mind that studying in detail even one book is a very large amount of work, since you should look into each proof, each exercise and there should be exactly 0 point that you don't understand. I think the journey is worth it, though.
>>15445807Since I'm dealing a bit with set theory, truth tables/logical operators, and formal grammars, it seems like discrete mathematics is probably a good idea to learn as a whole.
>>15433578>mfw its msets all the way to the bottom
>>15433578do solids that intersect themselves count as platonic solids
I know it's consistent with ZFC that there exists sets of different cardinalities whose powersets have the same cardinality, and that their non-existence is implied by a weak form of GCH. Is it consistent for a larger set to have a strictly smaller powerset?
>>15446809strictly smaller than the smaller set's powerset, that is
>>15446809No, a surjective map from X to Y will induce a surjective map from P(X) to P(Y)
>>15444820How did you get your PhD offered? We're you grades that good? Would love to hear.
>>15445725yeah, you read more proofs
>>15445406it's a losers' thinking.Von neumann et al never gave a fuck what they would become, they did what they were good at.By thinking that way you're making sure you never become one of "them".
It's crazy how something as beautiful as math can be easily turned into something painful by a shitty teacher/system.
>>15446809iirc you can force N<X<R such that PN and PX both have the cardinality of R. The non-existence is ruled out by CH.The powerset operation is monotone and growing (already the cofinality of the image is growing)https://en.wikipedia.org/wiki/Easton%27s_theorem
good sources for mathematical modeling and dynamical modeling pls thnks
>>15447121Humans will always be subjective no matter what i bet even that one tao guy has his biases.
i wonder if ill ever ba able to publish a thesis and get a phd
>>15445406>The way that I see it, I figure that even if I don't become a luminary in my field of research, I can at least do my part for a larger group project beyond myself which benefits knowledge, extend study into a pet theory, or maybe have a little rule or hunch which gets some consideration in the literature. Maybe it's a sucker's thinking, but I have money at the moment, prospects for if I need to quit, and I'm not getting any younger.the phd thesis will crush you if you think like this
>>15447732>the phd thesis will crush you if you think like thisWhat’s the proper mindset for a thesis?
what is the strict (ie. in particular irreflexive) dual to a preorder? or more concretely; "an apartness relation is to an equivalence as an ??? is to a preorder". is there a common name for ??? here (especially in constructive mathematics)
>>15447739that science is not the be-all end-all in democracy. Even in a vaccum science has very poor intellectual success beyond sheer materialism. The general population just doesn't give a shit. the administrative side is a burden and the career oriented people too. A phd thesis is a thankless job and unless you are a genius, your work will be absolutely sterile (at least on the theoretical side of science, if you do an experience you may be able to turn it into some meme company) and nobody will care much.>>15447803preorder is really the most elementary relation i think
>>15447803if you care about constructive maths, always go on nlabhttps://ncatlab.org/nlab/show/preorder
>>15447803https://ncatlab.org/nlab/show/irreflexive+comparison
why was lawvere so obsessed with hegel?
>>15446998It's a European thing. If your diploma thesis advisor is pleased with you and has money for a PhD student, he offers you a position.
How important is geometry?
>>15448210What's your goal?Geometry is not very important to influencers on the Tik ToksGeometry is markedly more important to pure abstract +3 greased nerds called Mathematician
>>15447840>democracy>science has very poor intellectual success>general population>administrative sidewhy do you comment on topics you have no understanding of whatsoever?
>>15433578If you roll those dice do you really get a random result or can you always find a bias if you keep rolling?
>>15447996My answer is "Why wouldn't he be."
>>15445406>>15447840Wagie work is more "sterile". Even if you aren't a luminary you will miss the "noble air" of research work.t. left phd program with masters to be a wagie
generals usually have a link of some sort of resources for the topic so why doesn't this thread have it? like a collection of books or some videos or a website. do i just read the wikipedia page about math?
I fail my math midterm on series and series of function. How do i cope?
>>15440523Arguably stokes theorem.
How good are the Bourbaki textbooks? I can't remember them recommended ever.
