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[
"Not open is not closed. What other unintuitive naming conventions can you think of?"
] | [
"math"
] | [
"tng69h"
] | [
112
] | [
""
] | [
true
] | [
false
] | [
0.93
] | null | coconut is technically not a nut, but according to category theory naming conventions, it is. | Elliptic curves are not elliptical. Ellipses are not elliptic curves. | It always bugged me that elliptic curves - which by definition are nonsingular - can be supersingular. Iirc the term was coined to denote the curves being special in a sense. One of the possible generalizations to higher dimensional abelian varieties is actually called superspecial, which for elliptic curves coincides with the notion of supersingular, but alas, supersingular has been the de facto name and too well-established by now. | Jacobson defined a to be a gadget with the rules of an algebra, except that no mention of associativity is made at all. Then he defined an to be a that is associative. | Here's a bunch https://ncatlab.org/nlab/show/red+herring+principle |
[
"As of 2022, what is the most esoteric math subfield/literature you know of?"
] | [
"math"
] | [
"to0xve"
] | [
328
] | [
""
] | [
true
] | [
false
] | [
0.98
] | Obscure, weird, mind-bending, or all of the above! | Physics | Almost mathematics, including almost rings and almost modules. If only for the name alone. | Large cardinals seem like pure sorcery. Surreal numbers are fascinating and deserve to be better known. Unfortunately it's hard to do analysis with them. | I'm almost intrigued. | Perfectoid spaces. Not the most esoteric insofar as it's part of why Scholze got his Fields medal, but still. |
[
"A cool pattern of prime numbers"
] | [
"math"
] | [
"mg7yvg"
] | [
37
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.87
] | null | In practice you are constructing a series of numbers that are not divisible by 2, 3, and 5, and don't end in 3 or 7. So, until they are small numbers, it is not strange they satisfy that property. | It's the smallest natural number which can be written as the sum of 2 positive cubes in 2 different ways: 1729 = 1 + 12 = 10 + 9 | I think it's just a coincidence, though I'd love to be proven wrong. | Law of small numbers | Ramanujan! |
[
"Just a quick match question"
] | [
"math"
] | [
"mg0b2c"
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0
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""
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true
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false
] | [
0.4
] | null | Well, if the chance is 0,26% to get a knife in any given case, there is also a 99,74% (=0,9974) chance, it doesn’t contain a knife! So the probability of all n cases to contain a knife is 0.9975 For n=65 that’s around 0.85=85%. | Oh that was a lot higher than i thought for 1 knife. What if is repeated? So getting 2 knives in 65. How would you do that? And thanks for the quick answer. Edit: I just realized it is 85% chance NOT to get a knife. So nvm that first part about it being high lol. | Well, if you want k (here 2) successes (aka knives) from n (here 65) tries and the chance for a success is p (here 0.0026), hence q = 1-p being the chance of failure (here q=0.9974), the probability of exactly k successes is P = (n over k) * p * q . For the given values that’s about just shy of 1,2%. | Thank you so much for the help. It really because complicated compared to just the probability of it happening once. | Yep :D Just to make that clear: The ~15% from the first answer is not the chance to get exactly one knife. It’s the chance of getting atleast one knife |
[
"How to approach take home exams?"
] | [
"math"
] | [
"mglpp3"
] | [
13
] | [
""
] | [
true
] | [
false
] | [
0.79
] | [deleted] | Proof-based calculus is called analysis, sometimes "advanced calculus" in old books, though I doubt your professor is giving problems that are like solidly analysis because its typically a whole other course. Besides skills on proof-writing in general (i.e. chasing definitions), some useful introductory books for Calc 3-type analysis are: Though if these aren't the types of proofs your professor is asking for, we'd probably have to take a look at a few examples before being able to point you to proper resources, since a strong emphasis on proofs in Calc 3 is quite rare. A good intro book for analysis at the Calc1/2 level is Ross, Elementary Analysis. This could be useful for you as well. | This is probably not going to be a helpful reply but Calc 3 (as a course) is about application, not fundamental theory. If your professor is doing this they have no business in that classroom and should be scolded or removed. Your not learning, or at least being tested on, the useful bits and you are being tested on useless drivel (unless you go into math professionally). Take the advice already on this post but you should also go to your professor and ask him what he's playing at. If it doesn't resolve the problem go over their head. I went to a top 10 university for physics. The problem you just described is a waste of your time (again unless you're going into mathematics). I had an arrogant cuss of a Linear Algebra professor who pulled this crap on 2 of the three tests and all the physicists (70% of his class) complained to the head of our department that work, tests, and what we needed to learn were not aligned. It was corrected by the final. | Have you asked your professor how he would suggest you prepare for the exams? | Right here. In mathematics I find that we have this (I feel) toxic attitude of blaming the student (who is often ourselves) for everything that goes wrong. So we end up spinning our wheels in the mud looking for help anywhere we can when a professor slams us with unreasonable expectations. It's a huge waste of everyone's time, even the mathematicians. You gotta go to your professor and tell him how you feel. Maybe gently walk him to it when he sees how frustrated you are, or maybe more directly depending on what kind of person he is and if he doesn't listen, yeah maybe the higher ups need to hear about it to. Calculus 3 is not supposed to be that hard I don't care what university you go to. | I'll definitely check it out! He does say to understand the geometric and algebraic aspects of the material. The algebraic part is pretty easy, but the geometric part is what really gets me and everyone in my class. |
[
"Mathematicians Find a New Class of Digitally Delicate Primes"
] | [
"math"
] | [
"mgi4wa"
] | [
407
] | [
""
] | [
true
] | [
false
] | [
0.97
] | null | Is it just me, or is anyone else just totally uninterested in base-dependent results? Unless there's a direct link to something interesting in modular arithmetic, these kinds of results not really about the number itself... they seem to be more like than mathematics. | I was with you until I found about the Bailey-Borweni-Plouffe formula. It's as if certain grammars/bases are better than others at exploring properties of numbers, more concise, more regular... They reveal "something". Then, there must be some interesting mathematical ideas to find exploring them. https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula | Take a look at the numbers 294,001, 505,447 and 584,141. This is maybe the worst example of punctuation I've ever seen in my life. As for the result, it's obviously amazing that someone was able to it, but it's not all that surprising to me; once you get up into big enough numbers, essentially nothing is prime, so it's not too surprising that if you take perturbations that take an absurdly large number and make it even more absurdly large you don't usually wind up with primes. | I agree with you when we talk about a specific base, but if a result concerns all possible bases I'm inclined to consider it more a mathematical property than just a curiosity. | Interesting, I had never heard of that formula. Thanks! |
[
"Math podcasts?"
] | [
"math"
] | [
"mgkwgf"
] | [
47
] | [
""
] | [
true
] | [
false
] | [
0.92
] | I’ve listened to the breaking math podcast, and I find it to be awful quite frankly. It’s often clear one (or both) don’t have a clue about what’s being discussed, and it’s frustrating and misleading because I can’t trust what they’re saying. With that being said, I’m looking for a new podcast series to listen to. Specifically on the continuum, or really anything along that line. Thanks is advance! | I like the numberphile podcast. They sometimes go onto random tangents not about math but I sort of like that. You get a feel for how mathematicians think. https://www.numberphile.com/podcast | Three that I like are Ben, Ben, and Blue , My Favorite Theorem , and The Turing Podcast . | Joy of X is pretty solid, but it's a mathematician interviewing other scientists (Mostly mathematicians and physicists) https://www.quantamagazine.org/tag/the-joy-of-x/ Quanta Magazine has a news podcast as well, but once again it's a mix of Math, Physics, and Computer Science. https://www.quantamagazine.org/tag/podcast/page/3 I think it was mentioned in another comment, but Numberphile's interviews are also pretty solid as it is a lay person talking math with a mathematician https://www.numberphile.com/podcast | Among those I've heard, is most heavy on higher maths. There are at least some episodes in there that I have found very interesting, but also some episodes that are too far away from what I'm familiar with. | There was one female professor at ETH Zurich on here who has an interesting podcast and she is on this subreddit . Unfortunately, I forgot her name. |
[
"What makes a mathematical constant interesting?"
] | [
"math"
] | [
"mgcul1"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.13
] | What makes a mathematical constant interesting? Is it necessary for it to pop out in a lot of formulas? Or for it to have a special meaning? Truth is I think I have found a constant, and I wonder if it is worth sharing with anyone else. | What the hell is it? | But is it? A constant is not interesting because of the value that it has, a constant is interesting because of the that it has. What's this number do? | I mean that argument works for natual numbers, because natural numbers are canonically well-ordered (so that the existence of one uninteresting natural number implies that there is a least uninteresting natural number which is then interesting), but for real numbers I find the argument unconvincing. You can of course use the axiom of choice to get a well-ordering W on the reals. But I see no reason why the W-least uninteresting real number would be of any interest to anyone. | What's the constant? | I do not think the meaning is hidden in its value, but I can share it if you want to. |
[
"Is Serre's A Course in Arithmetic any good?"
