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[
"Please suggest a design for a random open-source Lottery"
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"math"
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"efeash"
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"Removed - not mathematics"
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true
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0.62
] | null | On another note, bitcoin doesn't "use too much energy to allocate coins" because the point of mining isn't money allocation but rather blocks creation. Randomly allocating coins would be the end of bitcoin because miners would stop creating blocks and transactions would take weeks to be validated. | Sunspots validated against what observation? Are you proposing a quorum of solar observatories that vote on the correct sunspot prediction? Because then, that quorum holds real control over the currency. Or, for weather and day/night considerations, perhaps you were thinking of relying on a single space observatory? (It seems unlikely that there can be enough of those with overlapping observations to make voting between them viable.) That observatory, and all the comms infrastructure between it and whatever network endpoint miners read its observations from, then become a single points of failure/takeover. This is beside my other, more fundamental objection: the lottery winner will, statistically, be whoever can generate the most predictions. So it's still a resource-intensive proof of work system, just one with less controlled characteristics. | It doesn't use the bulk of the energy for block validation either, just for reaching consensus on them. With better consensus algorithms, this can be improved. | You want something where there is not even a shred of doubt that it can’t be manipulated. So I wouldn’t use stocks or sports. Sun spots sound good. | No matter how the lottery is run, under your scheme, voting on the next block will still be computation resource intensive. It's just that instead of a proof of work with defined and engineered characteristics, control will go to whoever can generate the most lottery ticket instances. There's another weakness with a lottery scheme of they type you propose: dependence on external observations. With Bitcoin, all you need to validate a block is the blockchain up to that point. Think about the vulnerabilities inherent in depending on external data -- let's say, the state of the stock market at some defined point in time as reported by some selected authority. Finally, think about the task of accountably (ie in a decentralised way) selecting the lottery winner from a number of entries that is only governed by the speed at which entrants can generate tickets. The last thing a currency needs is to be limited by network bandwidth. Having contemplated these matters, you might come to the conclusion that, for all Bitcoin's downsides, Satoshi Nakamoto came up with something pretty clever. |
[
"Self-study mathematics"
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"math"
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"efgsml"
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"Removed - post in the Simple Questions thread"
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true
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0.33
] | null | It depends on your mathematical background and what you are interested in. If you have enough background for college /university level mathematics i would suggest to learn linear algebra and real analysis. Those two are much of the foundation of modern mathematics, and makes you ready to further study complex analysis, group theory and in general abstract algebra and even some topology. Books i like for these fields are Linear algebra done right by Axler, Understanding analysis by Abbott, Principles of analysis by Rudin, Introduction to linear algebra by Strang | Would it not make more sense to start from where you are right now? If you have studied geometry/algebra you can go on to trigonometry/calculus and so on. It is always helpful for recommendations to know where you are starting from. | Would it not make more sense to start from where you are right now? If you have studied geometry/algebra you can go on to trigonometry/calculus and so on. It is always helpful for recommendations to know where you are starting from. | Khan Academy is what you are looking for. It has some of the best contents available on topics like calculus and linear algebra | It’s great how enthusiastic you are about learning math! Definitely keep in mind that you will never gain a complete understanding of math. |
[
"Schrodinger Equation"
] | [
"math"
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"ef2t5x"
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2
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"Removed - post in the Simple Questions thread"
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true
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false
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1
] | null | I am not sure where literature is on this but I will list the things you will need to understand it. I have no idea where you are right now, so feel free to ask about things. You can probably get to some understanding as long as you know some calculus, but the less you know the harder it will be. That being said, you can treat this like a road map more than a list of what you should already know. Non-quantum math stuff: *A strong understanding of calculus and algebra, including some knowledge of multi-variable (which isn’t too hard to pick up) *What eigenvectors if linear transformations are *First order differential equations (air resistance or cooling are good examples) *Second order differential equations and complex exponential (oscillators) *The real wave equation using partial derivates, especially understanding all of the parts of a solution (waves in a rope are a good example) Quantum stuff: *What a wave equation represents *What an operator is and what the total energy and kinetic energy operators are *What each term of the Schroedinger equation means, which will help to understand where it comes from *what an eigenstate is and why we care about them *How to get from the time dependent to time independent Schroedinger equation and why that is useful Then you are ready to solve the Schroedinger equation. Edit changing my claim about large element based on other responses: O ne you are here, the textbook I used in university was called “Quantum physics of atoms, molecules, solids, nuclei, and particles” by Eisberg and Resnick. I though it was helpful for this and other quantum topics but I don’t have experience with others. I know a pdf copy exists online You will need to use and understand a few more concepts to do what you want. *Gradients and laplacians *spherical coordinates and how the laplacian works for these cases *The single electron case To keep going is a little beyond my pay grade (as I am more math focused and still working on my degrees, so I don’t have as deep of an understanding of these cases) but I’d still be happy to answer some questions. Best of luck! | Yeah and I know there is distortion but I have no clue how to account for that | Yeah and I know there is distortion but I have no clue how to account for that | Unfortunately, your submission has been removed for the following reason(s): /r/learnmath books free online resources Here If you have any questions, please feel free to message the mods . Thank you! | I think understanding the solution of the hydrogen atom would already be ambitious for a high school student. Multi-electron species require numerical methods or approximation schemes, building on the exact (-ish) solutions available for hydrogen. |
[
"Why is 3/3=1?"
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"math"
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"ef6rhg"
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0
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"Removed - post in the Simple Questions thread"
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true
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false
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0.35
] | null | FYI this question is best suited for the “simple questions” post, but here’s an answer: 0.999(repeating) really is the same thing as 1. You can google “0.999... = 1” for a bunch of discussions about this | did you put in infinitely many '9's? | .9999... repeating is not rounded to one, it is precisely one. .999... - 1 = 0.0000... or just 0 | it does equal 0.999... forever, but that number is also equal to 1. There is no "last little bit", an infinite string of 9s is the same thing as bump the first 9 and carry the 1. Same applies to any terminating decimal. 0.123 = 0.1229999999.... PS This belongs in Simple Questions or r/learnmath . Merry christmas | They are wrong. |
[
"What to do to stay in shape?"
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"math"
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"ef3jwt"
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1
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"Removed - post in the Simple Questions thread"
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true
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false
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0.53
] | null | i'm finding the workout and orgasm metaphors a little cringey | oh you mean differential equations don't make him literally ejaculate? Well in that case this language is not cringey at all. Thanks for setting me straight. | Of course man, anytime | you mean differential equations don't make him literally ejaculate? The e in e stands for ejaculate | I could’ve sworn he was joking ... right?? |
[
"Recommendations for Undergrad Complex Analysis Text"
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"math"
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"ef1kcw"
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60
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"Removed - post in the Simple Questions thread"
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true
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false
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0.96
] | null | I think most people like "Visual Complex Analysis" by Needham. | ok i consider libgen to be online available, so u can get freitag - complex analysis, it covers a huge range of complex analysis with an extra chapter on analytic number theory as application. (it was recommended in the 4th term of my physicist math) | Needham is great alongside an additional rigorous textbook. I'm not sure I would rely on him alone though. Still, hands down one of my favorite books, and I think it is a must-have. | Not afraid to mention a bit of cohomology How does this work? In my mind, cohomology would put the work firmly at graduate level. There are conceivably students who take their first complex analysis course alongside their first group theory course, let alone algebraic topology. | The undergrad course I took used Complex Analysis and it's Applications by Brown & Churchill and I found it very hard to study on my own. |
[
"Who is able to reach the formula of the bisectors of 2 lines that meet on the plane ?"
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"math"
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"ef0gig"
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"Removed - incorrect information"
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false
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0.17
] | null | If anyone is interested on all the demonstration i can search a way to share it, but it’s a little long, 3 pages | Just convert them into linear equations and then set thrm equal | Unfortunately, your submission has been removed for the following reason(s): If you have any questions, please feel free to message the mods . Thank you! | use the "reply" link under a comment to reply to an individual comment | Would u be able to find the equation by yourself? |
[
"What's this procedure called?"
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"math"
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"ef2bqd"
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0
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"Removed - post in the Simple Questions thread"
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true
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false
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0.2
] | null | It would X * (+/-percentage) called Using exponents. Easier to write this than complain about the simple nature of your question. | I don't think there is a single standard term for this, but "negative compound interest" should get the point across. | If they spent 30 seconds reading the sidebar they'd know there are several places that better suit the question. If they can't bother to give that small amount of respect, why should I bother to answer their question? | That helped. Thanks. | Unfortunately, your submission has been removed for the following reason(s): /r/learnmath books free online resources Here If you have any questions, please feel free to message the mods . Thank you! |
[
"Quadratic Formula Alternative"
] | [
"math"
] | [
"efrs3h"
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0
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"Removed - incorrect information"
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true
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false
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0.35
] | null | I’ve seen this so much, just all over the place. And yes, it’s great, I teach it to my students, yadda yadda. But it is not “new” as this man himself mentions several times - just completing the square. | That’s the thing - I still think there is a lot of pedagogical value to teaching this method alongside completing the square, and letting students pick up the similarities. Also, of course my students (that haven’t seen the quadratic formula) HATE the guesswork that comes up with factoring quadratics - so this is a very welcome and comforting method for them too. | Ah, darn. Funny thing is I do vaguely remember completing the square, but I guess the way he did it was different enough to where I didn't notice it. | Yeah, I figured everyone would either downvote it or someone would act too gatekeepy | Unfortunately, your submission has been removed for the following reason(s): If you have any questions, please feel free to message the mods . Thank you! |
[
"I just thought of something"
] | [
"math"
] | [
"efkkrn"
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0
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""
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true
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false
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0.25
] | [deleted] | 1001 = 7*11*13. | Is 10001 a prime number? | https://www.wolframalpha.com/input/?i=factor+10001 Are you just trolling? | Didn't think about that crap | ...you didn't think about whether those numbers were actually prime, before saying that they were prime? |
[
"Is there a math ebook on Springer that is less than £5 (or £10)?"
] | [
"math"
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"effkg7"
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189
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""
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true
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false
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0.94
] | [deleted] | Make sure to avoid this website, it has many free ebooks and is definitely not what you are looking for: https://en.wikipedia.org/wiki/Library_Genesis | OP is buying a physical book for £39.99. They need to spend £40 to get the discount. The idea is to purchase a cheap ebook in addition to the physical book to reach this amount. | Presumably to meet the £40 sale threshold | Presumably to meet the £40 sale threshold | Presumably to meet the £40 sale threshold |
[
"Moser's Algorithm and the Lovász Local Lemma"
] | [
"math"
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"efv9kc"
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32
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""
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true
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false
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0.89
] | null | Slightly related. I've always wondered how people speak about results named after them. Turns out, Lovász simply called the lemma "local lemma". We didn't even know it was invented by him, he presented it as simple tool for the thing we were learning. It was only after the fact that we realized the rest of the world called it "Lovász local lemma". | Does graduate student count? :) My impression is that this kind of stuff wouldn't come up too often outside of academia. I mean, people use SAT solvers in industry, but they are more sophisticated than Moser's algorithm and don't require LLL conditions. | I had the same experience when he taught our class about his theta function. Humble guy and a thoughtful teacher. | Lots of computational algorithms would use it. As it says in the article, satisfiability problems, graph colorings, tree codes, integer programming etc. But more importantly, does it matter? No one thought Fourier series would have any use outside of heat equations and it helped with DSP centuries later. So many other areas of math, like number theory, has no use at the time they were worked on but now have great applicability with cryptography etc. Besides, sometimes math is fun for its own sake - whether or not there’s any applicability. To quote Hardy, “A mathematician, like a painter or a poet, is a maker of poems. If his patterns are more permanent than theirs, it is because they are made with ideas.” | Possibly dumb question, in the proof of the entropy compression theorem, what is the n in the bound on |f(Y)|? |
[
"Stuff I learnt in 2019: papers, code, math"
] | [
"math"
] | [
"efm6tt"
] | [
477
] | [
""
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true
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false
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0.96
] | null | How can you read some many books so quickly and yet retain all the information you read? | So just read more is your method? | So just read more is your method? | Grothendieck didn't invent sheaves and sheaves weren't invented in algebraic geometry. | Thanks for this. I get hung up on exercises a lot and that impedes my progress with the maths books I'm reading ;) |
[
"Merry Christmas everyone!"
] | [
"math"
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"ef95cu"
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108
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""
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true
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false
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0.93
] | null | Merry Christmas! It reminds me of the last few pages of Contact from Carl Segan. | I used a "template" from this post , changed the 8 to a 9 and run a python script that would change the the numbers that were not 1's or the first 4 numbers to random numbers until it was a prime, I used the Miller-Rabin to test if it was a prime. | I used a "template" from this post , changed the 8 to a 9 and run a python script that would change the the numbers that were not 1's or the first 4 numbers to random numbers until it was a prime, I used the Miller-Rabin to test if it was a prime. | KAZE NO YOU NI | The musician in me is forcing the mathematician in me to read the non 1 numbers in a Christmas song i simply cannot remember the name or lyrics to. |
[
"Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+..."
] | [
"math"
] | [
"ef2war"
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648
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""
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true
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false
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0.98
] | null | Kinda weird that he linked 3blue1brown's basel problem video and not the video covering exactly this series for calculating pi. Anyway, good video. Liked grant's approach a bit more, but I think this is more accessible to someone that's not familiar with complex numbers. | I just learned that the 2019 point occurs before the Feynman point. | Yes, but the identity predates calculus, and does not require it to show. There is value in having multiple perspectives. | Yes, but the identity predates calculus, and does not require it to show. There is value in having multiple perspectives. | Mathologer gave a talk at the Australian maths society conference three weeks ago (he is a maths prof at Monash uni). One very surprising thing he said was that he does not watch YouTube. |
[
"Thanks to everyone on the Math subreddit for your regular discussions of Numberphile - just released this for Pi subscribers"
] | [
"math"
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"ef2vh6"
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414
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""
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false
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0.97
] | null | I’ve heard some people suggesting that you ask YouTube to lock Numberphile’s sub number to 3.14 million+, like they did with the 301+ video Either way, seriously congrats Dr. Brady! It’s no exaggeration to say that you (and your peers, with you very much as a leader) have worked to revolutionize education and the pursuit of knowledge. Hats off and Nail and Gear’s proudly flown for you! | Awesome channel. Quite amazing growth and I can’t wait for the tau million sub milestone! | Might be bit of a wait. ;) | Merry Christmas Brady :) | Thank you. |
[
"How do you find interesting Math papers on arXiv?"
] | [
"math"
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"eff1uq"
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35
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""
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true
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false
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0.83
] | First, I know Mathematics is a huge subject and "interesting" is just a relative term. But whenever I try to go to arXiv there doesn't seem a way to browse papers and find ones that might interest me, it is just browse by new and by recent, any tips on browsing the website in a better way and finding Math papers that might interest me? | i subscribe to the RSS feeds of the fields i'm particularly interested in; cf. https://arxiv.org/archive/math , which has links to the 'new' page for various arXiv categories. | The way I find interesting papers, not just in math but in general, is by googling for something that interests me, then following references (i.e. backwards in time) and citations (i.e. forward in time). Browsing arxiv, while nice way to spend time, is too overwhelming with too many papers on too many subjects. It's like looking for a needle in a haystack. | The arXiv is intended to be a record of every math paper that gets published, so if you want to successfully use it to find stuff, you need to have some concrete idea of what you're looking for. I use the Mathematical Subject Classification to search the arXiv for the specific tags I'm interested in. | You subscribe to an RSS feed by adding a link to a feed reader ( https://en.wikipedia.org/wiki/News_aggregator ). The feed reader then regularly polls that link to check for new items, and displays them. RSS feeds allow one to have articles 'pushed' to you (à la email - you don't have to keep refreshing multiple Web pages to discover new items) without having to actually provide personal details (such as an email address). | You subscribe to an RSS feed by adding a link to a feed reader ( https://en.wikipedia.org/wiki/News_aggregator ). The feed reader then regularly polls that link to check for new items, and displays them. RSS feeds allow one to have articles 'pushed' to you (à la email - you don't have to keep refreshing multiple Web pages to discover new items) without having to actually provide personal details (such as an email address). |
[
"What Are You Working On?"