>>15450664Not super nice to read and, at least for the foundational stuff, outdated (in the sense that their framework were not adopted). I started them once - e.g. in their logic they define \exists from a perplexingly strong choice operatorhttps://en.wikipedia.org/wiki/Epsilon_calculusAll their texts are very terse, although it might be that Dieudonne's analysis texts are worth it, I'm not sure.https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9
>>15451046>Not super nice to readOh, that's a shame.>All their texts are very terseNot necessarily a bad thing I reckon.
What are your favorite "problem books", /mg/?for me it's the Kaczor & Nowak Analysis series
>>15451480>Indian editionGood morning, sir!
>>15451480Lovász: Combinatorial Problems and Exercises,Valean: (Almost) Impossible Integrals, Sums and SeriesBoth can boost you far, I believe.
>>15450589>>15440523This anon. The integral of the differential is the original form. Fundamentals theorem of calculus is a special case of stokes theorem
>>15440968>modern "infinitesimal" calculusIt' 60++ years old and I don't know anybody fluent in it
I enjoy my job and it pays well but it's a dead end. I have an opportunity to do a PhD at a good uni (by Australian standards, it's about 50th in the world) in the maths department, the topic is loosely scientific machine learning which I'm very interested in. The supervisor is great. I'm a 30 yo boomer, I would be dropping down to half time at my work (2 days a week) and doing the PhD at 80% full time load (3.5-4 years total). Does this sound viable? After an untaxed PhD scholarship + another scholarship I'm eligible for, my income won't be significantly lower than it is now working full time anyway. Would appreciate advice from more experienced mathfags
Is [0,1] really uncountable? The proof seems to be fucking wrong for some lists. For example take the list[eqn]0.900000 \ldots \\0.080000 \ldots \\0.008000 \ldots \\0.000800 \ldots \\\vdots[/eqn]then[math]x = 0.988888 \ldots [/math] and[math]x' = 0.899999 \ldots = 0.9 [/math] which is in the fucking list.
>>15453114The list you give is not a complete list of the numbers in [0, 1].
>>15453121This doesn't change that the diagonal argument used to prove that a list of all numbers in [0,1] doesn't exist is logically wrong.It will fail for all lists of the shape[eqn]\begin{matrix} 0.& a & 0 & 0 & 0& 0 & 0 & \ldots \\ 0.& *& 8& *& *& *& *& \ldots \\ 0.& *& *& 8& *& *& *& \ldots \\ 0.&*&*&*&8&*&*& \ldots \\ \vdots \end{matrix}[/eqn]Where [math]a \in \{0,1,2,\ldots,8\}[/math].Maybe one of those list does in a fact list all numbers in [0,1].
>>15453154Any of those lists couldn't include a number that doesn't have your choice of diagonal digit in it. A number whose decimal expansion doesn't have an 8 in it and isn't 0.a00000..., for example.
>>15453154The proof is saying that if you claim your list is actually a list of all [0,1] numbers, THEN you get a contradiction.What you've done is conjured up a list where the diagonal argument is not a contradiction. That's fine in and of itself, but it doesn't interact with the proof at all, because you haven't shown that your list satisfies the PRECONDITION of the proof.Only if you can show that it's possible for one of your specially designed lists to _actually_ index [0,1], then it would contradict the diagonalization argument.You've disproved the argument in general, in a vacuum. But the argument only works if you assume the list actually indexes [0,1], it doesn't work in reverse, and it doesn't break in reverse either.
>>15453164Re-reading my post, it's not very well written. Let me try again.The diagonalization argument says if you have a list AND you suppose it indexes [0,1], then you run into problems (contradictions).You've found a list with a special structure (the a, 8, 8, .. thing) where the diagonalization argument doesn't create a problem, but if you can't show the list with the extra special structure you added actually indexes [0,1] and actually exists, then you're not attacking the real diagonalization argument, you're attacking a made up variant with a structure that you haven't shown actually exists (and in fact, doesn't exist)A problem is your book shows real numbers in the list like 3, 5, 8, but of course you can't assume any property of the list, otherwise you'd have to prove the list actually has that property. The real diagonalization argument doesn't do that, so it just works. You added some extra condition that happens to be impossible, so you've only shown that assuming an impossible case, then [0,1] is countable..