] | [
"math"
] | [
"mg5ea9"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.81
] | I'm looking for a book on number theory, and this one seems to be recommended pretty highly. Can anyone who's read it vouch for its quality or lack thereof? | It is a well written book, but for a particular audience. There is definitely a reason it is in Springer's Graduate Texts in Mathematics line of books. The first part (which is more than half of the book) is algebraic in nature and will assume you are readily familiar with basic notions in abstract algebra. The second part is analytic in nature and assumes you have a passing familiarity with complex analysis. The algebraic part is mostly about quadratic forms, which makes it an unusual introduction for the algebraic side of number theory. The analytic part is a more standard introduction to L-functions and modular forms. Overall I would say the book feels like a slice of interesting topics and less a book intended to make you feel like you have a broad sense of the subject. But for a graduate student who has taken the core coursework, it is an elegantly written book with a lot of clarity on its topics. For an undergraduate who has taken a few upper-level courses it could be a pretty challenging read. For better or worse, the book does not have exercises. This is dangerous as it makes it a little too easy to believe you understand what is going on when you should probably think about it more. | As with all recommendations, it depends on your background, of which you have told us nothing at all. If you can't read the first page easily then the book is not meant for you (yet). | This could mean a lot of different things depending on what classes you took and what university you went to. https://www.math.purdue.edu/~jlipman/MA598/Serre-Course%20in%20Arithmetic.pdf See if you can understand the first chapter--if you do and the contents seem interesting, get it. | Look at Ireland and Rosen's "A Classical Introduction to Modern Number Theory". It's longer than Serre's book, and covers a different range of topics, but while it too is in the GTM series I think it is more immediately readable to a general upper-level undergrad. | Back in the day this was _the_ introduction to modular forms. But things have moved on since then, now we have books like Diamond-Shurman for example. |
[
"Which Math Theorems Remain Surprising, No Matter How Familiar They Are?"
] | [
"math"
] | [
"mgmq9j"
] | [
25
] | [
""
] | [
true
] | [
false
] | [
0.97
] | null | Quadratic reciprocity. Among the results that are common to see in an undergraduate math curriculum, it is by *far* the least intuitive in my opinion. Why p being a quadratic residue mod q should have anything to do with q being a quadratic residue mod p is beyond me (yes, yes, I've seen proofs, please don't reply with an abridged version of one). | The theorem that /u/ben1996123 mentioned requires that the function has the complex numbers as both domain and codomain. | From the perspective of the Hilbert reciprocity law, which resembles a multiplicative analogue of the residue theorem, quadratic reciprocity becomes the "only" global relation among Hilbert symbols. When you input two different odd prime numbers p and q into the Hilbert reciprocity law, only three Hilbert symbols are not automatically 1 (symbols at the primes 2, p, and q, since p and q are positive), so what gets spit out is a relation between p mod q and q mod p being squares. If you allow "negative primes" to be used then the relation gets more complicated. With the residue theorem, a meromorphic function analytic away from three specified points can be chosen to have any residue at all at one of the points, or even at two of the points, but the global residue theorem makes the third residue completely determined by the first two. So locally there is a certain amount of flexibility provided all the residues satisfy a single relation described by the residue theorem. | I love this comment largely for highlighting that “proof of thing” does not always equate to “the ‘real’ reason thing is true” | One could say you are quite stoked about this theorem... |
[
"Weak Law of Large Numbers"
] | [
"math"
] | [
"mgcdzm"
] | [
11
] | [
""
] | [
true
] | [
false
] | [
0.93
] | Does there exist a sequence of random variables that are dependent and their variances are not finite and the Weak Law of Large Numbers still holds for them? | Im inclined to say yes, without providing much in terms of a proof. Consider two distributions with infinite variance, one with mean=1 and the other with mean=-1. Let’s also assume that the law of large numbers holds for both of these distributions. Start by selecting a number x(1) from the distribution with mean=1. Let M(n) be the sample mean after n trials. So, M(1)=x(1). In general going forward, if M(n)<0 select x(n+1) from the distribution with mean=1, otherwise use the distribution with mean=-1. I would conjecture that as n->infinity, M(n) would probably be get really close to 0. Why is that? Well, suppose we found ourselves at a point where M(n)=100. We would start selecting from the distribution with mean=-1 as long as M(n) stayed positive. But we assumed that the LLN holds for the distribution with mean=-1, so over the next N samples, the mean is probably going to be -1. Further, there probably is some N>>n such that M(n+N)<0, at which point we will switch back to the distribution with mean=1. | Take X {i-1}. Then Y_i is dependent, first absolute moment finite and WLLN holds, and the variance is infinite. Is that the type of example you are looking for? Edit: sorry, my formatting is bad, but I am sure you get the idea. | If we have an i.i.d. sequence with infinite variance and finite absolute expectation (such as the t distribution with 2 degrees of freedom), then the weak law of large numbers holds, but the variance of the random variables is infinite. An i.i.d. sequence of random variables is a special case of a dependent sequence of random variables. In general for i.i.d. random variables, we need E|X_i| finite for the weak (and strong) law of large numbers to hold. It is very well possible for E|X_i| to be finite and the variance to be infinite. | Don’t the infinite variances will cause a divergence of the M(n) then? If M(n) = 100 then we pick a value from negative mean distribution. However, the infinite variance of the distribution would get us to a situation where M(n+1) = -‘very large number’. Again the positive mean distribution would get us to very large positive number and etc. I don’t mean that you are wrong but I’d say that it is not so obvious. | i.i.d stands for “INdependent identically distributed”. I know that it is pretty easy to construct independent sequence. |
[
"The Mathematics of Dishwater"
] | [
"math"
] | [
"mg7kb8"
] | [
439
] | [
""
] | [
true
] | [
false
] | [
0.95
] | Where do you do your math? Ever since undergrad, I never really wrote much down. I remember in graduate school, I would sit down on my couch and I had a particular corner of the ceiling I would like to gaze at while I thought about my dissertation, and now as a professor and parent of toddlers during this pandemic, I find most of my mathematical thinking happens while doing chores. For instance, I just connected continuous POD modes from fluid mechanics to the SVD of a particular Gram matrix my colleagues and I have been toying with lately, while doing dishes. (As much as I wish I could claim that it was the fluid dynamics of the laminar flow of my faucet that inspired me... it wasn’t.) I know I’m not the only one out there that works like this, and I imagine most mathematicians do to some extent. I know that Poincare had particular inspirations on bus rides to and from work. | r/Showerthoughts but for math? | Sadly this is not the case for me. I need pen and paper to write my thoughts and draw things as my mind tends to divagate and lose track quickly. (Even now I should write a nice little proof I have been fighting with for a while). | r/showermath | I guess I'm in the middle. I muse on big-picture ideas in math anywhere that isn't my desk: on a walk, birdwatching, etc., but I really delve into the ideas at my desk. I can't do the latter when up and about, and I can't do the former when I'm sitting down. I need to do both to be productive, though. | I didn't say anything about no drugs. |
[
"Which area of mathematics did you neglect the most during uni?"
] | [
"math"
] | [
"mgf0hu"
] | [
99
] | [
""
] | [
true
] | [
false
] | [
0.98
] | Obviously, mathematics is quite diverse, and no one can be expected to know about all of it, so I am wondering: Which area of mathematics did you focus the least on during your education? Where are your largest knowledge gaps? (Of course this question is restricted to areas that you could reasonably expect to see during a mathematics degree - there is little point in everyone answering inter-universal Teichmüller theory) | For me, it has to be differential equations. I'm close to the end of my Master's and have managed to not take a single course on them. My analysis course did prove the Picard-Lindelöf theorem and differential equations did briefly show up in my complex analysis course a few times, but that is essentially all I know about them. | Probability Theory. Yes, introductory probability was compulsory for all math majors, but as someone who used to have a disdain towards "applied" (read: computational) math, the way it was taught really put me off probability. (Now that I'm finishing my masters I have a different perspective and find that applied math can be really theoretical as well) I'm studying PDEs now and am learning that there are a lot of connections between them and probability theory. I wish I had not given up probability so early and taken a proper modern probability course. I think a main factor in killing my interest in probability theory was the plethora of artificial word problems with a unnecessary story fluffed up in an introductory course. If more examples have been drawn from 'real mathematics' - e.g. random walks, my interest may have been kept longer. | Combinatorics, I'm truly ass at combinatorial problems. | Complex analysis. For some reason I was not required to take any course in it, even though plenty of later courses stated it as "recommended previous knowledge". | Anything geometry. Went to a good American university for my undergrad that for some reason didn’t have a differential geometry course or anything like that when I was there and my PhD advisor (at a different university) never advised me to take anything in this direction. Now I’m an associate professor at a European university and fairly actively publishing, but it seems standard for everyone here to have knowledge in this area and has caused a few awkward moments when I didn’t know any more than that the word “manifold” sort of means something like “surface”. |
[
"Author order matters.... prove me wrong"
] | [
"math"
] | [
"mgfk47"
] | [
141
] | [
""
] | [
true
] | [
false
] | [
0.85