] | [
"math"
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"efi5dp"
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18
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""
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true
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false
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0.85
] | This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on over the week/weekend. This can be anything from math-related arts and crafts, what you've been learning in class, books/papers you're reading, to preparing for a conference. All types and levels of mathematics are welcomed! | Trying to understand tensors for physics! It came up first in my Electricity and Magnetism class and I haven't been able to quite grasp what it is, and after A few different explanations that didn't quite work for me, I realized this was about to be a trip. Here's the YouTube channel (playlist) I'm currently working my way through. | Told my supervisor I would try to understand (g,K)-modules and their uses over the break because it's highly relevant to my research, so there's that. I'm taking a course in algebraic topology next semester, but apparently the prof is retiring after this semester and can barely write on the board anymore. Also we're using his text, and in the first 2 minutes of reading it I found several errors, including his claim that the trefoil knot is isotopic to the unknot... So, I'm also trying to teach myself some algebraic topology from reputable sources before this wild ride of a course begins. | The trefoil is isotopic to the unknot. In order to define knot types you need the slightly more subtle notion of . Without this extra subtlety all knots would end up being equivalent. | Doing some stuff with bi-exact groups, which are a generalization of hyperbolic groups... they are confusing 😅 | I’m waiting on my copy of Prime Numbers and the Riemann Hypothesis by Barry Mazur to be delivered. |
[
"Questions about mathematics as deductive system"
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"math"
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"efd7t8"
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14
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""
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true
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false
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0.81
] | Hello, I have a few questions about mathematics as a deductive system in which mathematical statements follow as a logical consequence from axioms and other statements. In particular, consider the real number system and Euclidean geometry. Is doing arithmetic with numbers and working on theorems that follow deductively from axioms of Euclidean geometry simply a "game of logic"? The real numbers and Euclidean geometry are useful and in real life. Why is that so? One idea I could think of is that axioms of real numbers and Euclidean geometry are not chosen without any meaning. They are set up with the aim of mathematizing natural observations. So how do the axioms eventually lead to mathematizing natural observations? Say the axioms are literally descriptions of our world. Then I guess the deductions of the axioms will also describe our world closely. What happens if we derive from the axioms a statement that is different from observations in our world? (Have this occurred in Euclidean geometry?) Edit: Thank you for the replies, they helped me a lot. So before looking at a proof (at least in basic Maths), I should think intuitively why the theorem is true with natural observations or my experiences. Many times, I have a problem in doing so. Below are 3 basic theorems (in Euclidean plane geometry and Real number system) that I have a problem in doing so, and can you suggest what I can do, or what are your thoughts about why the theorems must be true intuitively. My thoughts about theorem 3, In contrary to theorem 3, we can want to have a number x that is less than 1/n for any positive integer n. Then, people in the past would have come up with a different standard list of axioms for real numbers. However, we do not want to have x in our real number system, hence leading to the question, "Why wouldn't we want such a number x to exist in our real number system?" I think, suppose (in real life) a physical object U representing the unit (i.e. 1), and no matter how small another physical object V is, we try to cut U into sufficient parts, whereby one part is smaller than V. But, can we, or no? If we can cut U into sufficient parts, whereby one part is smaller than V, then this idea leads to we should not have x in our real number system, and if we can't, then we will have the x in our real number system. My problem is, I can't decide on this. | The idea that math is a game of logic is most often attributed to David Hilbert. Imo, it's a much better starting point for understanding philosophy of mathematics than, say, Platonism, which hopelessly leads you to go in circles thinking about math. However, I never understood why people say it is a game of logic. Numbers are meaningless symbols with certain rules given by certain axioms, but they can be interpreted in meaningful ways when you apply them as a model of the world (like when you say something is 3 feet). I feel like this point would present no challenge to a formalist, yet I've seen one quote that presented this as a refutation of formalism for some reason. You might find my post here interesting where I basically repeat what you said at the end of your post. In particular, the thing that distinguishes the real number system from other ordered fields like the rationals is the completeness property (or, equivalently, the least upperbound property). The completeness property is chosen precisely because it mathematizes our intuition like you said. | One idea I could think of is that axioms of real numbers and Euclidean geometry are not chosen without any meaning. They are set up with the aim of mathematizing natural observations. they are. Geometry's etymological roots are literally "earth measure". The whole point is to take our natural understanding of behavior in the real world, codify, and see what follows. So how do the axioms eventually lead to mathematizing natural observations? Depends on the axiom. Often, it's quite literally just a description of what we've experienced. In the case of geometry, that's pretty much all axioms. For instance, one axiom is that given two points you can construct a line. That's it. That's just something we "know" from real experience. That systems with axioms like these represent reality well should not surprise us: it's exactly what we were trying to do when we wrote them! If they weren't useful, we wouldn't talk about them. Other times axioms seem a little less natural, but have developed over years or more of simplification and trial and error. Group axioms, set theoretic axioms, a lot of them have a long background where it started out as something bulkier or less formal and over years of tweaking what we have today is where we landed. And then there's the axioms that we don't so much observe to be true as we wish them to be true. For instance, the completion of the reals is something we have not really observed*. Rather, it is an immensely desirable property that happens to be missing from the rationals and so we simply create a better set that has it. But are the reals really real? The full set of reals is, for all practical intents and purposes, overkill. But it's so incredibly versatile, its axioms wonderfully succinct, and its properties so nice that it's earned its supremacy in virtually all mathematics and scientific fields. does this thing we made up end up being so incredibly useful for understanding reality? Did we just get lucky? Honestly, I don't know. This is one of the most famous questions in the philosophy of mathematics, with a name dating back to a paper from 1960: the of mathematics. It has been tackled by many great minds, with very little consensus. If you ever do find a satisfying answer, please share. * It does come from our though. Like many axioms, completeness is an assumption that observed behavior in finite situations ought to carry over to certain infinite situations. | The idea that math is a game of logic is most often attributed to David Hilbert. While this is true, I do think it's important to note that Hilbert's formalism was strictly a tool and he personally wasn't a formalist. It's late as fuck so I'm just going to cite two sources : "Hilbert Vindicated?" by Hintikka and "The Last Mathematician from Hilbert’s Gottingen: Saunders Mac Lane as Philosopher of Mathematics" by McLarty. One can also look through his motivations in much closer detail by looking at "Hilbert's Programs and Beyond" by Sieg. | unreasonable effectiveness of mathematics The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960). New York: John Wiley & Sons, Inc. Copyright © 1960 by John Wiley & Sons, Inc. It can be found at this link. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html | Is doing arithmetic with numbers and working on theorems that follow deductively from axioms of Euclidean geometry simply a "game of logic"? Depends on what you mean by "simply". Certainly mathematics is a "game of logic", but oftentimes we're trying to work towards some goal in real life. The real numbers and Euclidean geometry are useful and accurate in real life. Why is that so? Because we invented the real numbers and Euclidean geometry to be used in real life. If they weren't accurate, we probably wouldn't be using them as much. One idea I could think of is that axioms of real numbers and Euclidean geometry are not chosen without any meaning. They are set up with the aim of mathematizing natural observations. Yes, that's exactly it. So how do the axioms eventually lead to mathematizing natural observations? Because the axioms are just natural observations that we take for granted, phrased as mathematical statements. What happens if we derive from the axioms a statement that is different from observations in our world? (Have this occurred in Euclidean geometry?) This is literally how non-Euclidean geometry was founded. Essentially, people started with the first four of Euclid's postulates, and then, instead of the fifth one (equivalent to the statement "given any line and any point not on that line, there is through that point parallel to the original line"), they took one of the other two cases ("given any line and any point not on that line, there is through that point parallel to the original line", and ("given any line and any point not on that line, there through that point parallel to the original line"). The first one leads to geometry, of which spherical geometry is an example (a straight line is a circle whose centre is the centre of the sphere, so any two straight lines must intersect in two points). The second leads to geometry. Both are perfectly consistent, and have been studied by mathematicians (even though the latter does not seem to accurately describe anything in the physical world). There's a general agreement in mathematics that your system should not contain any unnecessary axioms (i.e. axioms that can be proven from other axioms), and this is part of the reason why; by concentrating just on the necessary axioms, you can ask yourself what happens if such an axiom is changed. Does your changed axiom give a consistent model? Is the model interesting? Is it applicable to real life? |
[
"Homotopy Type Theory is the future?"
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"math"
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"ef793r"
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] | [
0.48
] | Reddit Mathematicians, do you think that Homotopy Type Theory (HTT) will have the same impact (or maybe even greater) then Cantor Set theory as a pillar for mathematics? (Disclosure: I'm not a mathematician) | No and that's fine. | It would be pretty hard to move mathematics away from its current set theoreical foundations for many reasons, probably the most important one is that there is no need for it and homotopical thinking doesnt seem to cover big areas of math in a way that favors it. I cant see it being as revolutionary as set theory was for mathematics, but I can see it maybe getting a more established niche among certain mathematicians. | Well, I’d say that most people don’t actually care about foundations. Moreover, I’d say that most mathematicians know what ZFC and Russell’s paradox are, but they don’t actually know the axioms of ZFC or how it works. Bottomline, people just use sets. So it’s possible that future mathematicians think of sets within HoTT rather than ZFC, but I’d bet that they wouldn’t actually know much HoTT. It’s a long shot though, since such a change would be met with friction. It would all depend on whether HoTT really does provide a substantial conceptual advantage over ZFC. I certainly believe it does and work like that of Jacob Lurie’s might help motivate non-logicians to appreciate the wonderful internal logic of infinity topoi. So... only the future will tell. | Is it tho. | No. |
[
"Yesterday I accidentally posted a broken link, so here is a working version for my first ever proof write up that Generalized Root Finding and Optimization is NP-Complete!"
] | [
"math"
] | [
"ef5am5"
] | [
8
] | [
"Removed - incorrect information"
] | [
true
] | [
false
] | [
0.68
] | null | There are uncountably many functions from R^n to R, and most of them are uncomputable, so these problems aren't well-defined. You should specify which countable set of functions you're considering, and how you're encoding them as inputs to your algorithm. Same goes for your regions Ω. | There are uncountably many functions from ℚ → ℚ also, so most of them are still uncomputable. You'd still need to specify which ones you're encoding and how (which will affect their size). | Why would optimization be O(n+m) for dimension n and "function size" m? If I make a grid in R on which I evaluate the function, I will have on the order of k points. And even then I have to restrict the function to be differentiable and |f'|<L, or else I can hide a minimum so that you wouldn't ever find it in 1D even. | I know that optimization has been done before, but in my Google Scholar searching I haven't seen anyone say root finding is NP-Complete. Im still an undergraduate so was considering sending this into an undergraduate journal if y'all think its worthy Any feedback appreciated :) | NP-hardness of "root-finding" as you call it is folklore, even when the function is an explicit (written as a sum of monomials) polynomial of degree 4 and there are no constraints on the domain. It's great you rediscovered a way to prove it though! See if you can strengthen your proof to get the above, and see if you can find interesting ways to go further. |
[
"Is there a visual/geometric interpretation of fractional derivatives and integrals?"
] | [
"math"
] | [
"ef2mb3"
] | [
34
] | [
""
] | [
true
] | [
false
] | [
0.89
] | So... I've started dabbling in fractional calculus a few days ago and although I find the results and formulas quite exciting, I've found myself wondering if there's an intuitive, geometric interpretation of the topic. My gut says no, but if anyone knows any literature on the topic, it would be much appreciated :) | There definitely is a geometric interpretation of fractional derivatives and I'll try give mine without trying to be rigorous (as I am aware there are plenty of complications in the story I'm going to try to tell). Also in part I hope that someone jumps on me and corrects what is untenable and perhaps sharpen how to think about this. There are two starting points for this: 1) The geometric interpretation of integer power derivatives and 2) The behavior of the objects in 1) when you generalized the exponent to the (non-negative) reals. Regarding 1), derivatives are tangent objects to a manifold (function, curve, surface etc). The first derivative is a linear, the second a quadratic, the third a cubic, the forth a quartic, the fifth a quintic etc. Each derivative locally approximates the corresponding algebraic curve at that point in tangency to the manifold. For a conic view of the first four derivatives in this picture see this discussion . Regarding 2), Our tangent objects are no longer polynomial curves, they are now curves with exponents in Q or R (as appropriate). Roughly and for our purposes, we will think of curves inbetween integer exponents as intermediaries (compare discussion of exponents ). Hence our tangent objects are curves that in the proper exponential way are between the integer order exponential curves. This completes my working intuition of fractional derivatives in simple geometric settings. This basic approach can also be found in this paper though somewhat more physically framed (see the capacitor charge depiction). In the context of simple quotients (square root for example) I have seen other geometric interpretation attempts but failed to find the one that I had in mind. I'm sure some googling will reveal those! | They have practical applications in physics, in particular quantum field theory when you want to do renormalization. We often deal with divergent integrals which can be regularized by analytically continuing into non-integer dimensions. | I'm not sure that dimensional regularization counts as a rebuttal to a claim that something is "deeply unphysical". Dimensional regularization is a mathematical trick that has little or no physical meaning, despite it finding applications in literal physics computations. Dimensional regularization is not fractional calculus, so I don't know if this example has any relevance to the current discussion. | There's this cool gif over on one of the Wikipedia articles https://en.m.wikipedia.org/wiki/Fractional_calculus#/media/File%3AFractional_Derivative_of_Basic_Power_Function_(2014).gif | Yep that picture gives the essential idea of step 2) in what I said in words. |
[
"UNPOPULAR OPINION: FIBONACCI NUMBERS ARE OVERRATED AND ACTUALLY USELESS"
] | [
"math"
] | [
"bwdii1"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.27
] | null | See: Fibonacci Flim-Flam . | The golden ratio does show up in weird places, like in the change in error for numerically calculating f(x)=0 using the secant method. That doesn’t mean it’s special I guess, but it’s cool that this constant pops up occasionally. | Fun fact: Given a triangle ABC (in Euclidean space), sin(A) + sin(B)sin(C) ≤ φ. How's for "weird places"? | You're right that the Fibonacci numbers, and especially the golden ratio, are overrated. There's a lot of silly pseudo-science around the golden ratio (for instance, I think the claim about the golden ratio being "beautiful" might be related to the myth that rectangles where the ratio of height to width is φ are the most attractive rectangles). Part of the problem, I think, is that people sometimes feel like they have to justify math to the public with contrived examples from "real life". Not only are these unconvincing, they're counterproductive: if you go around giving obviously fake examples to justify something being interesting, people are going to see through that and conclude that it's not interesting. That said, while the Fibonacci numbers are overrated, they're definitely not useless. The Fibonacci numbers are a great example of a recurrence relation and, as a prototypical example, come up unexpectedly in all sorts of places in mathematics. (My favorite example is their role in the proof of Matiyasevich's theorem solving Hilbert's 10th problem.) | Read the whole whole wikipedia article on them... https://en.wikipedia.org/wiki/Fibonacci_number There is a journal on them and related topics (old issues' articles are free to read on their site): https://en.wikipedia.org/wiki/Fibonacci_Quarterly |
[
"advice for an aspiring mathematician"
] | [
"math"
] | [
"bw8dfu"
] | [
10
] | [
""
] | [
true
] | [
false
] | [
0.75
] | null | If they were 1 in a billion, there would be eight mathematicians on Earth. | If they were 1 in a billion, there would be eight mathematicians on Earth. | You do not have to be a genius to do math . There's a lot of unhealthy imagery in the intellectual sphere about, as Tao says, the lone genius. It causes a lot of harm, up to and including people actually working in the field. The fact is that the vast majority of working mathematicians are people who are just good at math. They have a knack for math and an interest to pursue it, and the rest is years of hard work and participation in the community. Even a person who struggled with math but has a love of the subject and patience can make real contributions, and it sounds like you're already past that. Keep doing the curriculum. If you want to go past it, talk to your teachers, or even post on r/learnmath about good topics/books to help you explore farther. Don't be discouraged by not fitting into a mold that was always fictional. | Proof that this entire sub is bots. | At your age, your potential is practically limitless. Practice is key. Anyone who is good at a skill, whether it's playing a musical instrument, playing sports or video games, or mathematics, got there by putting long hours into learning it. Learn to accept that some parts won't be as exciting as others, but you'll still need to put time and effort into it anyway. Some things won't make sense at first, and familiarity and intuition will only come through repetition and practice. Luckily, you live in a golden age of information, and there are tons of videos and websites available that look at similar kinds of problems in different ways, which will be a great help if you get stuck along the way. For what it's worth, try not to think of teaching as settling because you don't think you'd be good enough. You should teach if you love teaching (have you helped classmates with homework, or enjoy telling family what you've learned, etc?), and make sure you're good at that, too (which will also require practice). Whatever it is you want to do for a living, make sure you love what you do. Best of luck to you! |
[
"Should I respect my own time or try to learn/study faster?"
] | [
"math"
] | [
"bwbbq0"
] | [
8
] | [
""
] | [
true
] | [
false
] | [
0.75
] | [deleted] | Depending how you're using those 2 hours, 2 hours a day isn't a low number. Most people have a fairly low threshold for how much serous thinking they can do in a day, especially on a sustained basis. Terry Tao has a great post about this. So I wouldn't recommend trying to force yourself to just learn math faster. The place where there may be gains is to ask whether there's less intensive but still useful things you can do to supplement your studying; for instance, at the level of math you're talking about, you might still be able to drill easier problems or review material even when you don't have more real learning left in you that day. | Study with people. This is way easier once you're in an academic setting but it's still possible outside of school. Definitely do not try to do your engineering degree alone. It's just not worth the struggle. Having people to talk to about a problem allows you to teach others (cementing your own understanding) and to learn from others. Groups also keep you studying for longer. As for now, I wouldn't worry about it too much unless you are really shaky on the basics. Most of intro calculus is just applying trig and algebra so you'll get lots of practice. The actual concepts of calculus are not hard. | funny, i always work way slower with others. which is why i haven't really done it much after one or two tries. | speaking of explaining my work, i basically narrate all my thought-processes in a lecture format anyway, so it works out. yes, mostly distracted, and mostly i find i get a better understanding of things really thinking and talking it through alone than doing it with others. generally others don't really care about discussing anything beyond the problems, anyway. not like i mind, i like working alone. | Hey I'm in a similar boat, relearning linear algebra since my CS course was fairly light on it. I think two hours of focused study is actually quite difficult. If your time permits, you could try chunk it up into several sessions that are maybe 30-40 minutes each. I know Khan academy tend to have 10-20 minute videos so something like watch two, go do some chores, errands or even just relax and try mentally go over what was learned and come back. You'd be amazed how well things stick with just a short break for contemplation. If I'm bored commuting I tend to do this kind of thing, helps you figure out what actually stuck from what you learnt and what you might need to pay more attention to. In the long run hopefully it leads to a better understanding. Hope this helps but I'm sure the other people in the subreddit have better advice! |
[
"Meet Olga Aleksandrovna Ladyzhenskaya: the Russian mathematician who pushed through the Iron Curtain"
] | [
"math"
] | [
"bwoxbh"
] | [
120
] | [
""
] | [
true
] | [
false
] | [
0.94
] | null | She visited the Courant Institute a few times. There’s a well known story about how she was offended, when Peter Lax fell asleep during a talk she gave there. A petition was then circulated, where the signatories attested that Lax had fallen asleep during their talks, too. Lax always sat in the front row and, during every talk, would fall asleep, snoring with his head tilted back. Often, he would suddenly wake up and start asking questions, making it clear that he somehow still understood the talk. | Fuck, this sounds like me, except my questions make it clear that I did not pay attention to the talk in the slightest. | Memories of Ladyzhenskaya by Morawetz. http://topo.math.auburn.edu/pub/2Olgas-proceedings/pa003-morawetz.pdf | savage | My academic great-grandmother |
[
"What are the differences between machine learning and data science in terms of math?"