Is infinity confusing? How do you define it? How is what he says about it false?
>>15452796it's fine, anglo countries have no problem with a 30 yo student starting a phd
>>15450664Bourbaki is really like an encyclopedia
i should work on my money making algorithm
>>15453667You shouldn't. There's a team of 12 PHDs who have been perfecting it for the last 20 years, and you're 1 person.You'll be earning negative dollars per hour if you pick an unfair fight in your disadvantage.
>>15453675why are you so mean? i just want to make a lot of money using math
>>15453292
>>15453683True criticism is future kindness.I also want you to make a lot of money using math. If I didn't care, I'd just sit on the sidelines and observe people fall for the unmarked trap that every new entrant in a field falls for.
>>15453292He has no formal system, he's just ranting angrily. Not that I vehemently disagree with him - it's just no strong base for a proper discussion.
>>15452796thanks mate the more advice i gather the more people are saying yeah it's not a bad idea.
I finished a calculus 1 course recently (also discrete math which gave me a bit of proof writing) and on a whim, I decided to prove out the derivative rules for myself since I was curious.When I'm going over the proofs, I feel more like a kid imitating an adult than anything. It begins to feel a little pointless because I'm not going into academic math (studying cs to write enterprise software) and I don't know how much mileage this is giving me for learning future math that I would be more interested in like linear algebra/combinatorics/statistics.Is there really any point to me proving out these tools that I'm using in computational-based undergrad calculus given the above?
>>15453954being able to do proofs is probably one of the most generally-useful parts of mathematicshelps you learn how to write out a coherent argument rather than just waving your hands with the old "it just works"
>>15453114their argument is wrong, yes. they should map all digits to a different digit that lies in {0,...,8}, which their given mapping does not do (lol) They probably want each digit to increase by 1 mod 10, except map 8 to 7 instead.
>>15453954> Is there really any point to me proving out these tools that I'm using in computational-based undergrad calculus given the above?It helps a good bit with critical thinking and better programming. Keep in mind that ~40-50% of your competitors in the market are gonna be Indians with diploma mill degrees who don't know about half the bullshit they'll spout in their interviews, and the majority of the rest have only somewhat of an idea. Programming is indeed mostly trial and error and learning to rely on a few core toolsets after experience, but when a problem arises, thinking outside the box like making proofs may be needed. In addition, you may want to take a serious look at data analysis as a possible alternative to programming, that may be where your bread will be buttered in the market. For example, I worked as a "technical consultant" teaching other impatient Indian and Chinese developers how to paste a three-line API snippet into someone's codebase and hating my videocall-filled life as a programmer who didn't program, while the data analysts used their software to track the incoming user metadata and calculated whether they were fraudulent or not, a bot attack was happening, etc. using graphs and statistics. Go ask the daily programming and tech workers generals on /g/, they should have some better insight.
>>15453980>>15454008ty
>>15453292..............another word for "Split" is "Schism".Just sayin'...
How do you calculate the density of a number? As in how much digits are closer to 9 than to 0.
>>15453292>hat says split>doesn't believe in Dedekind cuts
>>15453685based
Whenever I get the impression math nerds and even CS chads deal with too much abstract bullshit "pure math", I get violently brought down back to Earth when I see how business and econ majors literally just study calculating for semester after semester, obligatory assignment and in-person workshop after assignment/workshop.They just study how to balance a check book, and which the tools are.Then the climax, the "intellectual orgasm", is statistics, where you mindlessly regurgitate standard deviations, covariance, etc.It's seriously mind numbing.No beauteous links between that statistics to information theory and boolean algebra. No study of convex functions etc. as their own thing instead of just a tool.Studying business/econ is just training to become an Excel plugin.Math and many concepts of CS could almost be regarded as a separate discipline from that accounting garbage. Completing the triangle between "pure math" and "accounting/calculating" is the vertex of "algorithms/CS topics".