] | My training as a mathematician was in pure mathematics, where publications come slowly. I published the three papers that resulted from my dissertation a year or so after I graduated. Subsequently, I had started a postdoctoral position in engineering, where publications are submitted at a rapid pace and there are deadlines to meet for conferences. Thus, by the time I became a professor, while my academic training was in mathematics, I had come up in a rapid fire culture of publication in engineering and computer science. This results in a bit of a culture clash when I publish with other mathematicians, where they insist on alphabetical order of authors, but I insist on the main contributor being the first author, no matter where we publish. i especially insist on this when it concerns a paper by one of my graduate students. Papers are invariably referred to by the author in first position, even in the case of alphabetical authors. I want my students to get the credit for their work, even if their name is Zhang or Zhu. I feel that alphabetical order is a throwback to mid 20th century customs, where collaborative papers in pure mathematics were strongly discouraged. Some would say that writing a collaborative paper was like writing none at all, because you couldn’t be sure who did what. I see alphabetical order as the result of the community resisting giving individual credit for multi authored papers. what do you think? Am I missing something? is my interpretation wrong? EDIT: Thank you everyone for your very thoughtful responses. This has turned into a nice discussion. I assume the 20% downvotes are people that vehemently disagree with me, and truly, vote how you like. The purpose of this is to spark a discussion, and clarify the community's perspective on authorship. | I disagree with any attempts to change the alphabetical listing of authors. It is an outrageous attack on the dignity of the academic institutions. -- Aaron A. Aaronson | No joke: When we named our children, my wife argued that my surname should go first in the hyphenation so they would be earlier in alphabetical author lists. I'll let you know in 20 or 30 years if it made a difference. | Math PhD student here: I really like the custom precisely because collaboration is the norm. I have rarely worked on a project where it would have many any sense at all to suggest that one of the authors deserved more credit for it than the others. I appreciate very much that when other people read our papers, they will not be under the false impression that there was a “main” contributor because they know that mathematicians do not do first authorship. In fact, I have a former coauthor who has a similar experience to yours, but with the “opposite” problem (in a way). We published a math paper together as a part of an undergraduate research program, and obviously we were very pleased with this because that would look good when applying for grad school. Though I was planning on going for math, he was considering going more into computer engineering and CS. When he applied for funding, the NSF didn’t grant it to him, and one of the reasons that they cited was that, though he had a name on a paper, he “wasn’t even first author”, so it didn’t matter - not knowing or noticing that it was a math paper and that the names were listed alphabetically. I had, and continue to have, a big problem with this. The presumption of first authorship led someone to assume that there was really only one person who deserved credit for our work, and as for all the other names on the list, not so much. Even if math using first authorship, it wouldn’t have made sense for this project. Nobody deserved more credit than anyone else, and it would have been very troubling to have to pick someone to be first author and to receive more credit. If I had been first author, he wouldn’t have been given as much credit (or even much credit at all, as suggested by his grant application), and if he were first author, I wouldn’t have. And that’s to say nothing of the fact that in all likelihood, our faculty supervisor would have gotten first author because that’s just how things work out a lot of the time. I would probably be more or less okay with having the to list people as “main contributors,” but barring that, I do not like first authorship at all. You can’t avoid listing the authors in order, and so has to be listed first. I would not enjoy it if this led readers to believe that a single one of the authors was more deserving of credit for the work than the others, in no small part because I can’t think of many of my papers so far where it would be even close to accurate. I would rather the author order give a reader no information useful for inferring who contributed more than for it give readers an active reason to come to false conclusions. Also, as you say, math papers come out more slowly. If math did first authorship, and you were a contributor on a paper, but you get first author, it would be a pretty big deal that in one of the few things you published recently, you were given diminished credit. I can see how if you’re publishing papers at a more rapid pace, it’s not as big of a deal if you unfairly receive diminished credit on a paper here or there. If I only publish, say, two papers in a year, and on one of them instead of getting presumed-equal credit I get presumed-lesser credit, that’s really significant. This also makes me think that it would be super common for students to very often have their supervisors take first author as a rule. Some wouldn’t, but I’d have a hard time believing that nobody would do that. Also - and I don’t know this for sure, you’ll have to help me out here - if engineering papers are being produced so quickly, I imagine that each individual paper likely only represents a single idea - one that would be more easily attributable to a single individual. In math, it is good practice to try to get as much into the paper as you can (up to page limits and legibility, of course). In the papers I’ve worked on, the philosophy has been to not leave low-hanging fruit on the table. Once you have your initial “main idea,” and you get a result from that, you then think about whether there are any follow up questions you can answer in that paper too, or any implications you can explore. In my experience, this gives more opportunities for different parts of the paper being primarily attributable to different authors. Maybe I had the main idea that “cracked” the proof of the main theorem in Section 3, but someone else did the same for the proof of the main theorem in Section 4. It would be way harder to distill that process into saying “this one person is the ‘main’ author of this paper,” as the scope of the paper often means that there are many places where significant ideas and contributions were made by different people. | I have three main problems with 'contributional' order: 1) It says that it signals who contributed the most, but the signal can be really weak. For "X, Y, and Z", did X do almost all the work? More than half of it? A little more than a third? There's no way of telling just from the author order, but the system nevertheless demands that you recognize X as having done significantly more work than Y who in turn did significantly more than Z. 2) I know of too many examples where 'contributional' author order could more accurately be described as 'political' author order. Stuff like "This is Y's project but X is graduating soon and needs more first author positions, so this paper will be "X, Y, and Z" and the next one will be "Y, Z, and X" or "Z offered us their data if we make them first author" makes me put very little stock in author order as a signal of effort. 3) How do you measure and compare effort and contribution? If A spends 300 hours trying to prove something and doesn't get anywhere but stumbles upon an insight that lets B prove an important result in an afternoon, is that paper "A and B" or is it "B and A"? How do you compare the contribution between people who work on completely different parts of the project? Is A's 100 hours of theory equivalent to B's 100 hours of writing code? What if the venue values theory over experiments? That is, getting rid of B's experiments section makes the paper weaker but getting rid of A's theory section makes the paper a desk reject? | The rationale, such as it is, for mathematics authorship practice is given in http://www.ams.org/profession/leaders/CultureStatement04.pdf I think most mathematicians would agree that this statement is accurate when it comes to describing their collaborations. The issue is that, in engineering, the statement may not accurately describe the nature of progress in research. I have both engineering papers and mathematics papers (disjoint sets of papers: it is not the case that any single paper is simultaneously both). The rule is simple in such cases: if you publish in an engineering journal, use engineering practice. In a mathematics journal, use mathematics practice. |
[
"What is the meaning behind the second derivative notation (d^2y/dx^2)"
] | [
"math"
] | [
"kf89k4"
] | [
1
] | [
""
] | [
true
] | [
false
] | [
0.67
] | null | Wait till you see how they abuse the del and dot notation from calc 3 in more advanced classes. | The d means you’re doing the derivative operation twice. The x means that both the first and second time are with respect to x. It’s suggestive, but not necessarily to be taken literally. Edit: Maybe viewing it like this would help. d y/dx = d/dx(dy/dx). Analysis folks, especially physicists, love suggestive operation notation. | That makes sense. I originally thought the same about dy/dx being a fraction until I learned about differential equations. I was just seeing if the same was the case for the second derivative. Thanks | The operator for taking a derivative is often written ( d/dx ). Taking it twice and distributing squares gives ( d^2/dx^2 ). We frequently write the application of the operator ( d/dx ) to the function f as ( df/dx ) instead of ( d/dx )f, and similarly ( d^2f/dx^2 ) instead of ( d^2/dx^2 )f. | That actually makes a ton of sense. It’s just squaring d/dx then writing it in terms of y. |
[
"Anyone have a solution to this time problem"
] | [
"math"
] | [
"kf794a"
] | [
1
] | [
""
] | [
true
] | [
false
] | [
1
] | null | From 9am to 1am is 16 hours. So you can have a stretch every 1 hour and 26 minutes assuming you wanna do one at 9am sharp and one at 1am sharp. 16 * 60 = 960 for how many minutes in 16 hours. 960 / 9 = 106 for how many minutes per chunk of time between stretches. | There are 24 hours a day and 1am to 9am is 8 hours. You just minus 24 by 8. If there are 10 stretches that you're willing to do two at the beginning and at the end of the day, then you just need 9 chunks of time in between. Think about it, if you wanna divide a little area in parallel using 10 sticks, having 2 at both ends, then you only have 9 subareas. | So you can have a stretch every 1 hour and 26 minutes Sorry 106 minute is 1 hour 46 minutes, lol! 5pm or 1800 5pm is 1700, no? The second part I'm still a little confused Let me draw you a diagram to explain that. Hang on! | Try understanding this diagram ! Let me know if you have more questions! | Hi so quick thanks! How did you work out the 16 hours? Why did you divide it by 9 and not 10? |
[
"Complex number help!!"
] | [
"math"
] | [
"kf4ulg"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.25
] | null | This is actually very simple. But since we want to discourage this kind of posts in r/math and r/mathematics , I won't explain it to you. Consider posting this in r/learnmath or r/HomeworkHelp . | I realize it might sound rude, and I apologize for that. But the truth is that before posting on sub you should read the rules. And this sub's rules specifically state that homework questions are not allowed. But people keep doing it and many of the users who frequent r/math and r/mathematics expecting a different kind of post have the right to be mad or even confrontational. | You can post questions on this sub. That's not against the rules. I have done it. But the problem is the level of the question. What you asked is (and I apologize if it sounds patronizing) the equivalent of asking how to count in high-school, relative to this sub of course. I'm not saying your question is stupid. The reason I called it "simple" is that it actually is. It's a matter of doing some algebra to rearrange the equation, before or after the substitution. No fancy tricks involved. I'm sorry that you took it personally, but it was not my intention to upset you. | Thanks for the help! I will go and have a look. I would say that your message comes across a bit rude. As someone is trying hard to understand what I am learning, being told something is very simple with no explanation is not very encouraging. | This is not a homework question. I’m doing a bit of extra reading and I can’t understand some of the reading. I’m studying computer programming at university and I have a better knowledge of some area of maths than other areas. I have posted on this sub before. Maybe the sub should clearly state in the rules that this is ‘NOT’ the sub to post on when you have questions about a subject. I fully understand what you mean about this type of question. I’m sure to most people it is very easy. However, there is a nice way to speak to someone about it. I know I’m just one random person posting on this sub but when you are struggling to hear that a question I have is ‘simple’, it’s not very nice. |
[
"Are there two different functions that look completely identical graphically?"