] | [
"math"
] | [
"bwmt7r"
] | [
109
] | [
""
] | [
true
] | [
false
] | [
0.9
] | null | Machine learning is a subset of Data Science. | For someone who's been in both, my perspective. ML is more research oriented. Here, companies will implement papers and try their own alternate ideas. This tends to be focussed and quite labour intensive and can feel much like academia, but with actual management. My current role has me investigating the CTC loss function and filling in the gaps of negative results as well as trying strategies to improve speech decoding. DS is a free for all trying to apply statistical techniques to datasets. You will not believe the number of times I see people apply xgboost and get good results. I've seen someone use deep learning, when simple Fourier analysis would've sufficed. In many ways, it's returning to software engineering (which is where it belongs), but with a slight statistical bent. ML is far more maths intensive as you'd expect an R&D position to be. Trying new ideas, attempting fundamental changes on loss functions and rationalising the results are very maths heavy. I've never felt anything even close in DS, the knowledge of what the algos were and their properties was enough to get by. This is of course personal opinion and things are different in different companies. At the end of the day, these are just buzzwords and phrases that obfuscate rather than clarify. | It doesn't matter at all which term is older. Data science includes data analysis via machine learning and every other form of data analysis using scientific methods. | Close. But ML includes several topics, even linear regression - which does not mimic how humans learn at all. Neural networks are the only ML technique that were designed to mimic human learning. And the extent to which they do this is heavily debated. | Data science is selected topics from math, statistics, and computer programming. Machine learning is a particularly complicated specialization of these topics that allows computers to somewhat mimic the way that humans learn. |
[
"In what ways might an understanding of homotopy type theory be beneficial for a practicing computer scientist?"
] | [
"math"
] | [
"bwo97q"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | null | By ‘practicing computer scientist’ do you mean ‘mathematician specializing in computation’ or do you mean ‘software engineer’? | Learning it would be useful if you want to know more about: -Formal methods -Just how far you can get with constructive mathematics -Higher category theory in a "natural" setting -Synthetic homotopy theory I think much of the usefulness in practice to a computer scientist would not come from HoTT itself, but rather, the kinds of insights the mode of thinking HoTT provides. Martin-Löf Type Theory is already very elegant in the sense that once you "get" the idea behind recursion and induction, you can begin to see a lot of structures through the lens of logical harmony ( https://en.wikipedia.org/wiki/Logical_harmony ) -- Homotopy Type Theory just extends this with additional harmonies relating to isomorphism and equivalence, and those additional harmonies happen to encode synthetic homotopy theory. As far as practical usage, I think that you won't stand to gain much by learning it. However, you don't need to judge whether or not something is practical to learn it -- "it's fun!" is usually enough for me. | It'll help you annoy your friends | What little programming I had done kept running into problems that two objects a and b could compare as equal (a == b), but two objects F(a) and F(b) derived from them in the same way would not (e.g., because equality was undecidable on their type). This became particularly bad when F(a) and F(b) were dependent types and I couldn't even automatically convert from one to other because it involved knowing the equality a == b came into being. From what I understand, cubical types are meant to solve this one day. | Maybe it's easier to determine membership of a string of computations in a homotopy class than to optimize your cost function, i.e. calling a memoized result representative of that homotopy class might be better than performing even the shortest string of computations in that class. |
[
"Why the Proof of Fermat’s Last Theorem Doesn’t Need to Be Enhanced | Quanta Magazine"
] | [
"math"
] | [
"bwcpg3"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
0.58
] | null | This article seems deeply confused about...well, everything. A logical analysis of Wiles’ proof points to many steps that appear to disregard ZFC, and this is potentially scandalous Basically everything about this is wrong. Wiles' proof invokes some results in EGA which, in full generality, involve ZFC plus the existence of inaccessible cardinals. Putting aside the question of whether that makes it accurate to say that Wiles' proof uses inaccessible cardinals, there's nothing "scandalous" about that; at most, it places aspects of Wiles' proof at a well-understood step just beyond ZFC. Nevertheless — and this is what is missing from the standard philosophical account of proof — each of the 10 would readily refer to their own proof as Wiles’ proof. The fact that a mathematical proof can have multiple representations as a string of formal steps, all of which represent the same underlying proof, is well recognized by philosophers of mathematics, some of whom have spent a lot of time thinking about this and related questions. Indeed, the entire last part of the article makes a string of anodyne observations about the nature of proof and the complex relationship between proofs-as-done-by-mathematicians and proofs-as-formal-objects, and presents them as if logicians and philosophers of math have never seen them before. The closing accusation - that "many who follow number theory at a distance [are unaware] that a proof like the one Wiles published is not meant to be treated as a self-contained artifact" - is entirely unjustified. More broadly, the entire article seems contaminated by a common, but frustrating, misconception: that when logicians talk about formal aspects of a proof, that this is intended to cast aspersions or detract from the mathematical aspects, and that it's intended to be a contribution to the field of that proof. When logicians point out that Wiles' proof appears to go beyond ZFC, this isn't a criticism, it isn't an attempt to restrict the subject to a particular set of axioms, and it isn't an effort to distract people from further work in number theory. Logicians just think it's interesting and unexpected that a proof of an elementary statement should invoke such wildly abstract techniques; we want to know why: what is it about Fermat's Last Theorem that makes it natural to go so far afield to solve it? Harris is right that that's not really a question of number theory; it's utterly orthogonal to the work number theorists have done understanding and expanding on modularity. But number theorists don't own FLT: the proof also raises interesting questions in logic, and it's only right that logicians should be interested in answering those questions. Logicians who study that aren't doing bad number theory or missing the point of the proof; they're doing good logic, and investigating a different set of questions than the number theorists are. | What is the article trying to say? That the recent formalization push is unrelated to the actual matter of FLT? From what I recall, it's mainly science journos who have been thinking otherwise. The proof of the FLT may still be a convenient example for the discrepancy between the logic algebraists claim to rely on and the logic that they actually are using -- but that doesn't mean they are doing something wrong. The author also seems to get Gödel wrong. It does exclude the possibility of formalizing all existing proofs in mathematics. It merely says that there will still be unanswerable questions left. It's a bit disappointing to see a mathematician write about a different part of math while clearly basing his idea on the latter on pop-sci. | Yeah, I know who he is. Pointing out that he's a number theorist doesn't exactly defend him from the criticism that he's writing from a parochial view internal to number theory. (Unlike most number theorists, he's also done some writing about the philosophy of proofs, so he really ought to know better.) | Yeah, proof assistants are all over the place logically, and we aren't as finish with finding the "right" foundations for mathematics as an "intro to proofs" class would make you believe. But this isn't something the writer of the article even touches on. | Not to appeal to authority, but the author is a number theorist at Columbia—if I'm not mistaken. |
[
"Yet another favorites question: what's your favorite integer sequence? (Include an OEIS link if possible)"
] | [
"math"
] | [
"bwageo"
] | [
5
] | [
""
] | [
true
] | [
false
] | [
0.7
] | null | A261811 is up there for me because I came up with it. Been 3 years and I still have no idea how to work out general properties of this sequence (or even whether it's infinite). | A048648 is the orders of the stable homotopy groups of the spheres. It's relatively simple to define within algebraic topology, but it looks completely wild! There are patterns in it that researchers are still teasing out, and will be for some time. | It's a tie between 1,2,3,4,5,6,7,... and 0,1,2,3,4,5,6,7,... | Oh shit that's really interesting! I have been contemplating a few new sequences lately but I have a very weird problem... to make an account on the OEIS they want your real, legal name, but I hate my real legal name, but I can't change it because I don't want to hurt my mom's feelings (she selected it), so... honestly, in general, I need to choose an "academic pseudonym" that I'd be willing to keep, and it's kind of tough. But that's rather unrelated lol - I'd love it if you could just ramble on about whatever you HAVE discovered about this - and how you found it in the first place. Btw, I'm sure you've noticed this, but your sequence appears to contain most multiples of 4. It would perhaps be instructive to look more deeply at those multiples of 4 that it doesn't contain - and those numbers which aren't multiples of 4 but which ARE in the sequence, and see if there's any patterns in those - but you probably have already thought of that. Oh and also - did you ever notice the answer on Stack Exchange where the guy claims it somehow boils down to Bernoulli numbers? I can't make head or tail of it but maybe you can. https://math.stackexchange.com/questions/1417433/when-is-sum-n-0-infty-fracnk3n-an-integer | I like the sequence counting the number of regular polytopes in each dimension ( A060296 ): 1, 1, ∞, 5, 6, 3, 3, 3, ... A polytope is an extension of polygons and polyhedra to any dimension, and a regular polytope is one with equal faces and vertices. Regular polyhedra (dimension three) are called platonic solids, and there are 5 of them: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. There are infinitely many regular polygons (dimension two) determined by the number of sides (triangle, square, pentagon, hexagon, ...). The dimension zero polytope is a point in space and the dimension one polytope is a line segment. In four dimensions, you get 6 regular polytopes (try visualizing them?) and in every higher dimension there are only 3, roughly corresponding to the tetrahedron, cube, and octahedron. Alright, it is not purely an integer sequence, but almost all of the terms are integers. |
[
"How does one derive the integral representation of the Kronecker delta function as stated on it's wikipedia page?"
] | [
"math"
] | [
"bwqjll"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
0.67
] | [deleted] | It has to do with the integrals of monomials. When k is not -1, the integral of z is 1/(k+1) z , which is meromorphic, so the integral around zero becomes zero. When k is -1, the integral is the natural logarithm. When you loop around zero, you cross onto a different branch of the natural logarithm, so the integral is actually 2 pi i. This gives us a way to "check" if k is -1 or not, which extends to checking whether two integers are equal. | I think k needs to be an integer, so no, you set k = i - j - 1 and see what happens for different values of i and j. | I think I do mean meromorphic; it's not holomorphic if k < -1. The important thing is that the antiderivative doesn't have multiple branches. My wording probably could have been a bit better, I'll admit; when I say "so the integral [...] becomes zero," I mean the integral of the original function, which due to the lack of branches in the antiderivative becomes a value minus itself. | Oh ok,so when one evaluates this definite integral, one would take the limit as k approaches -1? | Oh right, thanks man |
[
"What Would Happen If All Encryption Could Be Broken? - Slashdot"
] | [
"math"
] | [
"bw9yin"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.38
] | null | Prove that not all encryption can be broken Proof: Take one-time pads with non-reused keys. Without the key, the encrypted message cannot be decyphered. Therefore there is at least one encryption scheme that cannot be broken. | Not really a math post... you could ask r/crypto or r/security if you're interested in the consequences of broken encryption or the likely community response (rapid rollout of post-quantum methods). | ∎ | A lot of crypto research actually follows that general form. Initial proofs of existence are extremely inefficient, they just validate the research direction. This was a silly example of course, but initial zero knowledge proofs were polynomial time in spirit but useless in practice, and now we're using them for actual monetary transactions. | I see. I can't say I know much about the field so I appreciate the explanation. |
[
"Tribonacci Numbers (and the Rauzy Fractal) - Numberphile"
] | [
"math"
] | [
"bwclyf"
] | [
444
] | [
""
] | [
true
] | [
false
] | [
0.98
] | null | Not needed --- you'd see the same thing over and over again by self-similarity. | I happen to work very closely to this area. I wrote my phd thesis on substitution tilings, including Rauzy fractal substitution tilings. Feel free to ask any questions. | Man, I wish they went more into the fractal | ... good joke, but not what I meant | 𝔉_𝔫 ≈ ⅠⅠⅡⅢⅤⅧ… |
[
"What is your favorite math topic and why ?"
] | [
"math"
] | [
"bwo498"
] | [
9
] | [
""
] | [
true
] | [
false
] | [
0.7
] | Please explain why it is your favorite area and whether you want to pursue that area in current or future research ? | Algebraic geometry because it can describe a lot without over-relying on local coordinates and puts singular objects on the same level as smooth ones, I say as I use local coordinates to do all my GRR calculations and struggle to find references for what theorems about curves pass over to various singular cases. | or connections on principal bundles ? | Dynamical systems, I like it cause it’s really great haha | Really like operator theory and Banach space geometry. Banach spaces are such cool objects! Enough structure to be manageable and enough flexibility to keep things interesting. | There is always one geometer ruining all of the fun (Just joking of course ;)) |
[
"I think I may have proved something cool about lattice points in metric spaces while trying to figure out something else!"
] | [
"math"
] | [
"bwqy8s"
] | [
13
] | [
""
] | [
true
] | [
false
] | [
0.79
] | I seem to have stumbled upon a proof that, in any metric space (at least, any metric space with a norm resembling the Euclidean one but possibly with signs changed), if you take two lattice points P and Q which are 1 unit apart, there exists no third lattice point R whose distance from P is a multiple of 2 and whose distance from Q is a multiple of 4 (or vice versa). This is because, given any space with N dimensions, with a norm which is the sum of ± the squares of the coordinates, if we translate, rotate, reflect etc so that P is the origin and Q is 1 away from it in the direction labeled zero (which preserves distances but simplifies everything): the equation for the distance of a point to P is ±x₀²±...±x_N² = (2a)²,and the distance to Q is ±(x₀-1)²±x...±x_N² = (4b)²,which expands to x₀²...±x_N²-2x₀+1 = (4b)². If the sign of the x₀ dimension in the norm is positive, this simplifies beautifully to (2a)²-2x₀+1 = (4b)², which clearly cannot be solved in integers because the left side is inevitably an odd number, and the right side is inevitably an even number. If the sign of x₀ is negative, it's the same but with 2x₀² added to the left side of the equation, which doesn't change the situation. I find this quite pretty. (As a byproduct, in a case of what might be called extreme overkill, this also proves there is no integer which is congruent to 1 mod 4 but to 0 mod 2.) EDIT: It has been suggested to me that "lattice point" is an ambiguous term, so note that I am referring to points in some space whose coordinates are integers. And I may have used the term "norm" when I should have said "seminorm", but for instance Minkowski space has a metric with negative signs in the distance function, which is the sort of thing I was referring to. | Have you studied Lie Algebras? This type of question feels like exactly the type of thing you’d hear when studying root spaces. | There's lots of common usages of the word lattice. I've definitely heard "lattice point" used to refer to points in the lattice Z /subset R . If you're going to be critiquing OP, you should be critiquing them for not making it clear what they meant, not by using the wrong terms. | But it's not incorrect terminology. It's one of the common usages of the term. Edit: After literally 30 seconds of googling I found an article using the term this way in its abstract: https://www.sciencedirect.com/science/article/pii/009731657990102X and I've read it as a definition in some textbook or other. It's definitely in common usage. | Yeah, "lattice point" is definitely a phrase synonymous with "integer point", especially if we're talking about R with the Euclidean norm. The person complaining here is just wrong. | Are you sure you need 2 and 4? Does your proof work for 2 and 2? You seem to have shown, even more generally, that the quadratic forms you are considering, once reduced mod 2 and viewed as quadratic forms over the field with two elements, are identical to linear functionals. |
[
"Question about the hidden subgroup problem. [Complexity Theory]"
] | [
"math"
] | [
"bwlk7d"
] | [
13
] | [
""
] | [
true
] | [
false
] | [
0.93
] | Please excuse my lack of in-depth knowledge. I'm familiar with complexity theory at an advanced undergrad level, and maybe slightly less familiar with group theory. I was reading up on the hidden subgroup problem - mainly on Wikipedia. It intrigued me. From what I gather, it's been shown that many decision problems can be reduced to problems of finding hidden subgroups. e.g. integer factoring & discrete logarithm problems are examples of hidden abelian group problems. Similarly, graph isomorphism seems to have been reduced to finding a hidden subgroup of S(n). My question is: can every decidable decision problem be reduced to a hidden subgroup problem for some finite group? I imagine this question doesn't have an answer, but it is fun to think about the possibility that group theory has a deep connection to complexity classes. Thanks in advance for replies! | It is suspected that the hidden subgroup problem, and all the problems poly-time reducible to it (as you mentioned, factoring, discrete log, graph isomorphism) are NP-indeterminate - that is, harder than anything in P, but not NP complete (and as such there will exist decidable decision problems in NP not poly-time reducible to a hidden subgroup problem). However, this is in the same standing as many of the other problems about the complexity hierarchy. We just don't have the tools to prove it that these problems are NPI, nor whether NPI is even non-empty (though if it were empty then P = NP, so you can form your own opinions). | Ladner's theorem shows that if P≠NP then NP-intermediate is non-empty. | Wait... complexity classes are a thing? | an every decidable decision problem be reduced to a hidden subgroup problem for some finite group? The answer is no. The hidden subgroup problem lies in PSPACE; to see this, note that one can try every single possible function and every single possible coset. Since there are decidable decision problems which lie outside PSPACE, by the space hierarchy theorem , not all such problems can be reduced to hidden subgroup problem. | In computational complexity, problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems is called NPI. Ladner's theorem, shown in 1975 by Richard E. Ladner, is a result asserting that, if P ≠ NP, then NPI is not empty; that is, NP contains problems that are neither in P nor NP-complete. Since the other direction is trivial, it follows that P = NP if and only if NPI is empty. Under the assumption that P ≠ NP, Ladner explicitly constructs a problem in NPI, although this problem is artificial and otherwise uninteresting. It is an open question whether any "natural" problem has the same property: Schaefer's dichotomy theorem provides conditions under which classes of constrained Boolean satisfiability problems cannot be in NPI. Some problems that are considered good candidates for being NP-intermediate are the graph isomorphism problem, factoring, and computing the discrete logarithm. |
[
"How to \"find\" a differential equation?"