>>15455178> Whenever I get the impression math nerds and even CS chads deal with too much abstract bullshit "pure math", I get violently brought down back to Earth when I see how business and econ majors literally just study calculating for semester after semester, obligatory assignment and in-person workshop after assignment/workshop.Welcome to the business world, anon :)Much of it consists of learning arbitrary tips from some other famous business guy’s podcast who gives them advice such as at least 2% the emails of any cold email campaign will be opened. Business relies more on instinct and building connections with other people, throwing shit at a wall until it sticks than pure concept. People are cattle by and large, you just find a way to lead them to your watering hole. How else would cattle be able to make money if business was about pure concepts? We all gotta eat, after all> Studying business/econ is just training to become an Excel plugin.What do you think they use to plug their numbers at the highest levels of government and business? And when you’re a number to them, how else do you think they look at you?https://www.youtube.com/watch?v=W7Jqwpnw9Lo
>>15453114You are correct. Very good job at finding the flaw in the proof. Personally, Rudin's proof is way cleaner than the diagonalisation argument, and it works for any perfect set in any Euclidean space. You actually have to increment the digits by 2 mod 10 (or 3 or 4, but not 1). If you generate a number this way, you are guaranteed to find a different number i.e., any two number which differs in any digit by 2 mod 10, must be unequal. See this for the complete proof https://youtu.be/CvalbBGhmW4>>154539810.9... and 1 are equal numbers that differ by 1 mod 10 in digits.
Hey i need some advice for grad school (Machine learning masters)I feel like im messing up my grad school dreams and im not sure how to go about what i need to do. I wanted to apply for grad school and get in by next year jan but i think the window has closed( I had to do some mandatory things that pretty much made it hard to attend to it by jan/feb).I also wanted to do the gre exam to help boost my poor gpa but with the windows closing on next year im not sure i have that luxury anymore.
If you put two small circles randomly within a big circle what's the probability the two smaller circles intersect? All locations within the big circle equally likely and independent for the two small circles.
Start by considering the probability distribution of a point on a line segment of length D. This is wrong because it's an outer circle and this give you the answer for an outer square, but whatever it's a startSay your two circles are centered at (a, b) and (c, d) inside that square.The PDF for a point on a line segment of length D is trivial, it's 1/D for all points of the segment.a, b, c, d are all picked independently, so you can combine their PDFs linearly I'm pretty sure (at least for a square, you can..)For any four points a,b,c,d, the probability that this is where your two circles are is D^-4.Now the distance between your two circles is l = sqrt((c-a)^2 + (d-b)^2))Let f_r(l) = { 0 if l > 2r; 1 if l <= 2r }Do the quadruple integral of f_r(l) times the joint PDF D^-4 over a,b,c,d, and I think that gives you the probability of two small circles intersecting in a big square.Can't be bothered to think about the circle case, I dum. Whatever.
>>15453954You will absolutely need to know the proofs in order to actually REAL WORLD use any calculus you learn or more than likely in differential equations. You are not going to be doing any of this shit by hand like in school, you will be using matlab or some ancient code base in pascal/Fortran and trying to determine how to numerically solve all of these things or update them. If you enjoy the applied side of math which will more than likely drop you in the PDE rabithole, do yourself a favor and go heavy on courses like numerical analysis, and CS 1,2,3 (programming 1, 2, algorithms), or if you're lucky any applied physics courses you can fit in with your electives.
>>15452639Green-Ostrogadsky theorem is also the analytic version of it; but it doesn't really "tensorify" as well. It can be generalized to nastier boundaries though.
why is inifinity a thing? if you measure the volume of the entire universe in planck (the smallest unit of measurement) its 4.65*10^185
>>15456739What gave you the idea infinity has anything to do with physical reality?
>>15456747like in calculus when you find the limit as x approaches infinity... im just using the size of the universe in the smallest unit as an example that inifinitys not really relevant in real situations
>>15456804> inifinitys not really relevant in real situationsWhat does that even mean? It's well know that any time an infinity appears in a physics theory that means there is a problem and usually you are trying to use the theory for something it no longer applies to. There are no infinities in nature but that doesn't mean you can't use calculus to model it.
>>15456739Infinity is the reciprocal of the infinitesimal. If the infinitesimally small exists, then mathematically so does the infinitely large.You mention Planck lengths- I don't think there's any proof that this is "the objectively smallest length", only that we would need new physics to describe interactions at scales smaller than this.