] | [
"math"
] | [
"kf52ym"
] | [
1
] | [
"Removed - incorrect information/too vague"
] | [
true
] | [
false
] | [
1
] | null | If two graphs are EXACTLY the same, then they are the same function. For a more interesting answer there are graphs that are the same other than at one single infinitesimal point. Like (x - 1) / (x - 1) and x + 1 for example, which are the same everywhere except x = 1. | What do you mean by "look the same"? If two functions produce the same outputs for every input, they are by definition the same function. However, a single function can certainly have several formulas that look very different from each other. For example, e and 1 + x + x /2! + x /3! + x /4! + x /5! + ... look very different, but they represent the same function. | how about the f(x)={x∈ℚ:1,x∉ℚ:0} and g(x)=1-f(x) | If your definition of "equal functions" means they have the same value at each input, then I agree with /u/PersonUsingAComputer that you seem to be asking for radically different-looking formulas for the function. Fourier series provide a nice example of this phenomenon: look up the sawtooth wave page on Wikipedia. On the one hand this function is piecewise linear and from that perspective can be described by a simple rule. On the other hand this function can be expressed as a Fourier series that involves infinitely many sine terms. Being able to write a discontinuous function as an infinite series of continuous functions was the source of a lot of confusion historically in mathematics (19th century). | Similar to your example, I understand that they are by definition the same, but I was just curious to see some cool looking equations that give the same output |
[
"I am just trying to figure out where this equation doesn't work anymore"
] | [
"math"
] | [
"kf1ar6"
] | [
2
] | [
""
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true
] | [
false
] | [
1
] | null | I might be wrong but I think the cancellation is wrong because you’re dividing the top and bottom by (10-10) and that’s the same as dividing by 0, which as we all know leads to many problems | The error stems from the assumption that 0/0 is a number. | Step 1 has a divide by zero which is not allowed. | THey cancelled 10-10 with 10-10, and that is exactly 0/0 | The reason you can cancel common factors from the numerator and denominator of a fraction is because x/x = 1, for any value of x other than zero. So, cancelling out is the same as multiplying (or dividing) by 1, which doesn’t change the value of the fraction. By cancelling out (10-10) in the numerator and denominator, you are implicitly saying that 0/0=1. So, what you have proved is that if 0/0=1, then 0/0=2. However, this would also imply that 1=2 (since they both equal 0/0). This contradiction, that 1=2, means that the initial assumption, that 0/0=1, is in fact false. |
[
"Pre Calc"
] | [
"math"
] | [
"kez8z2"
] | [
1
] | [
""
] | [
true
] | [
false
] | [
0.55
] | null | Khan Academy. | Have a look at the sticky post of resources at r/learnmath | Second this. A great resource | Paul's Online Math Notes - these are mostly for calculus, but some of the earlier sections (e.g. the Algebra and Calculus I sections) have review that may be helpful for a precalc course. To be honest I have no idea what's covered in a typical precalc course (the precalc course I took in high school wasn't organized very well, mostly I felt it was a review of algebra/trig with some random added bits), but hopefully this is helpful. These notes may also become more helpful if you go on to take calculus later on. | Second this, it’s an amazing resource. Very helpful when you’ll begin learning calculus too. |
[
"Any good discrete mathematics textbooks?"
] | [
"math"
] | [
"kewx9z"
] | [
14
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.95
] | null | Discrete Mathematics and Its Applications by Kenneth Rosen is pretty good. | Knuth/Graham/Patashnik: Concrete Mathematics. An absolute joy to read. Though more of a combinatorics textbook, really. | i liked it but at the same time hated the way they treated generating functions... | Oh really? I found the treatment pretty neat and intuitive. | i think that flajolet's take on it is better and more poweful |
[
"Formula to calculate the amount of times a nmber fits in another number"
] | [
"math"
] | [
"keu5zj"
] | [
0
] | [
"Removed - try /r/learnmath"
] | [
true
] | [
false
] | [
0.14
] | null | I mean, division is exactly the answer to your question. 9373167/7 gives you how many times you can subtract 7 from 937167. Then answer would be 133881 times. If you want to only get whole numbers as a result, round down. | What you are thinking of is the division with remainder. That is usually the way division is introduced to kids before they learn the stuff about decimal points. Can you remember how you have done division back then? | What you are thinking of is the division with remainder. That is usually the way division is introduced to kids before they learn the stuff about decimal points. Can you remember how you have done division back then? | 22/7 = 3.142... the "how many times" is the integer part of the result that is 3. To compute what remains you just do 22 - (7 x 3) = 22 - 21 = 1. Please post on r/learnmath since this on iams the wrong subreddit. | Well a computer can do it easily by Mod which gives you the remainder. Another way to get to the remainder is to divide and then multiply the leftover (ie 1/7 in case of 22) and multiply again with 7 to get 1. |
[
"Why does 1719265/40318 take 1,910 decimal places to repeat?"
] | [
"math"
] | [
"kequ7w"
] | [
0
] | [
""
] | [
true
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false
] | [
0.5
] | null | Let's look at a smaller example. Why does 1/7 take 6 digits to repeat? Well, to understand 1/7, you can write it as 142857 / 999999. The numerator is the repeating part, and the denominator is a string of 9s of that length (do you understand why this produces a repeating decimal?). We could only do this because 999999 was a multiple of 7. So the answer to the question "how long is the repeating part of 1/7" is the same as "how many 9s does the smallest number 99...99 which is a multiple of 7 have?". But for 99...99 to be a multiple of 7, that's the same thing as 10^n being congruent to 1 mod 7. So now we know that the period of 1/7 is the same as the period of 10 modulo 7 (in the number theoretic sense). The example you have noticed takes so long because the period of 10 modulo 20159 is large - it's 1908. So the repeating part has length 1908. (Why am I talking about 20159 rather than 40318? Factors of 2 and 5 work a little differently, because those are factors of 10. If you're trying to understand 1/14, you write it as 5/999990). | I believe that a number repeats with period p if and only if it can be written as m/n, where m is some integer and n = 10^p - 1 = 999...9 (there are n 9's). Now, obviously the fraction that OP posted is in a more simplified form than having a bunch of 9's on the bottom. But I think we can say that because 40318 divides 10^1908 - 1, it repeats every 1908 decimal places. (Off by 2 in the beginning since it hasn't started repeating yet.) Edit: well, Wolfram tells me I'm almost right : (10^1908 - 1) / 40318 is not a whole number, but it IS a multiple of 1/2. So, I think you can modify what I said to be correct, but I'm tired so I'll leave it to another kind soul to do. | Now that you point that out, that's exactly what happens! I had almost forgotten about it. If you're given a decimal expansion with period p you can do some simple but quite clever algebraic tricks to find a fraction that represents the number which exactly that peculiarity. You can perform a similar trick with periodic numbers that begin with an aperiodic string and recover a fraction that has a bunch of 9's and ends in a trail of 0's, basically by combining the decimal-to-fraction process for periodic and aperiodic rationals. | First of all, if it's a quotient of two integers it's by definition a rational number. You know that right off the bat. It may take thousands or millions of digits to see the period but it will eventually happen. As to why it takes 1910 decimal places to repeat and not 1909, I'm content with saying that it's because that's the way things play out for this number, and I would even dare to say there's nothing more to it, but maybe someone who knows more about number theory than I do can give a more elaborate answer. | Confirmed period of 1908 https://www.wolframalpha.com/input/?i=25909%2F40318 |
[
"Happy Pythagorean Theorem Day!"
] | [
"math"
] | [
"keq98o"
] | [
149
] | [
""
] | [
true
] | [
false
] | [
0.98
] | null | thats pretty neat | Wow! I caught this just before midnight, so glad to have seen! | West coast! Two hours to enjoy this | https://images.app.goo.gl/RVgpuueKBEKbHt4v6 | last comment on Pythagorean day of this year for my timezone |
[
"Recommendations for a \"Crash Course\" of Real Analysis"
] | [
"math"
] | [
"kelcjl"
] | [
4
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
0.83
] | null | Understanding Analysis by Stephen Abbott will give you very succinct and insightful motivations of the big picture. Since continuity can be defined using sequences, and a sequentially compact set is obviously defined by sequences, sequences can be a very powerful technique to get results in differentiation such as Fermat and eventually the Mean Value Theorem, which bridges the function with its derivative. As for integration, I’m not sure if sequences of points play a noticeable role, but I imagine series of epsilons can be useful to construct measure-zero sets so you can integrate some functions with infinite discontinuities. Read the uniform convergence part of Abbott and I’m sure you will get why it’s stronger. I noticed that point-set topology is absent from your topics, so perhaps that’s the missing intuition. | The term "uniform" is used because it is about having control over parameters like delta independently of the point being used. See https://math.stackexchange.com/questions/547510/clear-explanation-about-uniform-continuity/547518 . In particular, really understand why 1/x is not uniformly continuous on (0,1) or (0,1] but it is uniformly continuous on (1,infty) or [1,infty). I could have used an arbitrary positive number in place of 1 as an endpoint of those intervals and the same results hold. Read a that a uniform limit of continuous functions is continuous and figure out why how uniformity of the convergence is being used by figuring out with a picture why the proof does not apply to the functions x on [0,1], which have a pointwise limit that's 0 for x < 1 and 1 at x = 1. If there a uniform limit, then that limit would also be a pointwise limit, so the uniform limit would have to be 0 for x < 1 and thus is 0 for all x in [0,1] by continuity (a uniform limit of continuous functions is continuous). Convince yourself that x for x in [0,1] does not converge uniformly to the zero function on [0,1]. Forget the blather about "how ugly a function can be" and focus on the actual math that you have seen in your course. Master an understanding of why pointwise and uniform convergence of functions are not the same thing: uniform convergence implies pointwise convergence but not vice versa. | With uniform convergence, every point converges at a similar rate(you can bound how quickly any point will converge). | Here are a few big picture things. | This was the assigned textbook for my honors Real Analysis class this semester. Definitely an 11/10 recommendation! Enjoyed it so much, I’m planning on finishing the exercises in the book over break! |
[
"why is 5/4 equal to 75?"