] | [
"math"
] | [
"bwf8s7"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.68
] | Clarification. Let's say I record the sound of a drum hit with different levels of force. For simplification, a light hit, medium and hard hit. How would I exactly find the solution to the equation? First, I am sure someone has already solved it but I wanted to see the process of developing the equation. Second, here is where I thought some of the steps were. 1. Record the data vs time graph. 2. Take the derivative of the data. 3. Examine the phase portraits. 4. ....... 5. Profit! (Last two were a south park reference). I am aware I maybe completely wrong with the process but I am interested in knowing how it would actually work. Thanks in advance. | Usually you start from physical principles to derive the equation that models the system. You can google "derivation of the wave equation" or "derivation of the heat equation" for some examples of how this is done. The equation is supposed to predict the results of the experiment. Using the data to derive the equation is going in the wrong direction. Although the experimental data lets you check whether the equation's predictions were good. | Keep in mind that the wave equation is from more intuitively plausible descriptions of the forces involved in stretching an elastic sheet. E.g. the 1d wave equation is just what you get when you apply Newton's laws to a slightly deformed string. The wave equation will not really be precise enough to create a convincing synthesized drum sound (for one thing, the of the drum hasn't even been mentioned) and so there's a lot more physics to be understood here. A physically grounded analysis would proceed by looking at the basic equations underlying how different materials stretch and vibrate (and by that I just referring to Newton's laws and balancing out the forces), and the way that the air interacts with this. It's not necessarily by taking an existing simple equation off the shelf and just hoping it fits. Fletcher's is a good reference, discussing the difference between vibrations in pipes vs thin shells vs sheets vs strings vs reeds etc. The wave equation is just the simplest possible starting case, not a sufficient end goal. As you note, striking a drum harder and softer sound qualitatively different, not just being the same sound at different volumes. The basic wave equation applied to a freely vibrating sheet can't even account for this. | analytically, that is, estimate through calculation I think you mean "numerically"; an analytical solution would be some closed-form solution in terms of sufficiently nice-looking functions (maybe a series solution also counts as analytical). | analytically, that is, estimate through calculation I think you mean "numerically"; an analytical solution would be some closed-form solution in terms of sufficiently nice-looking functions (maybe a series solution also counts as analytical). | A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame. Due to the phenomenon of resonance, at certain vibration frequencies, its resonant frequencies, the membrane can store vibrational energy, the surface moving in a characteristic pattern of standing waves. This is called a normal mode. |
[
"The feeling of your entire grade hinging on one final"
] | [
"math"
] | [
"bwcgeb"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
0.57
] | [deleted] | Lol at the thought anyone here breezed through all of their undergrad. Anyways, the unfortunate reality is you've already said the main things. If your professor gave out a study guide/old test/hints, study those. If they seem to have a preference for types of proofs or proof styles, brush up on those. Otherwise, the midterm would be good or old tests. Still nothing? Find your professor during office hours and just ask him. "Hey professor, I'm really stressed about the final. What would you suggest I study?" They'll likely be more than happy to drop hints. Here's the bigger thing. Make sure you eat a good breakfast before and get enough sleep. It's even more important with math, because proofs are more of a creative thing than a computational thing. With enough practice, you can push through computations while braid-dead from lack of sleep. That'll never work with proofs. Timed, in class proof-based tests will always be stressful just because creative tasks don't lend themselves to tight deadlines by their nature. | Meh.. I’m sure some people here have breezed through. | [deleted] | Is it usual that the final exam doesn't make up the majority of the grade? All courses I've had in university were 100% graded based on the final exam. | Interesting, we have to get 50% in total on weekly assignments to get admission to our exams but getting more than that doesn't change anything. Maybe that's just a german thing, may I ask where you're currently studying? |
[
"Modern Classical Homotopy Theory"
] | [
"math"
] | [
"bwhspq"
] | [
9
] | [
""
] | [
true
] | [
false
] | [
0.85
] | I wonder what opinion people have on the text (the author is Jeffrey Strom). I currently study abstract algebra and category theory; after I'm done with linear algebra of R-modules and with adjoint functors I plan to start with algebraic topology (along other subjects). Although geometric texts like Hatcher resonate with most people, I'm personally more comfortable with abstraction and rigor (also, I'm fascinated with category theory). May seems to be written in a similar style to the book in question, but it is apparently too concise (even though Strom doesn't contain complete proofs, it is still designed as a textbook rather than a reference unlike May). I would appreciate opinions on this book (for some reason, it is not very popular hence there is little info on it). Are typos really that bad? 3-star review on amazon claims that some of exact sequences and definitions are wrong, which is worrying for a student, though I realize that a careful and a patient reading should bring them to light for anyone. | Strøm's book is not a typical textbook. It is a series of exercises and problems intended to guide the reader through a self-study, probably working optimally in a class using the Moore method. I'm hesitant to recommend it as your first exposure to homotopy theory unless you have a mentor willing to go through it with you. My first homotopy theory book was May's concise book, which I recommend. I don't agree that May is a reference book, it has enough narrative structure to be considered a proper textbook. If May is too concise for you, it might be an idea to use May and Strøm in tandem... | I second May's book. Although the last few sections are a broad overview of homotopy theory and try to do to much in not enough space (plus uses the steenrod algebra to do stuff with characteristic classes and cobordisms when there's a much simpler way as done in milnor iirc). | It is pretty concise, especially the sense of having (almost) no examples, and his proofs are occasionally difficult or omitted. He doesn't hold your hand as much as Hatcher, that's for sure. Tom Dieck might be the superior book. I've only taken a brief look at it, but it looked promising. His treatment of stable homotopy theory looked old-fashioned though, as far as I remember. | Yeah, I guess that's a fair criticism. For Characteristic classes Milnor is the best book. For cobordism (the cohomology theory) my personal recommendation is Conner & Floyd's . | Thank you for your answer. Can I use this opportunity to ask you about May? How concise it really is? How hard is it to follow his proofs? Actually, yeah, using more than one book is a good idea. I've also heard nice things about tom Dieck's textbook. |
[
"New Subreddit dedicated to a math concept"
] | [
"math"
] | [
"bwbr3i"
] | [
12
] | [
""
] | [
true
] | [
false
] | [
0.62
] | Hi everyone! I have recently become a mod of . Hexaflexagons are an interesting math concept about folding paper in which many mathematical concepts can be tracked. Even if you don't know what they are, you should immediately go and see. We also have our own chat for general discussion. | Weird flex but ok | I think he's just making a joke. | It's a reference to an internet meme which contains "Weird flex but ok". Google it. | Hexaflexagons were invented in 1939: https://en.wikipedia.org/wiki/Flexagon So if anything, it's 80 years "too late." And by your same standard, Vi Hart's video was 73 years "too late." | But where's the love for tetraflexagons? |
[
"Is mathematics invented or discovered ?"
] | [
"math"
] | [
"bwrpdu"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.2
] | null | I think that what you said doesn't really mean anything and attempts to fundamentally connect mathematics with reality tend to be either practical enough to qualify as physics or they wax poetic without really saying anything one could deny (or accept, for that matter). Inventing discovery (for the right definitions of 'inventing' and 'discovery'). Otherwise, you can smuggle mathematics into either category by being careful enough with your definitions. | https://www.reddit.com/r/math/search?q=invented+discovered&restrict_sr=on&include_over_18=on&sort=relevance&t=all | but the laws of maths and logic themselves are the ultimate truth Which laws of logic? Classical logic , Intuitionistic logic , Quantum logic or something entirely different? | It is, yes. | I did a short survey along these lines a year or so ago on this subreddit. The universe was built according to its laws but the laws of maths and logic themselves are the ultimate truth. This puts you in the Platonist camp. Not only do you believe that the objects of mathematical study are real, in some external sense ("realism"), you are positing them as . I'm sympathetic to parts of this viewpoint, but it has a whiff of the mystical to it that makes me blanch. And beyond that vaguely inspirational mystical fuzziness, I'm not sure if it leads to any significant philosophy or any better mathematics. Does the idea of an "ultimate truth" help us decide which set-theoretic axioms to adopt? Are there some large cardinal axioms required by physics, such that we could call those large cardinals "empirical," and beyond them only "theoretical" ones? Are they part of the "the universe" or not? Maybe the universe prefers category theory instead... So maybe mathematics is only a of the universe. This brings us to mathematical idealism, the idea that mathematical objects are ideas, not external entities. Sometimes when cornered mathematicians describe mathematics not as "the language" of the universe, but as "a language" of symbols and rules. This is formalism. Even more extreme, one can think of "mathematics" as the output of those we designate "mathematicians." This would be a social constructionist view of mathematics. It seems odd to think of "mathematics" as akin to "police work," but when you look at the history of mathematics, the conventions of discourse and of proof, there's something to this viewpoint as well. So I don't really have an answer. I'm sympathetic to parts of all the major philosophies. |
[
"Is anyone else really finding Math to be boring in school?"
] | [
"math"
] | [
"bwqf2g"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.2
] | null | have absolutely no clue how quadratic equation works What exactly do you mean by this? Presumably you've looked at an expression like 'y=ax +bx+c'. What do you want to do with it that you haven't figured out ? | Ah, and that you can put for the x a number and then it goes like this f(1) = ax * ax + 1 You didn't replace the 'x's with ones, but a part of the expression that was independent of 'x's. What does 'decode' mean in this case? | Come on man, you can do better than this. What do you not understand about quadratic functions? | I'm on a economy school and yeah we did use "linear equations" and I completely rocked it in the economy subject, but man in Math I have absolutely no clue, meanwhile in economy you can put one name to the other and actually calculate. Maybe I just really hate x and y. | I'm on a economy school and yeah we did use "linear equations" and I completely rocked it in the economy subject, but man in Math I have absolutely no clue, meanwhile in economy you can put one name to the other and actually calculate. Maybe I just really hate x and y. |
[
"What Are You Working On?"
] | [
"math"
] | [
"bwchap"
] | [
21
] | [
""
] | [
true
] | [
false
] | [
0.9
] | This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on over the week/weekend. This can be anything from math-related arts and crafts, what you've been learning in class, books/papers you're reading, to preparing for a conference. All types and levels of mathematics are welcomed! | Figuring out how to insert an animation into my Beamer presentation. Going to be presenting my master's on Wednesday, and then... I guess I'll find something to do eventually, but won't have any must-dos until fall. Other than helping to arrange the yearly maths department barbeque (I keep lobbying for it to be officially named \mathbb{Q}, but I haven't had any success yet.), I guess. | It'd be a very rational choice of name. | Can’t understand how that name hasn’t passed | I keep lobbying for it to be officially named \mathbb{Q} Please, this would make me so happy. You guys do hold it at a field, right? (Sidenote: I remember asking my Berkeley math friends whether they ever made a t-shirt that just said \mathcal . Unfortunately I think they haven't.) | did you mean to comment in the hexaflexagons thread? Otherwise, weird (hexa)flex but okay. |
[
"What's so bad about looking up the solution to a problem?"
] | [
"math"
] | [
"bw6w9u"
] | [
170
] | [
""
] | [
true
] | [
false
] | [
0.96
] | I'm studying for the Math GRE subject test and sometimes after spending maybe an hour on a certain tricky calculus problems, I have a tendency to look at the solution in the back in order to understand the 'trick' behind the problem. Is it that bad? During my undergraduate career, a lot of my professors were very anal about not looking up solutions but I think some of them were being quite ridiculous. What does think? | I think the questions on the Math GRE aren't worth your time to ponder, and there's nothing wrong with looking up the solutions. Learn the trick, make some similar examples, and practice until you can do them in your sleep. I think for some homework questions, though, you can learn a lot by pondering and trying incorrect methods of proving something. When I'm doing homework, I usually discover a lot of small things that I didn't understand correctly, and I find a lot of connections that I didn't previously notice. Homework questions usually have a lot more depth to them than GRE questions, and there's a lot to be gained from coming up with a solution by yourself. | Looking up solutions isn't necessarily bad. Looking up solutions and writing them down without making any effort to understand why they're right is. | honestly, i look up a lot of solutions. if i either know that it's fairly trivial, or if i know i won't be able to do it, i do think i get more efficient study out of studying the solution rather than being stuck for a long time. | There's a lot of advice in math discouraging looking at solutions because students have a tendency to do it much too quickly, and because students tend not to learn from those solutions. In particular, before looking up a solution, it's really important to have spent enough time thinking about a problem to have gotten familiar with the problem and seen how different ideas fail to work, so that when you see the solution, you'll have some context for how it fits in. It's also important to take the time to understand the solution, rather than just looking at it and moving on. How much time you should spend on a problem before considering looking at the solution depends on the subject. When you're studying for the math GRE, an hour might well be enough: it's material you (presumably) already know, what you're worrying about is the various small tricks it's possible to forget or not have seen before, and it's important to find time for a lot of problems. In a typical course, where you're doing the homework in large part to help you absorb the new material, working on the problem at least twice at least two days apart is basically the minimum before looking at the solution, and for upper level courses, more might be appropriate. | For real. Just wrote memorization to prove you can regurgitate what you may have learned in undergrad. Then as a grad student, your advisor is confused why you can’t think for yourself and come up with something original. 🤗 |
[
"Unintuitive results"
] | [
"math"
] | [
"bwo4na"
] | [
411
] | [
""
] | [
true
] | [
false
] | [
0.97
] | I have always loved the unintuitive results that maths sometimes produces and I just remembered one from high school that is super simple but still throws me: If you have a rope that goes around the equator then adding in just 2pi meters of rope will give you enough to suspend the rope a meter off the ground everywhere. What other unintuitive results are there that are hard to get your head around? | I think the Riemann Reordering Theorem fits: If you take a convergent but not absolutely convergent series, then given any real number x there exists a reordering of the series such that it converges to x. Totally blew my mind in Analysis 1. | Here is a list of the number of unique smooth structures (up to diffeomorphism) on the n-sphere | How does a 5-dimensional topologist put on his pants? One leg at a time, just like everyone else. How does a 3-dimensional topologist put on his pants? Jesus christ, don't ask. | I initially encountered that in my second semester in calculus while we were covering the introductory material on convergence and divergence of series. One day, the prof just mentioned it, as an aside without any details, and quickly continued on with his lecture as planned. My immediate internal thought was "Bullshit. I call shenanigans. How can there possibly be a way you can rearrange a series to get ANY value you choose, no matter how big or small? Impossible. Obvious nonsense is nonsense." I spent the next year or two 100% convinced that the professor had misspoke that particular day, or was confused and conflating two similar things, or misread his own notes, or something like that. It HAD to have been a mistake on his end, somehow. Fast forward a few semesters later, to my first analysis course. We proved the rearrangement theorem in class, and the proof was so crystal clear and straightforward as to leave no room for even the slightest of lingering doubts or confusion. My head was spinning for at least a week after that. "Holy shit. That thing my calc 2 prof said, way back when, was fucking TRUE." | The Monty Hall problem is a good one. |
[
"I study maths for two years to find out that I've basically learnt nothing in the past two years"
] | [
"math"
] | [
"bwlx7w"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.44
] | null | Spend a lot of time learning how you can use it in a useful way. People can learn all the equations they want and memorize a hundred different formulas, but what good are they if you can never figure out when to use one and which one to use? Try to solve real world problems using math. You'll quickly learn that there are gaps in your knowledge, and in order to solve the problem, you need to fill those gaps or discover a way to do it. For example, if you know the position of a comet at time T and T+10, can you figure out where it will be at time T+X? And, the bigger question: How do you verify that your answer is correct? There isn't an answer key to look up... | I've met people who know how to find derivatives and get straight A's but don't know that the derivative of position is velocity lol you would be surprised | If someone asks you what a derivative is, do you not give them the definition( lim x-> 0 of blah blah / blah )? If they ask what this means ,tell them that it's the instantaneous rate of change, and it can be used to model physical problems? | That is helpful in several ways. Thanks a lot | Thanks a mil! :D I'm definitely gonna give it my best to work this out |
[
"I want help for find the mistake"
] | [
"math"
] | [
"bwgkny"
] | [
0
] | [
"Removed - try /r/learnmath"
] | [
true
] | [
false
] | [
0.43
] | null | Screenshot how it looks like: https://gyazo.com/ecceed9b7829a40d377f911dfbb66c43 | The 2nd to last = is wrong | Edit. think of it as a vector and the 1 is the scalar weight and the +- is the direction. So by doing all of the other operations, you're changing direction but the weight stays the same | sqrt(1) has multiple solutions, both 1 and -1 are valid. Your error comes from the fact that sqrt(1) is not really equal to 1 or -1, in that they are not the same mathematical entity. In terms of sets, 1 is a subset of sqrt(1) and -1 is a subset of sqrt(1), but sqrt(1) IS NOT a subset of either 1 or -1. Does this make sense? Also, sorry about the lack of good mathematical notation, I am on mobile. EDIT: My explanation above treats the well defined function sqrt(x) and the set of values that are inverses to x as the same thing, which is incorrect. Thanks to u/fridolinvii for the catch! | Yes, you are correct. My statement is only true if we assume that we are using a NOT well defined function that gives the inverse of x Using the well defined function sqrt(x) the issue occurs in the equality of between (-1) and sqrt(-1 ), as you observed in a previous comment. Thank you for pointing this out to me! |
[
"histogram vs box plot for quantitative data?"