>>15456829>any time an infinity appears in a physics theory that means there is a problem and usually you are trying to use the theory for something it no longer applies toyeah, but what if you just... kind of ignored it? Wouldn't that be nice?t. QED
>>15456860You are partially right. The Planck length is the smallest distance that can be measured or described, it is not the smallest possible length. Space isn't quantized.>>15456916They are not ignored. They are removed using the method of renormalization.
>>15457021>They are removed using the method of renormalization.In other words, ignoring them and pretending that they don't exist, a fact which has been understood since the very beginning.
>>15457033Somehow it works
>>15457033Not true. That's like saying contour integrals ignore singularities in the complex plane. There are mathematical tools to deal with them. As for why renormalization works and what it, if anything, it physically means ... yeah, that's still an ongoing discussion.
Sup dudes, link me the best resource to understand how to prove a limit from first principles, particularly in 2 variables.Thanks.
24 | 0.1540 | x50 | 0.75if the first value is 24, the second one has to be 0.15! If the first value is 50, the second one has to be 0.75! Help me solve for x. If the first value is 40, the second is...
>>15457648If the left value is [math]a[/math] then the right value is[eqn] - \frac{2157}{8320} a^2 + \frac{15981}{832} a - \frac{32397}{104} [/eqn]You can check this by caluculating that[eqn] - \frac{2157}{8320} \cdot 24^2 + \frac{15981}{832} \cdot 24 - \frac{32397}{104} = 0.15 \\- \frac{2157}{8320} \cdot 50^2 + \frac{15981}{832} \cdot 50 - \frac{32397}{104} = 0.75 [/eqn]So[eqn]x = - \frac{2157}{8320} \cdot 40^2 + \frac{15981}{832} \cdot 40 - \frac{32397}{104} = 42[/eqn]
>>15457021>You are partially right. The Planck length is the smallest distance that can be measured or described, it is not the smallest possible length. This is not a theorem or anything, that's more a rule of thumb>Space isn't quantized.Don't talk as if QFT is the end all be all, we don't know shit
I'm having some trouble understanding Poisson's distribution formula.[math][e^(-μ) * μ^(x)] / x![/math]I don't think I currently have enough math knowledge to properly understand any proof.But If it is possible, I'd appreciate if someone could share at least an intuitive way to make sense out of what happens.Thanks beforehand for any reply.
>filtered by knuth>filtered by rosenfuck discrete math
>>15458171Do you know how to solve a differential equation?I'm not sure if there is some easy property that is evident from the formula you gave that lets you understand why it is the way that it is, but it's not too difficult to derive it from a few basic ideas.If you start with some ``process'' that generates arrivals independently over time (that is, one arrival that occurs has no influence on another arrival), and if for some very small interval [math][x,x+dx)[/math] the probability of an arrival in this interval is linear in its length ([math]P(\text{1 arrival in }[x,x+dx))=\lambda dx[/math]), the formula basically rolls out.Try it!
Is there a practical application of math theory in 2k23, i.e. are new mathematics being developed or is it just a history lesson?
>>15458171https://youtu.be/3z-M6sbGIZ0
>>15458253Probably the most notorious instance of "new mathematics" is the whole shitshow with Mochizuki and the abc conjecture
>>15458253>are new mathematics being developedyes>Is there a pratical application of math theory in 2k23that is a completely unrelated questionapplications for new maths are up to physicists and engineers and other math-adjacent fields to discover
>>15458244surely you wouldn't get filtered by epp
>>15458250I don't know how to solve differential equations. I guess I might still have stuff to learn to interpret a proof like that.I'm thankful for your reply, anon.>>15458264This one seems possibly helpful, as well. I couldn't grasp it yet, but I would also consider it a bit longer, and maybe come back, and asks questions again.Once again, thanks to both.
>>15458253There are applications but it won't be discovered until 2 centuries later.