] | [
"math"
] | [
"kejpa9"
] | [
0
] | [
""
] | [
true
] | [
false
] | [
0.29
] | null | sir this is a math subreddit | The only way I can possibly think to relate 5/4 to 75 is by the equation (5/4)•60=75. | 5/4 is not equal to 75. | 5/4 is the same as 1 1/4 which is 1.25 | Check out BBC bytesize, they will have a page on fractions that will help out out loads. |
[
"Best way to self learn"
] | [
"math"
] | [
"kentef"
] | [
7
] | [
"Removed - post in the Simple Questions thread"
] | [
true
] | [
false
] | [
1
] | null | Whenever you do calc1-3 (in the united states at least) there are no proofs in the mathematical sense. All the “proofs” are really just elementary derivations based on intuition rather than deductive proof. So in a sense, yes, they teach you the intuition, but they do not teach the math to you with deductive proofs. | I self taught bc calc using pretty much the internet. Khan academy is very effective. I also highly recommend paul’s online math notes. They are a useful reference. On youtube, black pen red pen is extremely good. Also ucdavis has some good websites devoted to integration techniques and lots of practice problems. Their other topics may be good too, but i mainly used them for practice problems. | Do they teach you the proofs on why it’s like that? Or just the implementation of how to do it. | You might like Spivak's Calculus , Honors Calculus by Pete Clark and Essential Calculus by Silverman in addition to the youtube videos. They all approach it from the perspective of a rigorous introduction. | Then is deductive proof really necessary for a good understanding of the subject, or to get a job, for example, software engineer? |
[
"Step function Fourier series visualized [OC]"
] | [
"math"
] | [
"kf50ga"
] | [
1053
] | [
""
] | [
true
] | [
false
] | [
0.98
] | null | You can even see a little Gibb's phenomenon! | Thanks for pointing that out. After Googling it, this seems to be the formal name for something I mentioned in my other comment. I've learnt something. | What are you doing step function!? | The first hundred or so terms of the step function Fourier series multiplied on top of each other with the opacity tapering off toward the higher terms. I thought this was a pretty way of showing how adding more and more terms eventually approaches the desired function. Geometry created in Houdini and rendered in Maya using the Arnold renderer. I'd be curious to hear if anyone has insight on some of the patterns that arise. For example, there seem to be certain amplitudes that align vertically and draw out horizontal lines. I also thought it was interesting that you can see how much more slowly the series converges around the inflection point. For those that don't know, you can learn more about this series from the 3B1B series on differential equations: https://youtu.be/r6sGWTCMz2k?t=268 | You should do the rest of the basic waves like triangle, sawtooth etc |
[
"A generalized improper integral definition to allow divergent integrals"
] | [
"math"
] | [
"kf6pgg"
] | [
23
] | [
""
] | [
true
] | [
false
] | [
0.85
] | null | Back in grad school (EE, not math) I was working with Green's functions where evaluating them required bringing a derivative operator inside the integral, which caused the integral to be non-convergent. We later used a weighting function, also passed inside the integral, that made it convergent again, but my professor asked me once "How do we know that we get the same answer?" I worked out a couple ways to show the answer the same, but it still bugged me: Why isn't there some derivative-like function we can integrate that gives the same answer as we get by taking the derivative of the integral? After a bit, I decided it made more sense to redefine the improper integral definition, and kind of played around with it for a while. Thinking about it off-and-on over the years, I eventually settled on the definition given in the paper, proved a bunch of properties, and talked my way into getting sponsored (or whatever the term is) to be able to post it on arXiv. I submitted it to one online journal, but it didn't really fit there and was rejected. I've looked around for other journals, but I never could find one where it seemed to fit well. So I'm posting it here. Hopefully you guys will find it interesting, and people seem to like original content. It shouldn't be hard to follow for anyone who's had college level calculus, but I've tried to be rigorous in all the steps. As a bit of an aside, I was pretty proud of the Z notation. Aesthetically it looks nice, but it was also important. I had been mentally keeping track of which integrals were using the updated definition and which used the regular definition, and it was a little confusing. Derivations were much easier to follow after I started using it, and then shortly afterwards I came up with Theorem 1 and everything fell into place. | Yes, the paper is cool, but as a physicist I cannot help but feel that I do a quick and dirty version of this every day. Which I guess is what OP was doing that motivated him to write this paper! | Skimming through, it seems the paper basically boils down to regularizing locally integrable functions, with the role of test functions played by this “termination” function. | I have a few question: What are your examples supposed to be examples of? The “termination functions” there aren’t différentiable, and you don’t show that the value you compute is independent of the termination function. Don’t you find it suspicious in your examples that you don’t need to take a limit anywhere? If there’s no limit needed, you can’t be capturing anything that these types of generalizations are supposed to capture. Again in your examples you seem to integrate by parts. You can’t do this in the Riemann or Lebesgue-Steiltjes setup as far as I can see; your cutoff function isn’t différentiable and also isn’t regular in the LS sense. You also do this in the body of the paper, and while it is perhaps valid there (you seem to never say what space you take f to lie in) it doesn’t hold for any of your examples. Don’t you find it suspicious that this very basic problem, that could be posed hundreds of years ago, would have a rigorous solution in terms of Calc I material that no one had found before but you? If you know some complex analysis, the Cauchy principal value is something that actually works to assign meaningful values to divergent integrals, although the singularités are of a different nature. Until you can explain what you mean by “différentiable” and prove that you can integrate by parts, there’s really no reason for anyone to read this. | If you are going to ask mathematicians to read your work, you need to write like a mathematician. Surely this is not news to you. (1) The delta function is a not a function. If you know enough functional analysis to phrase things correctly, do so. In particular you need to say which space you take the termination functions from. But again, the main point is: what are these examples supposed to be examples of? Your theorem 1 does not say what you think it does. The only way to parse it seems to be thus: Suppose the limit for some termination function z_1 exists. Then the limits for all functions z=z_1*z_2 exist, and give the same answer. But why should all termination functions for f be obtained by convolving their derivatives and performing your procedure? In general convolution does not have inverses. This question gets into mathematics of the second half of the 20th century already. As you want to allow the termination functions to be distributions, the natural setting to ask this problem is convolution of a tempered distribution and a distribution, and when you can get the Dirac delta this way. The answer depends on the behaviour of the Fourier transform of the tempered distribution. If you can explain which space you take the termination functions from, and that there is only one orbit under convolution, you would remove my objection to your examples. In its current state I'm not surprised that you haven't found anywhere that will accept this preprint. |
[
"How far can you get in math by doing random problems and generalizing them in as many possible ways as you can imagine?"
] | [
"math"
] | [
"kevgs1"
] | [
1
] | [
""
] | [
true
] | [
false
] | [
0.67
] | [deleted] | Look up Ramanujan. He got pretty far doing what you describe, although the fact that he was a Gauss-level genius probably helped. | What do you mean by "random problems"? My immediate reaction is no. Unless you develop some really deep insights into what's going on, it's unlikely you'll be able to imagine the most productive directions to head in. It's usually better, at least at the beginning, to allow teachers and textbooks to guide you on what problems to work on. Of course, if you become curious about a new direction, it can be worth pursuing it on your own for a while. | My immediate reaction is no How far can you get in math? No. | I think the trick here is: How are those random problems? Are you, say, dealing with 100 random 3rd degree polynomials, then moving on to 100 graphs, and so on? Then yes, I think you can find quite a bit of math. Instead, are you looking at a polynomial, then a group, then a graph, then the real number line, and so on? That can be problematic. Math is really just identifying and isolating patterns; if you keep shifting examples before you can do either of those things, well... | Not clear to me that solving 100 3rd degree polynomials will lead to learning new math. Could you elaborate? |
[
"I’m Above Average and So Are You"
] | [
"math"
] | [
"keldvl"
] | [
19
] | [
""
] | [
true
] | [
false
] | [
0.78
] | null | This is responding to superiority bias as if those surveyed claim that their performance is better than mean performance. But if you look at the actual surveys, they always explicitly ask subjects to compare themselves to the median. "Are you better than 50% of the population?" Obviously 90% of the people can't be better than 50% of them. Superiority bias is a real and very strong effect and it has little to do with anything you are talking about here. | EVERYONE IS ABOVE ABERAGE!!! YAY!!! | Regarding the last paragraph. Generalising the notion of an average to infinite sets, even countable ones, is obviously a little problematic, but I don't think it's unreasonable to say that the average of the set {2^(-n)|n positive integer} should if anything be 0. If we allow that, then all the children in the village might actually be above-average as long as there are infinitely many of them. | Actually, at least for the papers that introduced the phenomenon, the questions were directly about averages. See, for example, the paper "Global self-evaluation as determined by the desirability and controllability of trait adjectives". ( https://psycnet.apa.org/doiLanding?doi=10.1037%2F0022-3514.49.6.1621 ) | "Are you better than 50% of the population?" I think this is also what most people think of when they hear "are you above average?". Average is just a vague word when used informally. |
[
"Remember to thank your teachers/professors!"