] | [
"math"
] | [
"bwi9es"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.33
] | null | If you dont need to make comparisons a histogram is strictly superior to a boxplot. It simply has more information and can easily be augmented with summary statistics like the mean or median. | Say that you gather this spending data from your class and want to compare the typical spending of men and women. A histogram still sort of works for this. You can place the plots is a column, make sure they're scaled the same way, and they're easy to compare. But once you have more comparisons than that (say you want to compare spending based on people's ZIP code) comparing histograms becomes very messy. Boxplots are a very clean way to show the data and as a result they're easy to read even if they're squished to fit in a single diagram. | Plots have two major goals: Histograms show more information. Boxplots are more readable. Comparisons require you to present a lot more stuff. That makes histograms a problem, because they are complex they become harder to read when they are made smaller. Visually speaking a boxplot doesn't consist of much, in the simplest form just one rectangle and three lines, so squeezing a lot of them into the reader's field of vision doesn't harm how readable they are. You could compromise with a violin plot but they aren't very familiar and they have their own weaknesses. | thank you so much for this, is there any cases where a box-plot is superior? | thank you so much, so it mainly matters how much varying data there is? |
[
"Stats project data collection"
] | [
"math"
] | [
"bwlg1w"
] | [
5
] | [
"Removed - survey"
] | [
true
] | [
false
] | [
0.67
] | null | 3 | 3.14159 | 3.14 | 3.14 I know... | 3.141592653589793238462643383 Won me a piece of pie in the 3rd grade. I thought I was such a badass until this kid next year came in with the first 200 memorized. Jerk. |
[
"Someone teach me how to do this please"
] | [
"math"
] | [
"bwfrx9"
] | [
0
] | [
"Removed - try /r/learnmath"
] | [
true
] | [
false
] | [
0.5
] | null | Depends on the density of the liquid. Not a math question, more of a trivial physics question. | Head on over to r/learnmath , r/homeworkhelp or r/cheatatmathhomework . Those are where people will help you with this sort of question. I will suggest that without knowing the density of the liquid you cannot answer the question . | 25mg per ml | Still not a math question, but then just calculate the total mass of the tube, and divide by the mass used per day. So, (40*25)/15 = 67 days approximately. | What is the ratio of mg of liquid to ml of liquid? |
[
"Book recommendation for grasping the concept of infinity? (Introductory ofc)"
] | [
"math"
] | [
"bwapg7"
] | [
10
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
1
] | null | Rudy Rucker's is a great introduction to thinking seriously about infinity. Very personable treatment with rock-solid mathematics. Starts from nothing and builds to some pretty sophisticated ideas. | Early in my undergrad, I got a lot of mileage out of Mathematical Thinking: Problem Solving and Proofs, by John P. D’Angelo and Douglas B. West. I also recommend this Ted Talk , I show to my high school students. | It all depends on your level, but if you're still in high school you should check out The Art of the Infinite. It's super cheesy but if you're a highschooler/freshman in college wanting to start learning about this stuff it's a great introduction and very engaging! I love math texts that teach topics in a less dry way (I first learned about graph theory in high school by reading a biography on erdos) and art of the infinite is my favorite introductory material on infinity even if it's a little melodramatic. Besides mathematicians could use a little more teen angst in their writing. | What math class was the last one you took? | Unfortunately, your submission has been removed for the following reason(s): /r/learnmath books free online resources Here If you have any questions, please feel free to message the mods . Thank you! |
[
"I'm doing a Mathematical Presentation for my High School Calculus project on Wednesday. It's on the Golden Ratio"
] | [
"math"
] | [
"bwfmaf"
] | [
0
] | [
"Removed - try /r/learnmath"
] | [
true
] | [
false
] | [
0.38
] | null | The golden ratio probably | It wouldn't hurt to take a few minutes to address al the unwarranted mysticism surrounding the number. It is, after all, just a number with a handful of interesting properties. | Thanks | 👍 | Continued fractions, and how they can be used to measure irrationality in such a that the golden ratio is the most irrational number. If you go into detail you can probably fill 20 minutes. |
[
"Complex Gravity question"
] | [
"math"
] | [
"bw6aia"
] | [
1
] | [
"Removed - not mathematics"
] | [
true
] | [
false
] | [
0.6
] | null | I think you misunderstand. Objects do not have to have a greater the mass the further apart they are for gravity to act. Gravity is always acting. The force just gets smaller the further away, and smaller the object is in terms of mass. Google “the universal law of gravitation” if you’re interested more in this sorta thing. | It's not clear to me what you're asking. You have two things tethered together, they're rotating... and then what? You say this is a spaceship design. Are you asking if you can get thrust out of it? | The two things tethered would they be able to create a gravitational pull within each other? Thank you for your response :) | Everything in the universe exerts gravity on everything else, at all times. So yes. | Interesting :) |
[
"Asking for help and suggestions."
] | [
"math"
] | [
"bw61u4"
] | [
7
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.77
] | null | Book of Proof has a free PDF avaible online. It teaches proof techniques and might give you an idea of what this endeavor will be like. | You've had the standard 3 semesters of calc and probably a basic linear algebra course and an elementary differential equations course. You're ready for a proof course. They go by different names: intro to higher mathematics, intro to proof, mathematical structures. You can see what the math dept at your alma mater is using. by Joseph Rotman may be more suitable for self study. | Thank you. | Thank you. | Go the learnmath reddit and look at the sticky post there which has a lot of learning resources. |
[
"What is the proper formula for."
] | [
"math"
] | [
"bw6a7l"
] | [
0
] | [
"Removed - incorrect information"
] | [
true
] | [
false
] | [
0.26
] | null | All joking aside I would love to have access to the data to analyze this. It has to be somewhat predictable based on time, day, size of sub, pic vs. Text vs gif, etc. | (Screenshot or copy + paste) = repost repost • amount of subs posted in = karma | I am a physics major and I absolutely agree. It would be so fun to try out different possibilities to see what would have a larger pull or weight on the post. (I assume time of day is the largest reason though). | We could just ask gallowboob and skip the middleman. | ? |
[
"Algebra 2 in one night"
] | [
"math"
] | [
"bw71ub"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.47
] | null | KhanAcademy, PatrickJMT, maybe google algebra 2 review on google | KhanAcademy, PatrickJMT, maybe google algebra 2 review on google | Main things are inequalities, solving linear equations, graphing equations, polynomials and quadratics functions( quadratic functions and completing the square), logs and series/sequences. Make sure you know how to factor and graph, too. | Might not be worth cramming to try to pass. If you don’t feel like you can pass placement with relative ease, then your algebra skills might make your pre-calc and calculus experiences difficult. | Or you can try professor Leonard videos on YouTube, I think he has a list of videos for algebra 2 but I'm not sure, you can check it out. |
[
"Rethinking set theory"
] | [
"math"
] | [
"bwhlwj"
] | [
7
] | [
"PDF"
] | [
true
] | [
false
] | [
0.71
] | null | While I have many thoughts about what set theory is and what it isn't, I would like to point out that Leinster really undersells the consequences of finding a contradiction in ZFC. If such a thing were to arise that didn't also contradict his 10 axioms (in particular, function sets are still okay) then it would have to be because there is something going wrong with itself. A failure in restricted comprehension or replacement would mean something like "we have this first-order description, but we can't actually talk about the things satisfying the description". I'm sure many mathematicians find first-order logic both a basic and natural tool that they would be hesitant to lose. | I generally prefer links to the abstract page: https://arxiv.org/abs/1212.6543 . This lets you read the abstract and see other metadata like the author before deciding whether to read the paper, and it lets you find other work on the arxiv by the author. Finally, the PDF is one click away from the abstract, and not vice versa. | Something something LEM. Something something people mistake T for T+Con(T) (in the form of an implicit "natural" model) all the time, whatever they think T is. Something something intuitionism is what most mathematicians believe anyway. Etc. | Foundational questions aside, I think the biggest problem with many of these debates is that most mathematicians think of 'set theory' as meaning 'a set theoretic foundation for math', rather than a branch of math like any other branch. Saying that "the purpose of set theory is to provide a foundation for math" is like saying "the purpose of math is to get rockets to the moon". Many set theorists don't really care about whether ZFC is the basis for all of math, they simply fall back to that idea when they feel like they have to justify their existence to others. Certainly most mathematicians would find it silly if someone came along with a categorical system and claimed that it "is not to replace number theory; it is number theory", when basic pieces, like number fields or w/e, were missing. Without concepts like the ordinals/sets of ordinals or definability, there is no set theory. Maybe it's a minor, semantic difference; but the equating of 'set theory' and 'foundations' appears all over the place. And it ends up hurting both the field of set theory and the field of foundations. | I know Tom and he was high as fuck when he wrote this. |
[
"How Most Proofs Are Structured and How to Write Them"
] | [
"math"
] | [
"bwa1fq"
] | [
33
] | [
"Removed - incorrect information"
] | [
true
] | [
false
] | [
0.67
] | null | The attitude of wanting to make a flowchart to guide you through writing proofs is exactly what people have to get over. There is no way to make it like drilling calc problems where you don't have to think if you memorize enough algorithms. A proof is an explanation of why something is true, that meets a certain standard of rigor. Getting used to the standard of rigor is the hard thing that's unique to math. Structuring a proof is like structuring an explanation of anything else--the best way to structure it depends on what you're trying to explain. | That is perhaps the most complicated and backward explanation of how to do a proof that I've ever seen | This is not a well done explanation. It gets some incidental things wrong (we are pretty sure that P != NP). It makes many things very complicated. It also misses some basic issues; for example it starts talking about prime and zero divisors, but it doesn't talk about what ring that's in, which is absolutely critical. If they mean a ring in a general, then they should say that explicitly. | I think he's trying to explain how to come up with proofs, which is a different skill than writing them. Even writing precise mathematical statements is a challenge, as his utterly confusing "statement" shows. if x cubed equals itself and is no zero divisor, then it is not prime. Determine possible values. Is the first sentence supposed to be proven? Could I interpret this question to say that my only task is to "Determine possible values"? Also, variables are supposed to be introduced with their ambient set! Is x an integer? A complex number? An element of an arbitrary ring? Anyway, here is how to write a proof: State your assumptions clearly in correct English State clearly what you want to prove in correct English Set up an argument, with full English sentences, that explains how your assumptions inevitably imply what you want to prove. Nothing more. | Don't ever read any math posts on physicsforums, that site is really fucking cancerous. |
[
"I decided to look for patterns in prime numbers and possibly discovered a new fractal (Video inside)"
] | [
"math"
] | [
"w7rtyx"
] | [
86
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.8
] | null | While this looks cool, I believe it has little to do with prime numbers. By the prime number theorem, the n-th prime number is somewhat close to n*log(n). If you replace primes by a randomly generated sequence following a similar repartition, you will obtain similar pictures | I'm forced to agree. Each prime number is just a little bit bigger than the previous prime number, so of course the lines will never cross and if you just take a list of every natural number in order and discard some of them at random, doing the same plot with that list will create images which look similar to those above. | Cool concept! I do think you may want to dig a little deeper on some of the mechanics of these entities before you name it however ;) For example, it’s not clear to me if your construction is truly a fractal in the formal sense (despite the fact that it shared several visual parallels with common examples of fractals). Fractals are typically defined in terms of their Hausdorff Dimension , which must be different from their obvious geometric dimension (so like a fractal “drawn” in 2 dimensions has some Hausdorff dimension different from 2 and usually non-integer). Because of how Hausdorff dimension is defined, most of the objects we call fractals are uncountably infinite sets. However your construction appears (to my understanding) to be : every point in your construct can be put into 1-to-1 correspondence with the integers. You may therefore need to give this some consideration before calling Bryn’s Prime Path “fractal”. I would also point you towards irrational rotations . As others have pointed out, this behavior doesn’t appear to be unique to the prime numbers. I suspect that what may be causing the “fractal” appearance of your plots is that the algebra is shaking out such that you are either falling into or out of the dynamics of an irrational rotation, depending on choice of r. These are simple examples of chaotic dynamical systems, and so may be producing the “complex” (in the dynamical systems sense) character of your plots. See if you can derive an expression for the angle of rotation of successive points on the path (call it theta) in terms of r and try and figure out if there’s some critical values/regions of r which might change theta in some meaningful way (eg. from rational to irrational). It may be helpful to replace the nth prime with the approximation n*log(n), as others have mentioned. Nice work on the cool find, and I hope that you unearth some more interesting stuff about it! | Check out 3b1b’s video on prime numbers and spirals. Sadly I’m afraid your discovery has little to do with prime numbers | Somewhat related to this video: https://youtu.be/EK32jo7i5LQ Some questions for you: 1) If you do this process with other sequences of numbers, do you get similar looking pictures? E.g. you could try the sequence of even numbers, or the fibbonaci sequence, or even a randomly increasing sequence. 2) Based on the results of the first question, are these patterns unique to primes? Or is it just because the sequence of primes is increasing? |
[
"what produces better critical thinking skills philosophy or math?"
] | [
"math"
] | [
"w7lrlh"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.44
] | null | The term "critical thinking skills" is too broad and vague for anyone to answer this question with any definiteness. Especially if you're, as you state, looking for an objective answer. | Defining a term with even more vague terms doesn't really make it any less vague... (just in the first sentence you use the terms "quality" "intellectual standards", both of which can be interpreted in several different ways, giving several different answers. For example, if "high quality" just meant "low chance of being wrong", then math gets a heavy advantage, but if "high quality" means "applicable to more real-life problems" then another subject would gain an advantage. Furthermore, you mention "any subject", but in order to quantify whether something is "better", we need to know how these subjects are weighted. Are most subjects in real life, or are most subjects academic in nature? That changes depending on how you want to measure things.) We don't need to ask for much, you don't need to provide a proper definition to the standards of math or philosophy. All you need to do is provide very rough procedures for an experiment or a test in which we can measure some quantity, and we'll call that quantity "critical thinking". Then, we'll know what we're talking about. If you want a subjective opinion backed by argument instead, I'll gladly give one. | Any field gives critical thinking skills, it just depends where you want to apply your skills. A mathematician is going to be really good in detecting logical flaws in the reasoning, eg Wason's selection task. But if it's an argument about morality, a philosopher is obviously going to be better at it. A statistician would have better critical thinking skills than both the mathematician and the philosopher when it comes to analyzing the methodology in a research paper. Etc. | Mathematics trains you to be very very precise with your reasoning, and that’s one very important aspect of critical thinking, but it’s far from the whole thing. Philosophy similarly trains you in other specific aspects of critical thinking that mathematics does not necessarily — such as the evaluation of evidence and (unless you study more advanced mathematics) the construction of descriptive frameworks. There is some overlap, of course. And there are aspects of the skill that neither subject teaches directly. | And he told you why it isn’t clear. |
[
"Proposal for a New Unit of Angle Measure: “Tauradian”"
] | [
"math"
] | [
"w78iix"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.4
] | null | In no way whatsoever. | In what ways could this be useful? | There’s already a way to say this with one extra syllable: “two pi radians” | I understand the circle and trigonometry, I'm referring to OP's proposed definition and notation specifically. | I understand the circle and trigonometry, I'm referring to OP's proposed definition and notation specifically. |
[
"Ian ending on math terminology"
] | [
"math"
] | [
"w73bk0"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.3
] | null | The Jacobian matrix = the matrix of Jacobi, much like how the American flag = the flag of America. It’s just a construction in English for certain concepts and objects that are named after a particular mathematician. There’s no real reason afaik why sometimes it’s Gauss sometimes and Gaussian other times. | -ian and -an are suffixes in English (stemming from the Latin -ianus) which denote "from, related to, or like." An example from outside of math is "Dickensian," which means something of similar style to the writing of Charles Dickens. | Nearly every action or operator which is named after a mathematic ends in "-ian." There are probably as many as there are areas of mathematical study. My favorite is the Laplacian, which I have heard used for two completely unrelated actions. | I need to find an operator to name "The AcademicOverAnalysisian" | who need they Gaussi ate |
[
"I discovered that the series tan(n)/n^n diverges, what is the smallest fast-growing function that can make it a convergent series if there exists any?"
] | [
"math"
] | [
"w7tl18"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.42
] | null | The series tan(n)/n has been shown convergent for b=8, and is supposed to be convergent for b=2. So your series is convergent. See: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.112.5431&rep=rep1&type=pdf | WolframAlpha is not perfect | It seems very unlikely to me that the sum of tan(n)/n diverges. From here it appears to be very rare that |tan(n)| > n. I would expect that using an appropriate heuristic, there ought to be only finitely many values of n for which |tan(n)| > n . Do you have a source? | Looks like you found a mistake in Wolfram Alpha! Investigating further, it appears that Wolfram Alpha thinks the series diverges because of the Test for Divergence. It takes the limit as n goes to infinity of tan(n)/n^n, but incorrectly allows n to be any real number when it should be constrained to integer values. Then, since the real function f(n) = tan(n)/n^n has singularities at odd multiples of pi/2, the limit does not exist and so Wolfram Alpha decides that the series must diverge. The same logic would imply that the sum of sin(pi*n) diverges as well, even though all the terms are 0. Wolfram Alpha understands this but when you click on "Show tests" it says "series diverges" (which is inconsistent with itself). I submitted a bug report, so we'll see whether they fix it. | If you expand the step by step, you'll see that Wolfram doesn't know how to prove it, so it assumes its divergence. OTOH, if you ask to compute the value (from 1 to 100, to 200), it appears to converge |
[
"How much does school prestige matter? (Both for undergraduates trying to get accepted into a PhD program and for PhD’s trying to get tenure-track)"
] | [
"math"
] | [
"w7oyie"
] | [
111
] | [
""
] | [
true
] | [
false
] | [
0.93
] | null | It sounds like you're an undergraduate looking at applying to PhD programs. I've done application review for PhD applicants at a top 10 math department, and I can tell you that we did have a "Top-ranked university" checkbox on the form that would score you a few points for the prestige of your undergraduate institution. Much more important, though, are your recommendation letters and undergraduate classes. Prestige figures into these only in an indirect way. Top-ranked undergraduate institutions are more likely to have well-known professors who can write you letters, and are more likely to have tons of advanced classes available to you. But again, if you get very strong recommendation letters and do well in an array of advanced classes, that will help you regardless. I do think people underestimate the advanced coursework part of the equation. Some of the most unfortunate rejections I saw were people who had otherwise strong applications, but hadn't taken many of these higher-level (usually meaning graduate) courses. The logic was that we didn't have much of an indication that they would excel in PhD coursework. This also meant that we essentially never accepted students from small liberal arts colleges, no matter how prestigious. The only exception to that I saw was someone who had taken some advanced math courses at a nearby research university. | Undergraduate is 95% what you do and 5% where you do it. | For PhDs trying to get on the tenure track it matters tremendously. A huge percentage of professors is recruited from Ivy League and equivalent institutions. | Thank you for your very helpful comment. Can I ask, were there any courses that were particularly important to see on applications? Right now I’m planning to take number theory in my second year under my advisor, and hopefully do some research with him next summer, differential geometry, and real analysis. My school isn’t super great (R2) and no PhD program for math, but we have some cool professors that I’ve had the opportunity to talk to/befriend, so that should help with recs. | Just remember "getting into a top PhD program" is like getting into the NBA. It's a very small percentage of people. Just like those college basketball players with big dreams, it's best to have a plan B. Not trying to discourage you - just being realistic. |
[
"How do *you* approach proofs?"