Can someone tell me whether the null matrix can be considered a skew symmetric matrix? (I need to prove that it can be a subgroup of the addictive 3x3 matrices group
>>15459912A matrix [math]A[/math] is skew-symmetric iff [math] A^T + A = 0[/math].Have you tried calculating [math]A^T + A[/math] for [math]A = 0[/math]?
>thesis advisor suggest i should enroll into the phd program>later i find out that he only said because the college needs to fill out certain quotabruh
>>15460196how did you find out?
N00b here, how do I get Mathematica to evaluate this integral and not spit out a monstrosity? How do I get something neat like https://physics.stackexchange.com/a/176805/329254 ?[eqn]\int _{-\frac{L}{2}}^{\frac{L}{2}}\int _{-\frac{L}{2}}^{\frac{L}{2}}\frac{1}{\left(h^2+x^2+y^2\right)^{3/2}}dxdy[/eqn]
>>15460726Add something like ", Assumptions->L>0&h>0" to the end within the bracket, see what happens. If the "conditional" part vanishes you're in good shape. If it still looks like shit though, using Simplify[%4] on the output might help.
>>15460723it came to me in a dream
I'm interested in studying the Zeta function and its analytical extension from the perspective of the Euler Product Formula. (See the first example here: https://en.wikipedia.org/wiki/Euler_product#Examples )I would like to find the zeroes of analytical extensions of finite product approximations of the Euler Product Formula. (e.g. just using the first 3 factors, or just the first 10 factors, etc.). Can I do this using Python code? What libraries would I need? It would be especially nice if someone has already done something like this and published it on github but that is probably hoping for too much.And if I can't do it using Python code, is there another way?
>>15461124Thanks but it's still not great
https://scottlocklin.wordpress.com/2023/05/20/further-books-may-2023/>"I’m not going to pretend to be any kind of mathematician">proceeds to talk shit about all post-19th century math>"muh Arnold"Should his opinions on math and physics be taken seriously? I mostly know of this dude from his rants against quantum computing and string theory.
>>15437822Now I suspect this isn't the flavour you were expecting from an answer, but if you swapped functional analysis with harmonic analysis, then a satisfying answer may be Tate's thesis, where you fashion all possible completions of [math]\mathbb{Q}[/math] -that is, [math]\mathbb{R}[/math] and all [math]\mathbb{Q}_p[/math]- into a single object, called the ring of rational adeles [math]\mathbb{A}_{\mathbb{Q}}[/math].Now, just like you can do Fourier analysis on [math]\mathbb{R}[/math], and when your functions are [math]\mathbb{Z}[/math]-periodic your Fourier transform becomes discrete, and your Fourier inversion formula becomes a Fourier series [math]f(x)=\sum_{k\in\mathbb{Z}}\widehat{f}_k e^{2\pi i kx} [/math]), you can do Fourier analysis on [math]\mathbb{A}_{\mathbb{Q}}[/math], and when your function is [math]\mathbb{Q}[/math]-periodic your Fourier transform becomes discrete, and your Fourier inversion formula becomes a Fourier series [math]f(x)=\sum_{k\in \mathbb{Q}}\widehat{f}_k e^{2\pi i \Lambda(kx)}[/math] -don't worry too much about [math]\Lambda[/math], it's a sort of projection from adeles to rational numbers-
>>15461525>not a mathematician>should his opinion on math be taken seriously?..obviously not
this feels as if I were solving some extremely autistic and pointless puzzle... and it's weirdly pleasingthis whole chapter on permutations was pretty interesting.
>>15458773solve navier stokes, then we'll talk
>>15461498maybe add assumptions to the simplify if possible. Or replace ArcCot[x] with ArcTan[1/x] and try simplifying. Or just use your brain at this point and use arctan subtraction formula to simplify (might have to care about branches which might be why mathematica is not simplifying)
You mean you want to find the values s where [math]p^s/(p^s-1)[/math] is zero?I don't see how this is different than studying the individual factors p^s
>>15461961
>>15462125>Or replace ArcCot[x] with ArcTan[1/x] and try simplifyingI did, didn't work>Or just use your brain at this point and use arctan subtraction formula to simplifyi mean, that's not the point here, I have the solution alreadyi just want to get mathematica to solve it, that's the whole point of using something like mathematica right?