] | [
"math"
] | [
"kf3ngy"
] | [
31
] | [
""
] | [
true
] | [
false
] | [
0.84
] | as we all know 2020 hasn't been a great year for most of us. Being the time of giving, I'd like to remind everyone to send a quick email to teachers/professors just saying thank you for working their tails off to teach us during these difficult times! It doesn't have to be anything personal just a simple "thank you for all your work this semester" would make their day! and to any teacher/professors reading this by chance, Thank you for your commitment to your students and teaching. You all rock and deserve much more praise, especially for this year! | It’s really appreciated (faculty member here.) | Faculty here - please don't include in that email a request to round your grade. We hate that. | As a faculty member, these really do make a difference! | Oh shoot. I sent them right after finals. Hope they know I’m genuine and not trying to guilt a better grade | Hey - thanks. |
[
"Finding a metric approximating a semimetric"
] | [
"math"
] | [
"kf02f5"
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2
] | [
""
] | [
true
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false
] | [
0.75
] | I was reading up on Distance Geometry and came across the notion of a semimetric, which is a metric lacking the triangle inequality assumption. On Wikipedia, it reads that "semimetric spaces naturally arises from inaccurate measurements". To make this description more concrete, let ε > 0, and suppose that the space S is endowed with both a metric d and a semimetric d' such that, for all x,y in S, |d(x,y) - d'(x,y)| < ε. It follows then that for all x,y,z in S we get d'(x,z) < d'(x,y)+d'(y,z)+3ε, a sort of triangle inequality with controlled error. Is the converse true? Namely, if (S,d') is a semimetric space such that there exists an ε > 0 satisfying d'(x,z) < d'(x,y)+d'(y,z)+ε for all x,y,z in S, then does there exist a metric d on S and some 𝛿 > 0 such that |d(x,y) - d'(x,y)| < 𝛿 for all x,y in S? Can we find a metric with smallest 𝛿? Alternatively, instead of having a constant error ε, we could assume that the error on our measurement scales with the measurement itself, that is, |1-d/d'| < ε (with 0 < ε < 1). In this case, we get that d'(x,z) < (d'(x,y)+d'(y,z))·(1+2ε/(1-ε)) for all x,y,z in S. So conversely, if (S,d') is a semimetric space with an ε > 0 such that d'(x,z) < (d'(x,y)+d'(y,z))·(1+ε) for all x,y,z in S, does there exist a metric d on S and some 𝛿 > 0 such that |1-d/d'| < 𝛿? Can we find a metric with smallest 𝛿? This isn't related to my work and I don't know how to approach it, but I think it's an interesting question and I'd like to see what others think of it. Of course, if S is finite, any metric will work, in which case the problem of optimising for smallest 𝛿 is the only interesting part. Thanks. | I am not exactly sure what you are asking but a semi metric as defined by Wikipedia (satisfies some constant times the triangle inequality) has the property that if d is a semi metric, there exists some epsilon_0 such d is bilipschitz equivalent to a metric for every 0< epsilon\leq epsilon_0. I am not sure if this is helpful to you but it is in fact true. It's bilipschitz equivalent with coefficient less than or equal to 2 as well | https://epubs.siam.org/doi/abs/10.1137/060653391 | Given a function d'(x,y) that is a semi-metric satisfying your first triangle inequality with error ε (which I think you have backwards, btw), let d(x,y) be sup{|d'(x,z) - d'(y,z)| : z ∈ S}. d(x,y) is automatically an extended pseudo-metric (i.e. can have value ∞ and satisfies all of the metric axioms except d(x,y) = 0 ⇒ x=y). This doesn't rely on any special properties of d'. Note that since d'(x,x) = 0, we automatically have d(x,y) ≥ d'(x,y) for any x and y in S, which implies that d is actually an extended metric. By manipulating the triangle inequality with error, you can get the reverse triangle inequality with error: d'(x,z) < d'(x,y) d'(y,z) + ε. d'(x,z) - d'(z,y) < d'(x,y) + ε. |d'(x,z) - d'(z,y)| < d'(x,y) + ε, by symmetry. So, for any x, y, and z in S, we have |d'(x,z) - d'(y,z)| < d'(x,y) + ε, implying that d(x,y) ≤ d'(x,y) + ε. Putting this together we have d'(x,y) ≤ d(x,y) ≤ d'(x,y) + ε. I think you can show that this is optimal. | That sounds like a nice result. Do you know where I could find a proof of it? | I can look tomorrow for a reference. I have a pdf scan of a couple pages from a text that I can pm you. |
[
"How to self-learn music theory"
] | [
"math"
] | [
"kf2b6g"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.69
] | Hi! I'm interested in learning about the math behind music theory in my spare time. I don't really know where to start. I was an quantitative economics student, so my math background is 2nd year calc, 3rd year linear algebra (as well as a course on abstract algebra and a few on statistics, though I assume those aren't really relevant). I don't have any music background (can't read music, never took lessons to learn an instrument, and sang poorly in choir for a few months when I was 12), so all the music terminology is foreign to me right now. I'm not really sure where to start. Every time I dive in to a wikipedia article about it, there are so many music terms that I don't know, I just end up clicking link after link to get to "the beginning", and then run out of time or motivation. I'm not looking to become an expert, I just want to be able to keep up in conversation when people talk about mathy music stuff. Any guidance would be great. Thanks! | Read this book: https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-music.html . I've seen some presentations of connections between music and group theory (rather than with Fourier analysis and differential equations) and it looks utterly unimpressive. | Dr. Gareth Loy published a 2 volume set Musimathics: The Mathematical Foundations of Music. It may be of interest. http://www.musimathics.com/ | Definetely check out this blog: https://alpof.wordpress.com/category/music/math-music/ Other recommendations I can give are the following: https://dmitri.mycpanel.princeton.edu/geometry-of-music.html (Anything from Dimitri is worth it tbh) https://trace.tennessee.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=2692&context=utk_gradthes https://www.springer.com/gp/book/9783319429359 This short mini course is definitely worth watching: https://www.youtube.com/user/vornskr181st/videos | Like mathematics, I think music theory is often presented in a way that is much more complicated than it needs to be, which is a shame because they are both so much fun... Here are some notes on basic music theory with an eye towards helping others to play, improvise, compose, and the like. It isn't necessary to know any mathematics to understand music on this level. However, if you would like to associate music with mathematics, perhaps the study of Acoustics would interest you? Several have already mentioned the use of Harmonic Analysis ...In general, the Fourier Transform is quite useful. Another interesting avenue of exploration is that of " Pitch Spaces ", which might be considered a musical application of Group Theory. The work of Dimitri Tymoczko has already been mentioned, but it follows a rich history of musical mathematics reaching all the way back to Euler's Tonnetz Grid . If one is so inclined, study of the Tonnetz Grid can even lead into multi-dimensional musical structures ! Anyway, I hope this is helpful. If anyone would like more resources for learning music theory, or the applications of music theory within science and mathematics, please let me know. | Great, thank you. I'll have a look! |
[
"What do I get my in-laws who are mathematicians for Christmas?"
] | [
"math"
] | [
"kf24wz"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
1
] | My wife and I thought of a few books that we could find on Amazon (US): Puzzle books: General reading: We thought we would get 1 puzzle book and 1 for general reading. We were hoping to find something that's challenging/intriguing enough. Please drop your suggestions! Thank you. | i'd recommend against popular math books. If the book is about math and not loaded down with math then they are irritatingly vague and noninformative from a typical mathematician's perspective. | Get them regular presents, not math books (especially not puzzle books). Think about it by analogy: a surgeon probably would not want to get a non-technical medical book as a present. You could get them a visually appealing mathematical present from here: https://www.kleinbottle.com/ . That's assuming they don't already have one of those. Do you know the of math that the in-laws do research in? | I’m a biogeochemist not a mathematician, but honestly, I don’t really want gifts related to my work. For one thing my in-laws and family know absolutely nothing about biogeochemistry, so the likelihood that they would get me something relevant or interesting is slim, and for another, I have other interests. They have to have interests other than math, show that you’re paying attention and get them something else that they like. For example I have trained my family to buy me cooking tools, power tools, and coffee. I am never disappointed by these things that allow me to practice my hobbies. If I want a biogeochemistry book, I can use the university library or my employer, or colleagues to get it, hobby stuff comes out of my own budget. | Nope, buy normal presents. If they drink, a good bottle (wine, spirits) is always well-accepted and it is also something that can be shared and enjoyed together. Also easily regifted. I'm not a mathematician, but for the layman I'm close enough (I'm a researcher in CS, I've always be interested in recreational math and in everything scientific). During the years I've received a few pop math book which I already have or that I was not interested into. Edit: for professional mathematicians it can be even worse: many of them have no appreciation for recreational math. The most interesting book I've received was one very far from my knowledge, "The Man Who Mistook His Wife for a Hat" by Oliver Sacks. He has written a lot of fascinating books. Many people suggest things like a Klein bottle. To be honest, when I was a little kid (neeerd!) I would have loved it. Now that I'm much older I'm not that interested anymore in objects, even beautiful, that just "are": I don't like knick-knacks, but that's just my personal inclination (but my late wife agreed). | I agree that the non-technical math books are a bad idea because they are usually useless to a mathematician who would want all of the details most of the time. But I really disagree with “especially not puzzle books,” which implies that a book of puzzles is even worse than a pop math book. On the contrary I think many mathematicians do very much enjoy working on puzzles. A good puzzle doesn’t need to be rooted in technical mathematics to make you think about things in interesting ways. Many puzzles are just downright beautiful and give me a similar feeling to doing mathematics. |
[
"Any theorem provers with a deep embedding of first-order logic?"