] | [
"math"
] | [
"w7c4cw"
] | [
16
] | [
""
] | [
true
] | [
false
] | [
0.89
] | I'm an undergraduate studying math (hoping to continue on to grad school) and am curious how other people do proofs, especially when they're not sure where to start. Usually I just make a list of what I know from what's given and figure out how those can apply to what I want to prove. What about you? Do you do something similar? Something completely different? | Make sure you start with the definitions. Many proofs follow directly from them. Use theorems and lemmas only as needed. Also, try simple examples to see if they show you why something is true. Often the calculation and reasoning needed to show something is true for an example turns out to work in general. In other words, try all the dumb and obvious things first. Often that guides you to what you really need to do. | In addition to the other great advice here, I find it also helps to work backwards. That is, start from the thing you want to prove and work your way back until you reach something you already know is true. Then the proof comes from reversing the steps. It definitely doesn't always work, but I find it particularly helpful in analysis where you're doing a lot of epsilon-delta proofs. Additionally, it helps to think about the different ways to approach the proof at the outset. Should I try direct? Contradiction/contrapositive? Induction? etc. It's not uncommon for me to find that results which seem unapproachable through one proof method become straightforward when attacked from a different angle. | I am also an undergrad hoping to continue my studies. I do a similar thing to you, especially when I'm approaching something like "prove that [object] has [property]". I usually start with possible reductions, such as "it suffices to show for 0 by translations" or "it suffices to show for step functions", etc. sometimes I'd prove the weaker version of the assertion to get a grasp of what's going on like go for n=2 before proving for a general n, or start with drawings or specific examples. basically anything that will allow me to generalize from simpler cases | I usually like to write assertions like "object x is normal" into notation like "n(x)". Natural language sentences then take the form of simple logical sentences like "n(x) ⇒ !q(x')". Entire proofs take the form of substitutions and rewriting steps. I dislike that we have collectively chosen to communicate proofs and logical verbally. Like, why? Logic follows formal rules just the same as arithmetic and whatever else. "It is clear by recursive application of the summation rule under lemma 1.0.2 that three-hundred forty two plus negative twenty-eight reduces to three-hundred fourteen. Qed." | At this point in time I do a few things... first I ask myself if what I'm trying to prove is obviously true to me, and if it is I ask why. If it isnt, then I work on that a bit until it becomes clear to me. As far as the formal proof I think it is often the case that when framed appropriately the solution will fall out of the definitions. I usually just have to stay sharp while working through it. |
[
"Examples of \"close call\" conjectures"
] | [
"math"
] | [
"w7t92u"
] | [
102
] | [
""
] | [
true
] | [
false
] | [
0.95
] | I am looking for "natural" examples of the following: A reasonable example from physics would be the fine-structure constant, with c = 1/137, which was conjectured by before it was disproven experimentally, with ε ≈ 2 * 10 . I'm not looking for "artificial" examples, such as "take any conjecture about the natural numbers which was later disproven by large counterexample, and consider the binary number 0.d_1d_2d_3... where d_n = 1 iff n is a counterexample", or "consider the fine-structure constant divided by a googolplex". Admittedly I'm not drawing a mathematically well-defined line in the sand, but hopefully my question is specific enough. Does anyone here know any good examples of this? | Ramanujan looked at "close calls" to Fermat's Last Theorem. Specifically, he looked at solution to x +y +z =±1, and found ways to generate infinitely many such "close calls" . | This isn't a particularly high-profile one, but I like it anyway. In the binary look and say sequence , it was conjectured (not widely, but at least by one person) that the limiting ratio of "1"s to "0s" in the sequence is 5/3 = 1.6666... I found that the limiting ratio is actually ((101 - 10sqrt(93))a + (139 - 13sqrt(93))a - 76)/108, where a = (116 + 12sqrt(93)) . This number has decimal expansion 1.6657... | https://arxiv.org/abs/1001.3401 Conjectured value: 2.125 Actual value: about 2.125288 There is a conceptual reason for the "near miss" as well. | This relates to the part of the “1729 story” that nobody ever talks about. 1729 is the product of a Fermat near miss. 1729 = 12 + 1 = 10 + 9 - a near miss. | This is a nice one: https://en.m.wikipedia.org/wiki/Legendre%27s_constant |
[
"What Are You Working On? July 25, 2022"
] | [
"math"
] | [
"w7sm89"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
0.74
] | This recurring thread will be for general discussion on whatever math-related topics you have been or will be working on this week. This can be anything, including: All types and levels of mathematics are welcomed! If you are asking for advice on choosing classes or career prospects, please go to the most recent . | Any talk should contain exactly one joke and one proof, and the audience should be able to tell which is which. | 68 y/o retiree working through Intro to Linear Algebra to try to actually understand Eigenvalues so I can understand the math behind Quantum Mechanics. I’ve tried several times to approach it casually as an amateur and gotten scrapped off. I read someone’s comment that such difficulties are usually an incomplete understanding of the base material, which seems likely. This time I’m going chapter by chapter and working lots of problems to make sure I understand before moving on. I only completed Calculus 3 during my formal education years, as I was BA major (philosophy). I always wondered after If I could have gone on in math and engineering. Never had time to find out once I started working and raising a family. Now I am finding out. The answer is apparently maybe, but probably not if I had to do it along with 4 other classes in a semester. | I have some new results in discrete geometry from my masters thesis which are not published yet, so Im working on writing them down properly. They are basically answers to several questions noone asked, but hopefully they are still publishable somewhere. This would be my second paper and the first that has me as the sole author, so this is quite significant in my academic career, especially since I still havent decided if i want to do a phd. | It's squeaky bum time now: my exams begin next week, and I've not finished analytical mechanics. I just have to do the meagre amount of calculus of variations there's yet to do, and then I have to do ODEs and numerical methods, both of which really should have gotten more time than a week, but what can you do. I wish they would just give me my timetable so I know what has to be done first. | After a slight break, I've returned to homological algebra and I've finally built up enough machinery to understand the basic theory of derived functors. I first learned about Tor groups in December (?) and Ext groups a short while after so it's very satisfying to reflect on how much has gone into understanding why they have all of the properties I was promised so many months ago. There are still a few things that have gone unanswered, such as computing Ext groups in either argument and using flat resolutions, but I should be getting to those soon. I'm also glad that I learned some algebraic topology when I did. Aluffi applies the basic theory of derived functors to define group cohomology, so I'm able to see multiple interpretations of things like the bar resolution. I took a brief aside to read about some simplicial stuff to motivate the bar resolution and found it all quite fascinating. At the very least, I feel like I have a rough answer to my question a few months back of where all these chain complexes come from. I'd like to try extending some of these ideas to Lie algebra cohomology or Hochschild cohomology on my own and just seeing how far I can get. |
[
"Famous fast growing functions"
] | [
"math"
] | [
"w7559m"
] | [
11
] | [
""
] | [
true
] | [
false
] | [
0.71
] | Hey guys! I was wondering whether there are "famous" or well known functions in math which grow extremely quickly but after a certain point growth reduces to almost 0. Example: f(1)=34 f(2)=45678899 ... f(10)=16778889999999999887 f(11)=16778889999999999888 Edit: Thank you guys for all your input! | Is it OK if the growth goes negative? F(n)=TREE(n)/BB(n) eventually gets arbitrarily close to zero for large enough n, but the first few terms grow so fast that we don't have adequate notation to begin to describe their values. | I think sigmoids have the property you're describing. The logistic curve for example grows exponentially at 0 but is monotonic and bounded so this growth flattens out fairly quickly. nth roots (e.g. cube root which looks like an unbounded sigmoid) also have this property. You can find a point with arbitrarily big growth by getting closer and closer to the origin (the origin has an infinite derivative), but this decays off and before you get to 1, it grows slower than the identity function. | Goodstein sequences | Natural log goes from -infinity to 0 in one unit, but eventually grows slower than any x r>0. | grows pretty fast for large before eventually settling to zero. It shows up in the definition of the gamma function. |
[
"Where did your math education take you?"
] | [
"math"
] | [
"w7drnc"
] | [
113
] | [
""
] | [
true
] | [
false
] | [
0.94
] | For those who have a bachelors, masters or even a PhD. What do you do now. If you have a PhD, are you in academia. If so, how’s that going. If you have you bachelors or masters, what are you planning on doing next. Doing a PhD in math or moving onto other fields. For those who have moved on to other fields after their math education, do you still find time to study new pieces of math every now and then? I’m asking this as I’m not even sure what I want to do. Some days I feel strongly that I should continue on and become a Mathematician. But other days I feel like I should maybe go for a career that is a bit lucrative. This is because I’ve heard that it’s very hard to find an academic position in mathematics. More so that a lot of them aren’t that we’ll paid compared to what a person working in industry could make. If I ever do decide to go to industry, I’ll sure as hell keep studying mathematics on the side as that’s the thing I like the most. But to you now, where are you in life? | M.S. in statistics and I'm a research scientist in the field of global health (health economics, most recently). I work at a medical school where I build statistical models, write analysis software, and collaborate to publish papers in the field. The saying is that statisticians get to play in everyone's backyard, and it's the reason I joined the field. My career is a great mix of math, programming, and meaningful work for me. I'll be in the field for the foreseeable future. Lots of fun and challenging problems. | Have masters, going to do PhD. I’ll just go as far in math as I can get… | Software engineering. I may consider doing a PhD in theoretical computer science (satisfiability) once I get a million in the bank and family/children are stable, probably in my 30's. Complexity and computability theory are really just fields of mathematics. They don't get treatment in mathematics curricula, but then logic and numerical analysis do for some vestigial reason. I kind of dislike the distinction of physics, mathematics, computer science, and engineering at the institutional level. You should really just learn the ropes everything. There's so much low hanging fruit in elementary engineering that a research mathematician would benefit from and vice-versa. | I got a BS in mathematics and BS in computer science - I’d wanted the math major since kindergarten, but discovered programming in eighth grade. I decided in high school that programming would be a better paying career and I should double major. I’m two years out of college now and have been workin as a data engineer for the last three. I don’t use any of my theoretical math knowledge in my day-to-day, but still remember my classes fondly and regularly watch youtube videos about new math concepts. If I went back to college I would probably get an MBA or a masters in economics. | I have a BA in mathematics and philosophy. I work as a software developer now. I find that the programing puzzles scratch the math itch well enough, though it's obviously not the same. And, if you're interested, academic computer science is basically a field of applied mathematics. I've always had an interest in computers though, so this may not be the path for you. Good luck! |
[
"Is there any matrix multiplication/transformation I can do to shift all elements of a matrix in 1 direction?"
] | [
"math"
] | [
"mn30nv"
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1
] | [
""
] | [
true
] | [
false
] | [
1
] | null | I think left multiplication by [0,0,0] [1,0,0] [0,1,0] works. The 1s pick out the elements you want | Agreed :) | This works for the specific example /u/third_dude gave, but it won't do what they want in general (if the bottom row doesn't happen to be only zeroes). Specifically, At each step (every second), every "1" moves downward ( ). | how did you come up with that? I am interested in doing this for other sizes of matrices and want to learn how to do it myself. | Essentially assuming you want to transform matrix A to B using a matrix P (PA = B) in a manner shown above where the rows are shifted down, you have to select rows of A that will be the rows of B. So if you know the second row of B has to be the first row of A, then in the second row of P would be [1 0 0 ... 0]. Essentially this row vector is saying that take the first row of A and put in the second row of B. To give another example, say you want the first row of B for to be the last row in A then you would make the first row of P [0 0 ... 1]. Do this for all the rows of P in whatever manner you see fit and you can generate a P for your needs :). Edit: Also the length of each row in P or the number of columns in P will be the number of rows in A, and that should make sense from a matrix algebra viewpoint and this slightly more abstracted but equivalent viewpoint of matrix multiplication |
[
"Does anyone else find it funny when non-math people think you’re a “math genius” when you know how little you actually know?"
] | [
"math"
] | [
"w779mk"
] | [
1139
] | [
""
] | [
true
] | [
false
] | [
0.96
] | It’s classic Dunning - Krueger effect but at my high school students and other colleagues call me the “math guru” but I myself feel very incompetent at math. I struggled a lot with upper level undergraduate courses and even lower ones like Discrete Math. It’s one thing to teach high school algebra, trig etc but I wish I could explain how little I actually know to people. I’m not trying to sound arrogant or anything but it’s impossible to even describe to “non-math” people that I’m actually incredibly stupid when it comes to math. All the more reason to love the subject I guess | For what it's worth, the first step to truly understanding a topic is being totally overwhelmed with everything you don't know and doesn't make sense. | It's this way with every subject. I'm sure many of your fellow high school teachers struggled with upper division courses in their fields as well. It's probably exacerbated for math by people being math-phobic, though. | I don't really find it funny. It's a reality of how our society functions. Personally, 25 of us out of a starting class of 750 were able to take Calc B/C in high school, something that gets decided for us in middle school. Before you're even 18, everyone is highly stratified in terms of math education, and it's really easy to get trapped into thinking that people are either smart or stupid. Usually safest to stay humble. | So, it’s important to think about what they’re saying. If it’s clear to everyone that you’re much, much better than they are at the topic: (1) either they’re normal and you’re a genius, or (2) you’re mediocre, as you claim… and then they must be unthinkably ignorant Be nice to your friends and let them call you the genius. :-) They’re not in a serious sense claiming you’re the next Erdos, they’re just acknowledging your real abilities while maintaining some respect for their own. | I wish employers hired based on willingness to learn and excitement in the topic. Honestly, I’ve been completely overwhelmed so many times… naps seem to help. |
[
"Do you think mathematical knowledge is finite ?"
] | [
"math"
] | [
"mn2jgs"
] | [
1
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""
] | [
true
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false
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0.54
] | null | I know mine certainly is. Maybe my knowledge is infinitely small too. | Depends on what you mean by "knowledge". If it's all the mathematical truths known to humans then yes it's finite because there will only be finitely many humans. If you mean the number of provable theorems of ZFC then no (this is infinite). But not all of them are interesting. If you mean the Platonic universe of mathematical objects (that exist independent of our existence and outside of the approximation that we make with our axiomatics) then I guess that first depends whether you believe in it or not. | Depends on what you mean by "knowledge". If it's all the mathematical truths known to humans then yes it's finite because there will only be finitely many humans. Even there, you need to be careful about what you mean by "mathematical truths known to humans". It's fairly easy to produce infinite families of statements that we can prove all of true (for every integer n > 1, the statement "n has a unique prime factorisation, up to reordering the factors" is a statement known to humans, for example, and it's not hard to come up with similar uncountably infinite families). | Yes. I mean the total number of proofs trees physically written out by humans (or their machines in the future). | Memory is finite. If computers had infinite memory, that might mean that chips are infinitely small which seems impossible. |
[
"Some prime number thingie I've been up to lately, but I can't exactly say anything"
] | [
"math"
] | [
"mn0b5m"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.5
] | null | So with X1=5 and X2=11. 5+9=14 i failed again | So with X1=5 and X2=11. 5+9=14 i failed again | N1-X1=X3 N2-X2=X4 | N1-X1=X3 N2-X2=X4 | Oh I see, sorry, but thanks anyway |
[
"Interesting asymptote"
] | [
"math"
] | [
"mmuyp7"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
0.67
] | null | Actually, y=1 and y=(x-1)/(x-1) are not the same object. It has "indeterminant form" at x=1 or whatever you pick c to be in y=(x-c)/(x-c). They the same everywhere but that one place, but they are not actually the same. | This. Also, y=(x-1)/(x-1) does not have a vertical asymptote. | The functions y = 1 and y = (x-1)/(x-1) look the same (and are equal) for all x except x = 1. At x = 1, the former takes on the value 1 (as it does everywhere), while the latter has a little “hole” (it is not defined at x = 1). | Aight thanks bro! So it just means it LOOKS the same but they're just completely different graphs ah | Yep! I'm not sure where you are in school, but there are some really cool theorems about limits kinda about this. These are technically different functions, but y=1 and y=(x-1)/(x-1) have the same limits (even at x=1!). |
[
"Is there a way to pot an exponential or log graph or reciprocal graph, then workout the equation of the graph by looking at points on it?"
] | [
"math"
] | [
"mmlj17"
] | [
1
] | [
"Removed - see sidebar"
] | [
true
] | [
false
] | [
0.67
] | null | Sorry, could you be a little more specific? Are you asking that "if given a graph of a function and we know if it's exponential, can we determine what the function is as f(x) = something? " | Yes, if there was a physics experiment that covered plotted an exponential graph and you wanted to extrapolate or find values in between your plotting, is there a way to find out the f(x) for that graph so you can find the y of any x value of the physics quantity | There is something called a log plot, where you plot x values normally but the logarithm of the y values (or scale the y axis with logarithmic distances) in that case any exponential function becomes a line in that plot, where the slope of the line is equivalent to the factor A in the exponential exp(A*x+B) | Oh yes, I’ve heard of this before in a random YouTube video! Thank you | I’m not exactly sure what you mean, but if your data points are more-or-less on a straight line then there are standard formulas for the “line of best fit” (if your points are on straight line then you can find the equation more easily from just any two data points). exponential graph The method above is for when you expect y = m x + b. But if you expect that y = a e you can still use the line of vest fit idea by first taking the log of all the y values: y = a e ln(y) = ln(a e ) ln(y) = ln(a) + ln(e ) ln(y) = ln(a) + k x So now we have “y = m x + b”, where x = x (no change) and y = ln(y), and we can use the original line of best fit method on the data (x , y ) to find the constants m and b. Finally, a = e and k = m. or log graph or reciprocal graph You can use similar changes (x,y) → (x , y ) to turn other kinds of data plots into straight lines. |
[
"Set Theory & Topology, or Real Analysis?"
] | [
"math"
] | [
"mmhyho"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
0.81
] | null | Real analysis: this course covers some very foundational material. Set theory is interesting, but less useful for most areas of mathematics. As a sidenote, your description of set theory and topology does not list any topics from topology. | Set theory is, as far as I understand, more of a mathematical logic topic than anything else, and topology (while it is founded upon sets) is kind of its own branch when you're starting out (of course, later it has strong ties into differential geometry, analysis, pdes, etc). I don't see why the two are tied together, but you'll mostly be starting from foundational stuff in both of those classes. It sounds like you already have a lot of knowledge in analysis. It depends where you want to go from here: do you want to learn a new field from the ground up? Do you enjoy analysis? | If they're tying topology to set theory, then it is almost certainly Point set topology (also called general topology). It has a much different feel from the topology you use in analysis or in any of the various topological Fields like algebraic or differential topology. It's also a less active field in general not a whole lot of people do research in it anymore from what I have heard ( but I could be wrong about that). As for the second comment, definitely. It's all about where you want to end up. Also look at the other options you'll have afterwards and what opportunities each one will open up for you. If you want to get into algebra for instance it's really going to depend on what courses come after these two, because neither is super relevant on their own. Also look at the professors that teach the classes. I never really liked analysis, but during my masters I had a professor that was absolutely incredible and now I'm doing a PhD that involves a lot of analysis. A lot of times the teacher to make all the difference. | It's true that there isn't a lot of research in point set topology, but it's also true that it's a basic tool of mathematics that one should absolutely be familiar with when it comes up. If this were about choosing a research field, it would be a different matter, but I think there's a lot of utility to studying these topics at the advanced undergraduate level, which is what this sounds like. | Yes I agree. Sorry I did not mean to make it sound un-usefully, it is very important to know. |
[
"Radio Antenna Line Of Sight to Aircraft Distance based on altitude?"