>>15460726>>15461124Without those assumptions, this integral mathematica a long time to evaluate on my machine. Was it the same for the both of you? I also couldn't improve on the answer systematically through mathematica alone. Will try later.You may also want to post it on the Mathematica stack exchange, you'll usually get a very informed response in a short time. If you or your institution have support, you can always send it to wolfram directly, and then they are obligated to help with it. Why do this? Because often customer complaints end up as test-cases for the developers to work on. I don't work there, but in a similar situation, that was my procedure for building up test-cases and to prevent regression errors for future releases.
>>15462347I don't like this notation for the permutation.Normally you write (n, n-1 ,...,1) for the cyclen -> n-1 -> ... -> 1 -> nbut this problem only works if interpret it as the permutation1 -> n2 -> n-1...n -> 1That is the product of the floor(n/2) transpositions that swap k with n+1-k.
>>15462511Do you think the authors give a shit what you like as a notation?
learning surface integrals for the first time lads this is sick
I came up with an interesting math problem.The idea is that you have a road construction company. The square grid in picrelated represents a grid of cities - each dot in one square is a city. The government asks you to build roads such that the network of roads makes it possible to travel from any city to any other city. But building road costs money. So you want to minimize the amount of road that you must build to connect the cities.The problem is this. How long is the minimized road for the n:th grid? For example, if n=7, then the number of cities is 49. Also assume that the distance between two adjacent cities (horizontally/vertically) is one unit.I already put the correctsolution for n=2 where the road is marked with the red line. For n=2 you could solve the length of the road with some basic trigonometry but I have no idea how you would do it for the general case.
>>15462605Why isn't the solution for the n=2 case just an X of roads between the four cities, with the roads meeting in the center?Also, for higher n grids, is it ok if the road path from city A to city C passes through city B?
>>15462592Maybe something like this, i have no idea
>>15462626I actually solved it real quick. For X-shape the road is [math]2 \cdot \sqrt{2} \approx 2.82[/math] but it can be minimized to [math]\dfrac{1+\sqrt{3}}{2} \approx 1.36[/math] (the red shape in picrelated)
>>15462680What are the angles at the vertices of your n=2 solution? Is it just 120-120-120?
>>15462680I think you are right that that shape is better, but you are off by a factor of 2 in calculating its length: The actual minimum is about 2.73. In any case, how sure are you that the solution isn't something like pic related, just kind of scaling up your n=2 solution?
>>15462690Oh I made a mistake. The real solution is sqrt(3)+1 = 2.73 but it's still shorter than the X-shape.The function in picrelated is 2*a+b as a function of the angle x. So if you minimize this function and multiply it by two you get the solution and the angle.
>>15462719I was kind of thinking about that but even if it is optimal, you would still have to prove that it indeed is shorter compared to something like >>15462629
>>15462738Yeah I have no idea how to prove it. My n=3 guess has a length of about 7.46 and my n-4 guess has a length of about 13.66. But I think I'd have to do a lot of math to calculate the length of your n=3 guess, and even if that were bigger I don't have any idea how I'd rule out all other n=3 guesses and prove that mine is minimal.
>>15462738Or it could be like this also, like i said i have no idea.. i saw a video about it many years ago but i think drawing "diamond" shape in the middle solves the basic one so i just did that and i have no idea further
>>15462754Basically the final part is beyond me
>>15462605https://en.wikipedia.org/wiki/Steiner_tree_problem
>>15462754I found something related from the internet
>>15461777what about Arnold? Is he a meme?
Is there anything fun and math related I could do as a CS project?
You know what sucks? Being a mathematician and getting stumped by a problem in an undergraduate textbook. And when I ultimately solve the problem, I'll feel like a double retard for taking so long.
>>15433578Does anyone know the significance of adding summations together giving alternating squares? So far I've only found an application to Pascal's triangle.
>>15463548You should feel dumb for feeling dumb.Learn to be okay with being retarded. If you resent it, it'll push you away from putting yourself in situations where you'll feel retarded for learning something simple.But that's a big mistake.Very clever people sometimes have to go over simple stuff. Take it in stride. You only look like a fool if you complain about not being able to solve the simple problem.Just take it slow and take it in stride until you have a solid unshakeable understanding. It's fine.