] | [
"math"
] | [
"keyojh"
] | [
5
] | [
""
] | [
true
] | [
false
] | [
0.67
] | So I am currently trying to formalize some results regarding weak first-order arithmetics, requiring a proof assistant with some sort of embedded first-order logic, if possible, already equipped with a library featuring proofs of completeness, compactness, etc. However, while there are some metalogic libraries for, say, Isabelle, they tend to either exclude some quite important features (such as infinite sets of axioms), or are made for specific first-order theories. Does anyone know of a proof assistant/library that provides what I need? | I'm only somewhat familiar with Lean, so there's flypitch . Is that what you're looking for? | that looks pretty much like what I'm looking for, thanks! | Do you know lean ? By Leonardo de Moura from Microsoft | Well, that library in particular, although I don't fully understand their whole variable binding approach | Misread your comment fsr, yeah ik lean |
[
"Sharing a Introductory Group Theory Cheat Sheet I made"
] | [
"math"
] | [
"kf3fyj"
] | [
176
] | [
""
] | [
true
] | [
false
] | [
0.98
] | The link to the image: I have made a Intro Group Theory Cheat Sheet for my exam, since I couldn't find any online. Hope this is helpful for someone out there. The cheat sheet contains a typical Introductory Group Theory syllabus up to the Sylow Theorems. Here is the overleaf link for the cheat sheet incase you happen to find some mistakes (or styling issues) and/or if want to get a pdf for yourself. I have mainly referred to the Dummit and Foote book and occasionally Wolfram Mathworld for almost all of the definitions. I have only included matter that is in my syllabus because of this you may notice a few tiny sections missing which I haven't taken from the book. Let me know if you spot any mistakes I will try to edit and fix it as soon as possible. For something of this size I wouldn't be surprised if a few errors fell through. The document was entirely based off a template I found on overleaf Edit: Made a few changes. Added cardinality bars in RHS of Class equation, and corrected typos. | I would say that it is simply a matter of notation. I.N. Herstein's Topics in Algebra lists closure as an axiom. Joseph Gallian's Introduction to Abstract Algebra doesn't. Both are equivalent statements. One book equips the set in question with a binary operation, while another simply uses a function. | closure is not an axiom of a group. or rather you could say it is tautologically satisfied if you define a group formally as a set with an operation which is a function on pairs from that set to the set. It's only a necessary axiom to define a subgroup. Not a group. | i couldve used this like 2 weeks ago for my final | Amazing! This would of came in handy back in my undergrad days. Out of all the upper division math classes that I’ve taken, both group theory and ring theory were personally the most challenging. Also, to any aspiring mathematicians here, feel free to use the lecture notes I used when I took group theory. Stay safe and happy reading! https://boltje.math.ucsc.edu/courses/w14/w14m111anotes.pdf | Kernel of a homomorphism depends on the function you choose, rather than the Group itself. For example, for some f: G —> G, the trivial homomorphism (send everything to identity and be done with it), would have the kernel as the entire group. The identity homomorphism (send each element to itself) would have only identity element in the kernel. So saying Ker(f) is more accurate and helpful. |
[
"Career and Education Questions"
] | [
"math"
] | [
"kf1987"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.81
] | 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. | I would explore things you like and if you end up taking many classes in a single area then that isnt bad (i am doing this). Just make sure youre still able to take a lot of analysis, algebra, and topology. | About your second question: Assuming you're in the US, don't do it. In your application you can write that you are interested in such-and-such research areas, and that you might like to work with person A, B or C as a potential advisor. The exception is if you actually are familiar with the particular research area of a professor -- maybe you read one or two of their papers -- and you know that you want them to be your advisor. That would be an unusual situation. Grad school applicants are not expected to be at that level. If you do name-drop potential advisors A, B and C in your application, make sure the people you list: - are tenured or tenure-track, - are not about to retire or leave for another university, - are currently active in doing research and advising students. | I see. Linear algebra is certainly more fundamental, but whether a second course is necessary depends on how abstract your first course was. If the first course is mostly computational I would certainly take it. If it was already pretty rigorous, I would think about which class fits my research interest more. Another factor is if linear algebra class is undergrad, and you feel ready for the challenge, I would prefer a grad class. | I'm trying to figure out, realistically, what my options are for graduate school. I guess my first question is: What do top 10 graduate schools in math expect as an undergraduate? I know this is a very general question, but it seems like graduate courses are absolutely necessary, along with some research experience, and strong letters from tenured professors. It also seems like that many accepted undergrads have already figured out what they're interested in, like algebraic geometry or something, and they are at a fairly advanced level in that subject. But is there something extra here that I'm missing? It seems like there could be many, many students that meet this criteria, but there are only so many spots. I'm currently a junior and I've taken basically all of the undergraduate curriculum, and by the end of this year I will have the basic graduate algebra and analysis under my belt. I have done some research in a program over the summer (combinatorics) that I personally really enjoyed, but I don't know what graduate admissions would think about it. I don't think I know any subject at all very in depth, and I don't really know what part of math I like. It seems like many people going to these programs have read whatever the equivalent of Hartshorne is in algebraic geometry in their free time. Meanwhile, I think my interests are pretty diverse (outside of math), so I don't spend that many hours a day on mathematics. But do you need to do that to get into these programs? | I have done some research in a program over the summer (combinatorics) that I personally really enjoyed, but I don't know what graduate admissions would think about it. The other response is right on target. I just wanted to respond to this part. No one expects you to continue in combinatorics just because of your REU. They want to see that you liked the process of doing research and that you learned some interesting things. You should write about your experience in your essay, not from the perspective of "I proved the world's best theorem" but rather "I really immersed myself in this problem," and get a recommendation letter from the professor who supervised you. |
[
"What makes representation theory special?"
] | [
"math"
] | [
"kezh25"
] | [
62
] | [
""
] | [
true
] | [
false
] | [
0.93
] | My title is vague, but that’s the best way to summarize what I’m thinking. I’m a new math grad student finishing up a course on representations of finite groups. This is my first taste of rep theory and I’m enthralled. My first specific question: why are only certain categories studied in association with representations? The big ones seem to be groups, associative algebras, and Lie algebras. Was representation theory of, say, rings ever investigated? Why or why not? Besides the obvious answer that we get important results in the three categories that I listed. Was this known beforehand or were there failed attempts at further generalization? Second, even restricted to finite groups, representations seem to have a lot of important properties. The most striking one to me is this notion of induced representation - that a representation on any subgroup extends uniquely to that of the whole group. And of course it has many desirable properties like Frobenius reciprocity. Does this induction functor generalize to other categories, perhaps with a more abstract characterization? In other words, are there other functors which have these nice properties that induction does? I imagine any reasonable answer would have to involve adjoint functors (given the Frobenius formula). | People study representations of rings all the time. They call it module theory. Even representations of (discrete) groups is essentially modules over the group ring. A representation of a group G is just a linear action of G on a vector space, i.e. a homomorphism of G into GL(V). Similarly, a representation of a ring R is a "ring action" of R on an abelian group, i.e. a ring homomorphism R -> End(M) for some abelian group M. It is equivalent to say that M is an R-module --- it's the same thing. A lot of students view modules as "vector spaces except the coefficients are from a ring", but I think another valuable viewpoint is "representation of a ring on an abelian group". Any time you have a homomorphism X -> End(Y) from your object X into an algebra of homomorphisms/automorphisms of some other object Y --- that's called a representation of X on Y. Same as representable functors in category theory: those take an object A and represent it as a functor B -> Hom(A,B). This is not too different from the left regular action of a group/ring on itself. When viewed this way, Yoneda's lemma is "obvious". Module = representation. They are synonymous. | Damn, you blew my mind. I know about this view in the group case but somehow never connected it to rings and modules. | At the beginning of Fulton/Harris's text on representation theory, the authors write something cool in comparing representation theory to manifold theory. In the early days of manifolds, manifolds were always embedded in euclidean space. The advent of abstract Riemannian manifold led to a whole new understanding of what intrinsic and extrinsic geometry of the manifold, i.e. which properties depended only on the manifold and which properties depended on the embedding. In the early days of group theory, the only groups that were studied were subgroups of the symmetric group and subgroups of automorphism groups of a vector space. It wasn't until the 1900's that the abstract notion of a group was defined, and of course, thanks to Cayley's theorem, we know that every finite group is isomorphic to the subgroup of some symmetric group In both cases, both the abstract concept and the embeddings became separate routes of study: you could study a particular manifold, or you could study how to embed it into R^n, and likewise, you could study a particular group, or you could study how to map it into GL(V) Kind of a cool metaphor that I thought I'd share. | For Lie theory, the reason is very simple: The exponential map. Now, since a Lie group is a differentiable manifold with a group structure, we can talk about its tangent space at any given point. The tangent space of the identity of the Lie group is the Lie algebra of the given Lie group. (There are a few other formulations of Lie algebras using left or right invariant vector fields but that is not too important right now.) The exponential map is the map from the Lie algebra of a Lie group to the Lie group itself. This map is a local diffeomorphism. For nice enough Lie groups (connected, simply connected Lie groups), we have a very strong result: There is a one to one correspondence between the representations of a Lie group and its Lie algebra. [1] This is what makes it so useful: You don't need to worry about group representations anymore. You can work with Lie algebra representations, which are a lot easier to study (because linear algebra is easier than abstract algebra), instead of studying Lie group representations. This is such a useful correspondence that many introduction to representation theory books devote most of their chapters to the representation of Lie algebras. For example, the book by Fulton and Harris is around 500 pages and nearly 350 of these pages are about representations of Lie algebras. Also, there are some very strong machinery developed in Lie theory for dealing with representations of Lie algebras. There are constructions like universal enveloping algebras and theorems like Poincare-Birkhoff-Witt theorem that are very useful, to say the least. Reference:[1] This can be seen in, for example, "An Introduction to Lie Groups and Lie Algebras" lecture notes of Alexander Kirillov Jr. Theorem 4.3, p.40. | See also this classic MathOverflow question, Why is there no Cayley's theorem for rings? (Spoiler: There is.) OP there even has the same reaction as you :P My mind was blown too... |
[
"Important computations in algebraic geometry"
] | [
"math"
] | [
"kf22cm"
] | [
7
] | [
""
] | [
true
] | [
false
] | [
0.78
] | As someone relatively new to algebraic geometry, I was curious as to what more experienced folk think are computations/examples that are important to know for an algebraic geometer, especially one more interested in the concrete side of things. Examples of what I might include in this list are blowups and Cech cohomology. | Im blanking a bit, but the 27 lines thingy is pretty nice https://en.wikipedia.org/wiki/Cubic_surface#27_lines_on_a_cubic_surface which is at the end of Hartshorne iirc. I dont know if it'll count for you as its maybe a bit of a lenghtier project but Enriques'* classification of surfaces Grothendieck's theorem of the classification of vector bundles in the derived category of P ( or Atiyah's over elliptic curves, although cant remember how that goes tbh ). There's a bunch of cohomological calculations for different cohomologies. Grinding through some spectral sequence is something you just need to do at some point, otherwise they'll always feel like pure magic. Hope this helps a bit. | If it’s not an elliptic curve or a toric variety or in Cox/Liitle/O’Shea then you’re probably trying too hard | Riemann-Roch calculations with curves; inductive arguments with complete intersections; Cech cohomology of projective space; Picard groups of projective space, P^1 x P^1, affine space, blowup of a point on a surface; dimensions of moduli spaces for low genus curves (intuitively rather than formally); Hilbert polynomial for Veronese embeddings; lines on the quadric surface; twisted cubic things; minimal model program for surfaces. | On the more arithmetic side, using Grothendieck-Lefschetz to compute the number of points over a finite fields for many varieties by computing it's cohomology and the Frobenius action first. Taking a variety fibered over a curve (or DVR) where you know the generic fiber and you can try to figure out what sort of singularities you can get over the special fiber. Computations of etale fundamental groups for various varieties. Combining the last two points, computing monodromy actions of the generic point on the cohomology and so on. Computing intersection products on various concrete objects (surfaces are a good place to start). | Calculating (anti-)canonical divisors using e.g. the adjunction formula, formulas for K_X where X is a blow-up, etc. can be pretty important in a lot of 'concrete' examples. |
[
"Essential math topics that aren't commonly taught?"