] | [
"math"
] | [
"mmxxkz"
] | [
1
] | [
""
] | [
true
] | [
false
] | [
1
] | [deleted] | If you're asking how far away a plane will be when it goes below the horizon, then mountains make a huge difference. | having an answer ignoring the mountains would still be very helpful for us. | Even if your horizon is ocean in all directions, there are probably things like atmospheric refraction that significantly change the answer. I feel like you'll get a much more useful answer on a physics or aviation board, unless you're actually interested in this pure geometry problem. | the pure geometry was what I was thinking, but yeah that makes sense. ill send my question that way. Thanks! | This is a question for /r/askmath or /r/learnmath . With r being the radius of the earth (in meters) and d the straight-line distance to the plane, you can just draw out a right triangle centered at the building's location. You'd then use pythagorean's formula to come up with d + (r+2300) = (r+35000) . Solving for d is easy. If the building is above the curve of the earth then it's a different problem. |
[
"I found a function that when evaluated it equals a predefined tuple. Is this something new or even useful?"
] | [
"math"
] | [
"mn0r4b"
] | [
0
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.47
] | null | First and foremost, playing with mathematical ideas is good, and I never want to discourage it. From what I can tell, the function you wrote down picks out the nth element of a tuple when you input n by summing over the tuple while multiplying by a function which is only non-zero when the input is n. Is this new? No. That's not to say that anyone necessarily wrote this exact thing down before (although they probably have). It is not 'new' because it contains no new concepts or calculations. You asking whether it's new tells me you are years away from understanding what 'new' would even mean in a mathematical context. I mean this encouragingly! Maths takes a lot of building up knowledge, but asking whether something like this is new is like asking whether you broke the world record 100m in your school sports day when you were nine. No. No you didn't. Is it interesting? If you find it interesting then yes! It would not interest any mathematician. What your function is saying is as follows: "The nth term in a tuple is equal to the sum over all the tuple where I have multiplied all except the nth term by zero". Finding a kooky elementary function which is zero for the appropriate inputs is a neat thing to muse on though! | What you did is called interpolation: given some inputs x (here, the integers from 0 to l) and some outputs y , you want to find a (smooth) function such that f(x ) = y for all inputs x . I think your way of doing so is interesting, if a little convoluted: you end up with a very large sum that's hard to evaluate in practice. You might want to look up Lagrange polynomials for a simpler way to perform the same task. | Yeah, I can see the link to Lagrange polynomials and interpolation. Thank you for the link. I guess instead of just having simple linear function that equal to zero at certain values, I use cosine squared. Kind of cool then instead of an interpolation polynomial in the Lagrange form it is an interpolation polynomial in elementary form, or at least cosine squared form. | Yeah the function repeating is by design but it will never overlap itself. Yeah looking at the other interpolation it's just a really convoluted way of doing it. | Yeah the function repeating is by design but it will never overlap itself. Yeah looking at the other interpolation it's just a really convoluted way of doing it. |
[
"Erdös Open Problems for Cash"
] | [
"math"
] | [
"mn7bfq"
] | [
164
] | [
""
] | [
true
] | [
false
] | [
0.98
] | Is someone still offering cash for solving Paul Erdös's ? My understanding is Ron Graham was managing it. Unfortunately he died last year. If you don't know what I'm referencing, check out this Quanta article. | Presumably Ron's wife and fellow mathematician Fan Chung Graham would continue the tradition. | What was it like revolutionising algebraic geometry? | im being pedantic but its ő not ö | Yep. My guess is that lots of people needed to misspell it for a long time because ő is not part of Windows-1252 / latin1, the popular single-byte character encoding that encompasses German, French, Spanish, etc but not Hungarian. Before the rise of modern UTF-8, most software didn't have the ability to type or represent his name, so they would probably swap out ő with ö. https://en.m.wikipedia.org/wiki/Windows-1252 | The only reason I'm working on the Collatz Conjecture is because I want that 100 bucks. |
[
"Infinitesimals in the extended real number system"
] | [
"math"
] | [
"mn5wm7"
] | [
11
] | [
""
] | [
true
] | [
false
] | [
0.82
] | So can we use a dedikind cut to form infinitesimals much like the extended real number system has infinities? | Don't quote me on this, but I think that approach won't work because the reals are complete. Dedekind cuts essentially fill in the gaps in the rational numbers. But if you do the same with the reals you won't get anything new, in the sense that the structure you obtain is isomorphic to the real numbers | I can confirm that this is correct. Moreover, the reals are the Dedekind complete ordered field. (Although they are not the only Dedekind complete dense linear order.) | The idea of Dedekind cuts is that a real number r can be described precisely by telling me which numbers q have q > r and which have q < r. The next jump up would be to try specifying a hyperreal number by telling you all the information about which real numbers are smaller/larger than it, but the problem is that this doesn't uniquely determine a hyperreal--two different unequal infinitesimals will both be characterized by "less than all posiitive reals, greater than anything else" so the Dedekind cut idea fails. Simply put, Dedekind cuts rely on the fact that the rational numbers are "order-dense" in R, in the sense that between any two real numbers there exists a rational number, so the data of knowing which rational numbers are bigger/smaller than me completely determines me if I'm a real number. But there are hyperreal numbers that don't have any real number between them, so cuts fail. | The way to construct the infinitesimals is through the use of ultrafilters. | Now that you mention it, I remember reading that's one way to define the reals up to isomorphism. |
[
"Opinions on Calculus on Manifold by Michael Spivak?"
] | [
"math"
] | [
"mmkxki"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.71
] | I plan on reviewing calculus upto manifolds and Stoke's theorem. We used this textbook for my second year analysis class (which I didn't do too well), and apparently it's a horrible textbook? Should I use this to review or should I use some other textbook? Any textbook recommendations for this topic? (Currently an 3rd year undergrad) | You might consider by Munkres. I haven't read it myself, but he cites Spivak's text as the inspiration and I think it covers largely the same topics but in more detail. When I get around to doing, well, analysis on manifolds, I'll be looking at Munkres. | It's not a horrible textbook. It's just overly concise and not suited for beginners. I think Spivak does integration on manifolds hastily and leaves out a lot of details that would escape the attention of a beginner. I found Rudin's Principles of Mathematical Analysis (used it only for multivariable differential calculus though) and Munkres' Analysis on Manifolds (for integration) helpful when Spivak was too terse. | About 150 pages long and weighs about 600 pounds. | Spivak's book is quite terse, which some people like but most will not. I prefer by C.H. Edwards, which goes into far more detail, has lots of examples, and is simply a better book. | Book is terse. Seek qualified instruction before attempting it yourself. I am not a math guide. |
[
"What problems would be solved if a “Largest Natural Number” existed? What problems would arise?"
] | [
"math"
] | [
"mmgxh4"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
0.69
] | This thought came to me while learning about uncomputability, and how the HALTS function is uncomputable, as an endless loop would cause no value to ever be returned. For reference, HALTS is a function that returns 1 if the function has an end, returns 0 otherwise. This function will either return 1 or never finish operation, as the alternative suggests an endless loop. However, if we had some largest natural number, let’s call it Λ, we could say “if the TM provided to HALTS doesn’t end in Λ steps, return 0.” I we wanted to be more specific, we could say Λ is a number that is larger than any BB(n), n ∈ N, meaning it would be bigger than the maximum finite operations of any Turing machine. Of course, no Λ exists, as Λ+1 > Λ. This question is just asking what existing “issues” would have a solution (like the HALTS example). What problems would come up from this number existing? | Well, you kind of stumbled upon the idea of ordinals (see here) . There is actually nothing wrong with defining a number to be strictly larger than all natural numbers, as long as you're not calling it a natural number. Ordinals are useful in proofs in their own right but are not part of Peano arithmetic (not that this is what is commonly used in mathematics nowadays anyways). For example, this series proves Goodstein's Theorem (something which can be stated but not proven using Peano axioms!) using ordinals, and goes through their creation and useful properties in an organized way before doing so. | To be honest, set theorists identify omega and aleph null. | Careful! Aleph null is a cardinal and omega is an ordinal, so they aren't comparable in the way you imagine. In fact they are intuitively the same size (the cardinal of omega is aleph null). You might want to think of cardinals as size-y (in a certain sense of the word) and ordinals as order-y (in a certain sense of the word). | Everybody knows 24 is the highest number. | If you were to order the , yes, aleph0 would be in the omega-th position. So far, they are more or less the same, but after that it gets tricky. For instance, omega + 1 is a valid ordinal that is different from omega, but alpeh0 + 1 is still just aleph0. |
[
"Career and Education Questions: April 08, 2021"
] | [
"math"
] | [
"mmuwye"
] | [
8
] | [
""
] | [
true
] | [
false
] | [
1
] | This recurring thread will be for any questions or advice concerning careers and education in mathematics. Please feel free to post a comment below, and sort by new to see comments which may be unanswered. Please consider including a brief introduction about your background and the context of your question. Helpful subreddits include , , , and . If you wish to discuss the math you've been thinking about, you should post in the most recent thread. | It’d be nice to knock out as much of the Calc sequence as possible. But upper level math courses are drastically different than what you may be thinking. Just enjoy your time in HS for now. Take a Calc class if you get course credit and really want to jumpstart your major. | I'm in my penultimate year of undergraduate study. I'm thinking of taking a gap year between undergrad and grad school. I want to do a Masters, and eventually hopefully a PhD. But, after double majoring in computer science and mathematics and battling some significant health issues, I'm burned out. I want to take at least a year in between programs. I was thinking of trying to get a job in software development or data analytics, so that way I could build some savings (double majoring is expensive). I was wondering if there was anything else that I could/should do that would help strengthen my application in the long run while taking time off? | Took 3 graduate courses while in high school: 2 graduate algebra and 1 graduate PDE. Expect to spend orders of magnitude more work than on your undergrad classes, with no handholding. The hardest thing for me was that your proofs are no longer read simply for correctness, but also elegance and conciseness as the courses are for training professional mathematicians. | I'm not sure you'll be able to find a strong connection between high school trig and social justice, but you can definitely touch on things like statistics and machine learning if you're looking to do a "math in social justice" type of lesson. Look into Cathy O'Neil's book for some examples. | It's definitely a fad, and it is also something that is beneficial to know. I think there will be a lot of demand for ML-adjacent labor for at least a little while, but I think that a lot of people who don't yet work in that field don't really understand what this work will consist of. Most (and, eventually, perhaps all) of this work is closest to what would be called "data engineering": it consists of creating efficient and maintainable systems for creating, cleaning, organizing, and standardizing large data sets. Machine learning algorithms can be easily standardized, and they are largely independent of specific use cases, so there isn't that much demand for the creation of new and fancy mathematical models. What does change with every use case is the specific nature of the data; we still need people to do the dirty labor of turning real world information into a data stream that the algorithms can make use of. It's going to take a while to scrape every aspect of every day life for useful data, and the systems that do that will need to be maintained and updated over time, so I think that this kind of work will exist for a while. So I guess my answer to your question - is that it depends on what, exactly, you mean by "pursuing a specialization in ML". Do you mean getting a bachelor's degree in CS so that you can get a good job? Sure that's probably a fine career choice. Do you mean getting a Master's degree in ML? That'll probably also benefit you in the job market. If you're looking for fun and intellectual adventure, though, then I'd say that you should give at least a little bit of consideration to alternatives. A lot of people are leaping into this stuff with dollar signs in their eyes, and that's a bad sign for opportunity and intellectual growth. The world is probably going to change a lot in the next 10 years, too, so it's hard to say how hot and thrilling ML will seem if (hypothetically) you get a BS followed by a PhD in the subject. It could be a whole new world of exciting intellectual discovery, or it could be a career wasteland of boring, but decently-compensated, garbage work. |
[
"Intro to Topology or Number Theory?"
] | [
"math"
] | [
"100y6jo"
] | [
5
] | [
""
] | [
true
] | [
false
] | [
0.78
] | null | I suggest Topology. If you're ever interested in getting back into math, knowing Topology will open up a lot of doors. It is far more fundamental than number theory, and you'll understand many of the "why are we doing this" of other topics when you learn topology. Number theory is good, but if you know algebra and analysis really well, I think you'd benefit from learning more modern approaches. | Whilst number theory is fun, topology is fundamental. | If you are asking for advice on choosing classes or career prospects, please post in the stickied Career & Education Questions thread. | I’ve got an okay understanding of analysis. I’ve taken undergraduate and graduate real analysis courses and a course in complex analysis as well. Also had a semester in abstract algebra and will be taking an advanced linear algebra course. I feel like most of my math courses have been pretty analysis oriented. Thanks for your input! I’ll likely take it to an advisor and see what they think | I’ve got an okay understanding of analysis. I’ve taken undergraduate and graduate real analysis courses and a course in complex analysis as well. Also had a semester in abstract algebra and will be taking an advanced linear algebra course. I feel like most of my math courses have been pretty analysis oriented. Thanks for your input! I’ll likely take it to an advisor and see what they think |
[
"why there isn't a tartaglia's triangle for trinomials?"
] | [
"math"
] | [
"100rhgf"
] | [
2
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
1
] | null | There is , and likewise for arbitrarily many terms. | I'll take a look, thanks. | thanks a lot! i was in a debate with the professor if a trinomial pascal triangle exist, she continued to deny but i guess i was right. | thanks a lot! i was in a debate with the professor if a trinomial pascal triangle exist, she continued to deny but i guess i was right. | You could just try it yourself and see what it looks like. |
[
"Ok, hear me out"
] | [
"math"
] | [
"100pdo9"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.27
] | null | Nope.. | No | The 3’s go on forever, but each 3 is a tenth as large as the previous 3, so adding up all the 3’s only gives a finite length | You might check out this page . | Why was this removed? It doesnt break any rules |
[
"Why do you like math?"
] | [
"math"
] | [
"mmxbg7"
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542
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""
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true
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false
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0.95
] | I find maths really interesting because it seems to make up the fabric of the universe. Everything in physics has been mathematically consistent. If the universe didn’t have consistent laws of physics, the universe would be a chaotic mess. Yet somehow maths maintains order. I also like how, using axioms, you can build up maths to new levels, so from set theory to group theory, group theory to abstract algebra etc. Maths is like a giant graph of nodes, each topic connecting to many others. We can use axioms to expand the number of objects in our graph. Lastly, it seems really mystical when really deep and complex maths created by humans 100’s of years ago is found to describe the physical properties of the universe now. It makes me wonder what other secrets the universe is hiding, and how the universe came to existence. | The consistency of it. Math always functions the same way. The same process works the same way on everything. Unlike things like grammar and biology and even chemistry and physics, where everything has a bunch of exceptions. As Bob Parr so eloquently put it, math is math. | I like it cause the elegance can sometimes feel like magic and because solving hard problems makes me feel smart. I know that might be shallow but it's the honest truth. | There's a saying that goes something like this (cough cough my topology professor): "While the physicist studies the physical universe and its consequences, the mathematician is only limited by his/her imagination." To me, it's exactly this imaginative and creative part of mathematics, where underlying structures of seemingly meaningless but infinitely interesting objects are revealed that gets me. It always does. This creativity of math and its accompanying rigor is what makes it elegant to me and what makes me like it so much. | Mathematical truths are undeniable. In Math, once you prove something to be true, it is forever true. Period. When you study subjects like Humanities or Science, there is always this tiny possibility that your conclusion can be false, and people can strawman this like no tomorrow. But no, you cannot say that my mathematical proofs are influenced by my bias, and they may be wrong. There are no such thing as "may be wrong" proofs. You either show that my proofs are wrong, or they are correct. That's all there is to it. | I agree with the shallow part. This might seem extremely stupid, but one of the big parts I love about math is the "fancy" words. |
[
"awnser this mathmetitions!"
] | [
"math"
] | [
"100ai8l"
] | [
0
] | [
"Removed - low effort image/video post"
] | [
true
] | [
false
] | [
0.1
] | null | No | you're bad at typing uniformly chosen random digits | Looks like about 1 | I copy and pasyed | ok smoothbrain |
[
"I want to count up the Julian date added to the next."