>>15463765https://oeis.org/search?q=1%2C+4%2C+10%2C+20%2C+35%2C+56%2C+84%2C+120%2C+165%2C+220&language=english&go=Search
I heckin' love math
Has Shulman published anything since being canceled?
>>15433628eaT 4GEW
How do you guys store your work solving math problems? Pen and paper, latex? Personally I use Mathematica notebooks, with a lot of custom styles to get a "near" LaTex experience with the convenience of WYSIWYG editing and of course built in graphics/plots. That said I wouldn't say Wolfram does a good job explaining how useful their software can be for this, so I'm not sure it's very common. The question applies for both school work, and personal learning.
>>15463548>undergraduate = easy>graduate = hardthat's not how it works buddy
>>15464071me when solving optimization problems
>>15464266did i miss something?
>>15437641reeing old man just like people against zero, then negative numbers, then imaginary numbers etc or physicists that argued for luminiferous aether, then determinism etc
how many of you are doing an REU this summer?how hard is it to find small research projects in the fall?
>>15465091guess I did kekhttps://github.com/HoTT/book/pull/1101#issuecomment-1065813889https://groups.google.com/g/homotopytypetheory/c/L53FHA2gDGI/m/7TTK96UCAgAJ
>>15465081Like the one that I just posted?
>>15464266http://home.sandiego.edu/~shulman/papers/index.html
>>15465292that's more of a combinatorial/discrete problem i was thinking more about regular optimization problems with smooth and convex/concave functions
Bros, how do I catch up in calculus? I barely passed differential calculus in college by the skin of my teeth. Now, we got integral calculus. I feel like I have a very shaky foundation in calculus as a whole. How do I catch up? I don't really have the time to start from scratch.
>>15465772See, I typed a long comment and accidentally pressed ESC, so fuck that.But the TL;DR is sign up for fucking Khan Academy right now.Go back from the start to the point where it's easy and you're bored to tears, and speed through it at 1.5x.It takes less than a day to fast forward through the stuff you're supposed to already know.But then, ACTUALLY stop and slow down when it stops being able to follow without any effort.You can pass with just the foundations and not knowing any advanced tricks. But the other way around where you memorize a couple tricks while being hopelessly confused is a disaster.So even if you don't have time, any time you spend on the foundation is more effective than stumbling around on more advanced topics and not really understanding shitOpen your search bar, search some online course site, and watch that shit for the next several hours. Right fucken now, punk.That's what I'd do.
[eqn]1 = \left( 1 - \frac{2}{3} \right) + \left( \frac{2}{3} - \frac{3}{5} \right) + \left( \frac{3}{5} - \frac{4}{7} \right) + \ldots = \frac{1}{3} + \frac{1}{15} + \frac{1}{35} + \ldots = \left( \frac{1}{2} - \frac{1}{6} \right) + \left( \frac{1}{6} - \frac{1}{10} \right) + \left( \frac{1}{10} - \frac{1}{14} \right) + \ldots = \frac{1}{2} [/eqn][eqn]1 = \sum_{k=1}^\infty \left( \frac{k}{2k-1} - \frac{k+1}{2(k+1)-1} \right) = \sum_{k=1}^\infty \frac{1}{4 k^2 - 1} = \sum_{k=1}^\infty \left( \frac{1}{4k - 2} - \frac{1}{4(k+1) - 2} \right) = \frac{1}{2} [/eqn]
>>15466062congratulations, you just learnt about the Riemann rearrangement theorem.
>>15466067The series is absolutely convergent by the comparison test[eqn] \sum_{k=1}^\infty \frac{1}{4 k^2 - 1} < \sum_{k=1}^\infty \frac{1}{3 k^2} = \frac{\zeta(2)}{3}[/eqn]
>>15466269Oh, right. At least the first one is technically correct mathematically; that one's just wrong, because the claim that the first series in that one sums to 1 is blatantly incorrect.
new thread, please
>>15466431Make your own thread.
New thread>>15466521>>15466521>>15466521