] | [
"math"
] | [
"100bsjv"
] | [
1
] | [
"Removed - see sidebar"
] | [
true
] | [
false
] | [
0.67
] | null | Pretty sure it’s n+1 not n-1. Your equation gives that the sum of natural numbers up to 1 is zero. | Turn them into pairs? (365+1)+(364+2)… = 366*182 +183 = 66,795 | This simplifies to S = (n(n+1)) / 2 where S is the sum of natural numbers up to n. So, n in this case would be 365. Apparently this formula was first discovered by Gauss (as a young child) when a teacher who thought they were clever gave him this problem thinking it would take a long time to solve. He had the same intuition you did and simplified the sum to the final formula. | Oh wait, I remember this formula, we learned it in math class for the number of unique handshakes between n number of people | Closed form for sum of arithmetic sequence |
[
"Software to produce math illustrations"
] | [
"math"
] | [
"100hqg1"
] | [
42
] | [
""
] | [
true
] | [
false
] | [
0.88
] | Math and physics books often have really simple but good illustrations. What software do they use for that? | TikZ. | https://personal.math.ubc.ca/~cass/graphics/manual/ | Maybe manim? | That's so cool. I worked in mathematical typesetting/prepress for seven years and never heard of this book. One of the sets on the cover looks familiar to me from the continuum theory courses I took. Maybe there'll be some tricks in this book to help me come up with graphics for the first few iterations of the chains used to make the pseudo-arc and the BJK continuum. I suppose the only other comment I'll make is that if you want to use a PostScript graphic with pdflatex, you'd need to turn the PostScript or EPS file into PDF first, but for that there's Ghostscript. But, it's been a while, maybe pdflatex know spawns a process for turning input PostScript files into PDF for inclusion in the final PDF file. Thanks for the link! | And PostScript doesn’t support transparency IIRC, which can be a deal breaker for many. But it’s loads of fun! |
[
"What books belong to a personal maths library"
] | [
"math"
] | [
"100mj68"
] | [
24
] | [
""
] | [
true
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false
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0.78
] | I studied maths, theoretical stats and probability, and computing in my youth. Now I have been working the industry for many decades and am still collecting and reading some classical quantitative book for self enrichment. During this journey, I have been building a personal maths/cs library of books. I can read grad school level material pretty well and looking for suggestions for text that cover a topic and its fundamental results well instead of very specialised discussion that only 10 people in the metaverse can understand so I can expand my library. What’s your suggestion? | This obviously highly depends on your interests, but usually if you Google something along the lines of "what is the best book for <insert subject here>", you will get many Mathematics Exchange posts about it. | I’ll go ahead and list some of my favorite graduate level math books which seem to fit your description of broad books well suited to a physical library. ‘Algebraic Geometry’ by Hartshorne is a classic and concisely covers most of the foundational results in the area (with the exception of analytic results in complex geometry). It’s an extremely hard book to read though. ‘Real and Complex Analysis’ by Rudin is also a classic for good reason, covering many of the essential results in measure theoretic real and complex analysis. ‘Abstract Algebra’ by Dummit and Foote is an amazing algebra book with great exercises. ‘Commutative algebra with a view towards Algebraic Geometry’ by Eisenbud is fantastic and has great appendices. ‘Introduction to Smooth Manifolds’ by John Lee is a very comprehensive and readable introduction. It also has great appendices which provides much of the needed background. I might add some more later on but these are the probably the physical books I cherish the most. Some other books worth mentioning since they are considered classics are ‘PDEs’ by Evans and ‘Algebraic Topology’ by Hatcher. I mention them here since I haven’t read Evans at all and I don’t like Hatcher’s book very much, but it’s definitely the most popular book on the topic. | Elliptic PDEs by Gilbarg and Trudinger. Any book by Stein and Shakarchi. | Lang - Algebra Lang - Algebraic Number Theory Neukirch - Algebraic Number Theory Knapp - Lie groups beyond an introduction Spivak - A comprehensive introduction to differential geometry Kobayashi, Nomizu - Foundations Of Differential Geometry Conway - Functional Analysis Rudin - Functional Analysis Folland - Abstract Harmonic Analysis Helgason - Differential geometry, Lie groups and symmetric spaces (dense and a bit more specialized compared to everything else on my list, but builds everything from the ground up, not an easy read (at least not for me)) | Here are some good reads that I know of, mainly for probability theory. Any recent book in the St Flour Summer School series: good to explore a certain topic with (almost) zero prior knowledge. Frieldi & Velenik - Statistical Mechanics of Lattice Systems for an overview of numerous models Revuz & Yor - Continuous martingales and Brownian motion is a classic. Lawler's books on random walks are great as reference books but may not be what you're looking for. Rudin's book on real and complex analysis is great for a solid basis on analysis and measure theory. Çinlar's book Probability and Stochastics is also great for probabilistic foundations. |
[
"How do you make the complex plane an intuitive step?"
] | [
"math"
] | [
"1014iy4"
] | [
66
] | [
""
] | [
true
] | [
false
] | [
0.84
] | The way I was taught, and how nearly every book and video introduces complex numbers goes something like this: "x +1 = 0 has no solution. But what if it did? x = √-1. We call this number i. But i can't be anywhere on the number line... so how about we just put it zero! Draw some lines and violà! We have the complex plane!" However, the step of putting i above zero always seems to come out of nowhere. Of course, once you get to explore more of what i does, it makes sense why we put it there. But how do you go from i +1=0 to putting i above zero as a logical step to take? | If you take a shape on the xy-plane and change every (x,y) point into (-x,-y), you get a shape that is a 180° rotation of the original. So in some way "⨯ -1" is a 180° rotation. If you do this you have ⨯ -1 ⨯ -1, which is just ⨯ 1, which means a 360° rotation of your original shape is exactly your original shape again. What if you want to do only of a 180° rotation? Of course that's a 90° rotation, but is it in terms of multiplication? Well, whatever “⨯ (?)” you're doing, it must be that ⨯ (?) ⨯ (?) = ⨯ -1 in the same way that ⨯ -1 ⨯ -1 = ⨯ 1, so now you have the famous x = -1. Indeed, if you describe a point as a complex number z, then around the origin. If your shape is just the single point (1,0)---which we can also think of as +1 on a horizontal number line---then multiplying by i should give us a "number" that must be 1 unit away on a vertical axis (corresponding to the point (0,1) on the xy-plane). | As others pointed out. You start off with R You explain things through geometry and linear transformations. You then show how certain rotations/reflections (like -1) correspond to real numbers. From here, you show how there is a "number" we can replace these transformations with. We can then show that the 90⁰ rotation is algebraically the same as i, and from here, you R explain how we can view it as C and use i to replace these transformations. Geometry to most people is "intuitive" while algebra/analysis tends to not be. Since complex numbers correspond very well to these geometric transformations this would be a good way to teach it intuitively. However, the caveat is that it might even be harder for students to accept complex numbers as "numbers". They will view imaginary numbers as instead transformations in R (which isnt "wrong") | if you decide to interpret i as a 270° rotation instead Or said perhaps more intuitively, if you decide to measure rotation clockwise instead of counterclockwise. | It's really not much different to adding √2 to Q and closing the field (i.e. Q[√2]). It's pretty clear that all elements you end up with are of the form a+b√2. While in this case, they all fit on the number line, it would not be the easiest visualization. Instead, we observe that Q[√2] is a two dimensional vector space over Q, and thus we draw two Q axis, the 'ordinary' rational axis and the √2 axis. | This is the right answer. |
[
"Taking effective breaks inbetween studying"
] | [
"math"
] | [
"100lkc0"
] | [
106
] | [
""
] | [
true
] | [
false
] | [
0.94
] | I'm a 5th year master's student. I feel that at most, I can effectively put in like 2 hrs for my thesis daily. It's just a reading project of sort and I don't have to attend any classes so I'm pretty much free all day. After taking breaks(playing games mostly), I usually work at like 50% efficiency for the rest of the day(that would be like 2 more hrs). How to take breaks effectively so that I can atleast do math 2 more hrs efficiently? Also how much hr do you normally put in doing math effectively? | I can only do 2 hours of serious intellectual work a day. Henri Poincaré used to do 2 in the morning and 2 in the evening and he was a giant. It's just super exhausting to run your brain hard. | Okay two tips based on the fact that computer screens are awful to read from: Print Look into the accessibility options on your laptop to have your computer read to you (very easy on a Mac) During my masters I could read my 100 page thesis once per day off of a screen, to make edits, but if printed it or uses text to speech I could read it twice. That was 2x efficiency. You’d be surprised how much eye strain affects ability to concentrate. Next tip: Keep track of what’s distracting you. When you’re doing your work, distracting thoughts will enter your mind and most of us just push them down… that doesn’t work because they come back stronger. When studying keep a pad of paper beside you and take a note every time you find yourself distracted of what the thought was. That evening address the distractions and they will occur less the next day. Tip 3: turn all of your screens grayscale. You want fake life to be fake and real life to POP! Tip 4: change up your study spot. Rotate through coffee shops, and different places on campus etc. | Playing games doesn’t rest your brain. | Working in a lab is different, and honestly MOST people doing 10 hours in a lab every day aren't working super efficiently. That's not really a slight though. It's hard dividing out lab work, reading, writing, and thinking! I have experience in bio, but I'm guessing the situations in chem and physics are at least somewhat similar. | Yup, exactly it's crazy difficult. Ik there is no comparison but my peers in other depts(phy,chem,bio) are basically working 10 hrs a day in lab and I'm here calling it quits after working a couple of hrs lmaoo. |
[
"I made a documentary about Ted Kaczynski's (unabomber) PhD Thesis"
] | [
"math"
] | [
"100aflc"
] | [
311
] | [
""
] | [
true
] | [
false
] | [
0.93
] | I made a post here earlier this month linking my video, but the auto-moderator never unlocked it so I'm following up with text. I created a two and a half hour documentary about Ted Kaczynski's PhD thesis titled "Bondary Functions", going into the weeds of the proofs as well as discussing his mathematics career biographically from what we know from secondary sources. I think this is the deepest dive available on the internet to date. The math is mostly real analysis with topological arguments, some measure theory sprinkled in, a lot of focus on Baire class functions. Anyways I've seen many discussions here over the years about Ted's math career and hopefully this answers some of the lingering questions from those threads. Thanks for watching. | Kaczynski's PhD thesis titled "Bondary Functions" I sure hope it isn't! That'd be quite the embarrassing oversight. Jokes aside: thanks for this--looks very interesting. | *Better known for other work | You know who else had whack spelling and making authorities wonder if it's troll or unintended? | fuck | half of the fun of watching my videos is spotting whack spelling and wondering if it's troll or unintended |
[
"Why are special constants typically around 1?"
] | [
"math"
] | [
"100bfx5"
] | [
39
] | [
""
] | [
true
] | [
false
] | [
0.79
] | Pi, e, Feigenbaum’s constant, Euler’s constant, and lots more I’ve seen throughout the years. They’re all around the same order of magnitude. Of course there are exceptions, but I’m wondering if there’s any reasoning here. | I of course understand what you mean on a practical level, but I don't think you can "prove" some particular numbers are special or important, since "importance" is not a matter of strict logic. I might as well ask why you think 3.1415926.... is "around" 1. After all, the word "around" has no rigorous meaning. All but one of the numbers you listed are greater than 2, so they are not even within 1 of the number 1. :) Your question reminds me of a quote of Ron Graham: The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. There is also an issue of selection bias here: you are remembering small numbers, but were you also trying to keep track of big numbers that came up in math that you studied? Perhaps the whole issue is affected by the kind of math you have seen. Google "5077 elliptic curve" or "Ramanujan 1729" and you'll see why those 4-digit numbers are quite famous in number theory. | Since the question is vague, I hope you'll allow me some liberties in my vague answer. For ratios like pi or the golden ratio, maybe we are more likely to take ratios of things that are "close" in size, meaning that their ratios will be "close" to 1. Euler's constant is the difference between two curves. The reason why the difference is taken is because we expect the areas to be "close," so that means their difference should be "close" to 0 which is "close" to 1. We might also expect some numbers to be "close" to 1 if they are about the same order of magnitude as a parameter and mathematicians normalize that parameter by setting it equal to 1. I don't think I have a good explanation for e or Feigenbaum's constant. That is, an a priori reason why motivations might lead one to discover a number close to 1. This might even get into metaphysics: a large constant for e would mean our universe explodes or something, lol. | Not sure I have a solid answer to your question but here are some thoughts: 1) It makes sense to say that a number is "around 1", contra other comments in this thread. Order of magnitude (some mathematicians call it "scale" instead) is just too useful of a concept in practice, even though the number 10 is kind of arbitrary. I would shamelessly say that all the numbers you listed have scale 1 when talking to another mathematician, even though it's not a precise constant. 2) Presumably so many interesting constants have scale 1 because of the anthropic principle -- a constant can only be interesting if there is intelligent life around to measure its value, so if a constant is interesting, then its value cannot obstruct the evolution of intelligent life. Imagine living in a universe with highly curved geometry, so pi is absolutely enormous. The laws of universe would be pretty fucked up! 3) However, I think there is some confirmation bias in your original post. Plenty of constants don't have scale 1, especially if the "reason" why the constant is finite is a compactness argument. Particularly infamous examples include busy beaver and Ramsey numbers. I felt inspired by your OP to see if I could get a crude upper bound on a constant that appears in some work of mine, and was able to get an upper bound of 9.8 times 10^(267) -- this could be optimized a lot, but I doubt that the constant in question is anywhere near scale 1. | I also think this is the reason. We've always mostly been investigating simple and intuitive phenomena, so we've been encountering relatively small constants. For example, people would take a circle, or a square, and investigate simple ratios in these rather than study polygons with googolplex edges. We just haven't been doing nearly as much stuff that can produce constants which are on the other of magnitude of the Graham's number (but we do this stuff as well, that's why we know of the Graham's number in the first place) | https://en.wikipedia.org/wiki/Strong_law_of_small_numbers?wprov=sfla1 |
[
"My sister passed out; Math suddenly feels meaningless"
] | [
"math"
] | [
"zh7j6l"
] | [
31
] | [
""
] | [
true
] | [
false
] | [
0.73
] | null | My condolences. I don't know you nor her, so I won't be saying stuff like do X because it's what she would have wanted. You know those things better. All I can ask is, please make your best judgment going forward. | God, this comment is so annoying. Read his post thoroughly. He found out, because of the passing of his sister, that there is more to life than maths. And that all the time he spends on maths, he could have spent with her. There is no way of getting this time back, so he feels sorrow. Now you understand? OP I feel for you bro. Don't be too hard on yourself, brother. | Please take care of yourself ❤️ it's easy to blame ourselves, but many things in life are out of our control. | Keep going for her. | Sorry for your lost. FYI "to pass out" means to go unconscious, like when one goes to sleep, or when one faints. |
[
"Is Euler's \"Foundations of Differential Calculus\" good fo learning calculus?"
] | [
"math"
] | [
"100k0t1"
] | [
135
] | [
""
] | [
true
] | [
false
] | [
0.84
] | I mean it should be. After all, it is written by the master of us all! But I want to know what is it's level? Is it for beginners? What are the prerequisites? | If you want to learn calculus, pick a modern book | Dude, reading differential geometry books from the early 20th century is already difficult enough (and not because of index notation, I am physicist, I live and breathe indices) because 1) they use terminology that is often completely different from today's; 2) they use archaic language that is understandable, but not often easily understandable; 3) pre-Bourbaki mathematicians had a completely different concept of what constitutes a mathematical proof than the post-Bourbaki ones (by today's standard, those works are not at all rigorous); 4) they often assume analyticity and other nice properties when less regularity conditions would be more appropriate; 5) they use notation that has been largely replaced or avoided in modern works. I sincerely doubt that calculus would be any different to DG in that regard. I have also read texts on an assortment of topics like Pfaffian systems and analytical dynamics from a time that postcedes Euler (usually mid 19th century) and they are close to unreadable. It is super interesting to read them to see how things were thought-of at the time, but for someone trying to learn a subject for the first time? Screw that, pick up a modern book instead. | You'll find that the texts written a long time ago do not resemble how we do and think about math today. Learn from a modern text, then read that one if you are curious about the early approach. | It is a good book. But, I do not recommend it because it spends all of its time on things that'll be covered in a few pages of a standard book. Now, if you want to spend a lot of time on _why_ stuff like infinitesimals and differentiation being a fraction is justified, you _might_ read it. But even then, the modern treatment of such ideas are quite different from how Euler justified it (though it was completely correct for his time). I think the main value of the book is it's historicity, and its proving of certain rules (like product, sum, quotient of derivatives, Taylor series expansion of a power, etc). If you want to start learning calculus, I recommend Richard Courant's book. | lol imagine being a hipster and the thing you choose to be a hipster about is . We've come a long way in the last few hundred years with respect to how to push information into our monkeybrains. It's silly to think that a decent textbook in a modern context won't blow away any of the original texts in terms of raw educational utility. Maybe you have an preference for those texts, which is fine, but they clearly aren't the best way to learn the material. |
[
"Recourses for self studying Stochastic Calculus?"
] | [
"math"
] | [
"zh572w"
] | [
27
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.92
] | null | I found that oksendal is a very readable first pass Stochastic Differential Equations: An Introduction with Applications (Universitext) https://a.co/d/cen5Jmi Or you could read study material for actuary exam MFE | Quant trading doesn't use much stochastic calculus anymore. It's mostly been replaced by various forms of AI. Study that if "breaking into quant trading" is your ultimate goal (more transferable as well). | It depends - banks and derivatives market makers use stochastic calculus to price their derivatives. ML is more commonly used in investment side (like prop trading or funds). | I’m also getting into some ML and DL but I’ve been told that hard applied maths is still the kingmaker for most strats. | by Steele is very readable. |
[
"Help! My brother keeps saying the imaginary unit i “doesn’t exist.”"
] | [
"math"
] | [
"1012h3w"
] | [
455
] | [
""
] | [
true
] | [
false
] | [
0.9
] | For context: My older brother who’s highest level math education is a B in college algebra keeps trying to tell me whenever it comes up in conversations, that the imaginary unit doesn’t exist, because you can’t have a number multiply by itself to equal -1. On the other hand, I argue it does exist, as it is used to solve and (with the help of the complex plane), visualize many real-world problems in physics. And that it being called an “imaginary number,” was coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless. There is a big difference in math, between “doesn’t exist” and “isn’t real.” So what should I say to him? | Tell him that -1 doesn't exist because you can't have -1 people in a room. | Ask him if the matrix 0 1 -1 0 exists. He won't get the point, but he won't get any point you make about this quite honestly. | You could take it further and convince him the real numbers aren't real: we never have sqrt(2) of length that we can measure precisely as that would require infinite precision. After that convince him to be a finitist. | Just ignore him honestly. As a brother, I can say there’s a nonzero chance he understands what you mean and is just messing with you to be annoying | I don’t think concepts in math rely on your brother to exist. Moving on. |
[
"Math book recommendations for a high school student?"
] | [
"math"
] | [
"zgxanj"
] | [
1
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
1
] | null | If you want to brush up on highschool stuff then I like Sheldon Axler's Precalculus 3rd ed a lot. It has the solutions manual right after the section exercises with full explanations so you get to see the thought process and approach. Oh I'd say also try out David Cohen's Precalc 6th edition if you felt like Axler skipped through too much algebra, which I only realized afterwards. Cohen includes an appendix with review material on factoring and other basic properties of stuff like exponents/nth roots/fractions etc. | Do some past papers and practice. When you stuck maybe watch khan academy videos to relearn sections from the papers that you are not able to do. In my experience so far I have not found a one-size-fits-all book. | Oo one book im reading now that explores a lot of topics is the introduction to mathenatical crytography by hoffstein pipher and silverman. idk if its suitable for op since its undergrad and if youre not interested in crytography you can skip those parts; but its great because cryto covers so many different areas of math and a lot of it is covered in the book (notably tho calculus is missing, more emphasis on number theory, vector/lattice spaces, linear algebra, at least the parts im looking at in detail). and its cool to see applications of trapdoors in math!! | There's a bunch of geometry books I saw recently that don't use more than high-school maths... some of it will probably be familiar, but some of it should be new, and I think having a good understanding of geometry can be really helpful! Let me know if that'd be of interest and I'll send some titles | Guys, Can I recommend this? The most beautiful book on mathematics. I hope it goes well for highschooler though. |
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