] | [
"math"
] | [
"kes6hl"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
1
] | I'm wondering what areas of math people think are incredibly useful/powerful/interesting/etc. to know but that don't necessarily make it into an (undergrad) math curriculum. For example, my undergrad math major requires the Calc seq., Dif Eq, Linear Algebra, Real Analysis, & Abstract Algebra, which feels pretty "slim". Some initial thoughts of mine that may or may not be a good answer to my own question are number theory, topology, combinatorics, and graph theory. Interested in hearing what other people would add to this and why. Thanks | Lie groups/algebras. At the undergraduates level, I think most math topics are fairly self-contained and there's not much overlap, but Lie groups and Lie algebras blend many of these concepts (group theory, representation theory, topology, geometry) together in a natural way. Better yet, many of the ideas can be concretely motivated with small 2x2 or 3x3 matrix examples and basic multivariable calculus! The more I progress in my research, the more I realize just how many branches of math I have to draw from, and I think seeing the connections these branches is something every student should see much earlier. | You might find the book: All the Mathematics You Missed: But Need to Know for Graduate School by Thomas A. Garrity interesting. | Perturbation theory / asymptotics. | Idk about "essential," but definitely something that is pretty cool is "foundations" like set theory, ZFC, well-ordering theorem etc. In the same vein, propositional logic and first order logic, and maybe even things like computability though that starts to get into more computer science I feel. | Complex analysis actually blew my effing mind. |
[
"Struggling with functional analysis"
] | [
"math"
] | [
"kermzg"
] | [
19
] | [
""
] | [
true
] | [
false
] | [
0.85
] | I am a second year grad student (PhD program) and am following the analysis route. One thing I'm having trouble getting past is functional analysis - I've found that I really don't like it. Thinking about abstract spaces doesn't really do it for me, and arguments involving biduals and weak topologies make my eyes glaze over. I can never remember whether the dual of l1 is l infinity or vice versa, or whether I'm allowed to assume norms are equivalent. I loved measure theory, harmonic analysis, and a few topics in convex geometry I've studied. These feel like they have concrete geometric ideas I can visualize. Will my difficulty to work with the above functional analysis ideas interfere with analysis research? If yes, what are the most important areas of FA to understand before moving on? | In what setting are you unsure about being "allowed to assume norms are equivalent"? A very special feature of finite-dimensional spaces is that all norms on them are equivalent, but the drastic failure of this for infinite-dimensional spaces is (sort of) what gives functional analysis its significance, e.g., all the different L -norms on C[0,1] have completions that are the genuinely different spaces L [0,1]. You say your eyes glaze over when hearing about weak topologies. Have you actually with such topologies, or asked the instructor why that concept is important? The Banach-Alaoglu theorem is a pretty big deal since unit balls in infinite dimensions are never compact for a norm topology, but become compact in a different topology in the setting of that theorem. You might be interested in the following quote from Conway's : "Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. The common thread is the existence of a linear space with a topology or two (or more). Here the paths diverge in the choice of how that topology is defined and in whether to study the geometry of the linear space, or the linear operators on the space, or both." So you see, even within functional analysis, two specialists can find their eyes glaze over when they listen to the other person explain their research! Actually, this is not some specific to functional analysis. Within any highly developed area of math two experts may not understand each other's work. | Functional analysis provides a abstract framework for a lot of problems in analysis, so it's useful to know the main results albeit not necessarily in detail. The important thing is to be able to apply the theory as needed, and to know what should hold - often the abstract result you need simplifies in the context of a concrete space. To better understand and appreciate the theory, I suggest you try to look at concrete spaces and see how things behave there. Things like the Riesz representation theorem for continuous functions, interpolation in harmonic analysis, the weak topology for Lp, convergence of distributions, spectral / Fredholm theory for elliptic operators, the list goes on! Another suggestion it to look at Brezis' book It covers a lot of classical topics from the PDE and convex analysis viewpoint, which provides applications and geometric perspectives respectively. I particularly like chapter 4 on L spaces, and the exercises therein. Edit: I should also add that I'm a grad student working in (elliptic) PDE, so my perspective is biased in that direction. | Actually, this is not some specific to functional analysis. Within any highly developed area of math two experts may not understand each other's work. How do you learn to communicate with other Math experts I imagine such a problem is solved for fields like Computer Science, Physics, Economics, etc but it seems Pure Mathematics has it the worse in terms of communication ? | If you liked measure theory you could take a look at the different kinds of convergence for measures and how they relate back to weak and weak* topologies and motivate some stuff this way. | Username checks out |
[
"If you intend to have children, do you plan on teaching them advanced math from an early age? If you did, how successful were you?"
] | [
"math"
] | [
"keymb0"
] | [
116
] | [
""
] | [
true
] | [
false
] | [
0.94
] | Hi, It's my goal for my (future) children to teach them a bit of Calc, Stat, and CS earlier rather than later in life. My reasonings are that my children are probably going to go to a public school. As someone who went to one I'm pretty disappointed in how, well, was taught, and had a pretty rough transition going into college because of that. However, I do have a significant amount of control over those 3 things as someone who graduated in Applied Math, and could definitely structure a "semester" of programming, stats, or calc for my kids to learn. Has anyone else had these same kinds of thoughts? For those of you that did and went through with it, how did it go? | Not a parent but a kid of parents who taught a little bit of everything from an early age. I think that's a good idea to an extent; if the kid's obviously uninterested and actively resisting, I would say perhaps to stop. Another thing to consider is that it can be really difficult for some things to click for some people. My parents introduced me to math early and encouraged me towards it, but I still went on to struggle in it because of factors they couldn't control. In other words, cool in theory, as long as you don't overdo it. | Doing it right now. Had practice, no way was my youngest brother going to fail / suck at math... went on to do calculus / statistics (on path to accounting - guess loved numbers and $$). haha There are researchers who focus on early years and mathematical development. Douglas H. Clements, Arthur Baroody, Julie Sarama Here is an image I found in a research book https://1drv.ms/u/s!AnIsoUfPRWeZhGcTFRIGn6SpVtd7?e=e0TKgR Early on I am pushing conceptual thinking: less / greater measurement heavy / light time (days, morning/evening, slowly working with a clock) algebraic patterns (ABABA or AABB or); cause / effect (input/output) proportional reasoning (1:1 or 1:2 - nothing fancy yet). Goal is for my son is to identify patterns Geometry shapes (their properties), comparisons (differences / similarities); making groups Numbers: I count using fractions, decimals, and when using numbers I try to use base-ten language (to have meaning beyond just digits); even negative numbers. [oh and to understand 'zero'] I believe in exposure, familiarity and as many connections as possible to everyday context (playing, cooking, driving, etc.). So we touch / see / play (and I count anything and everything). II rhyme and chant (to try and make it catchy - have amazing memory) My big push though is: explain? why? and to constantly ask questions (can't wait for him to drive his teachers crazy, lol) The difference now is that I have a teacher background and have all kinds of games / visuals / analogies I can use that I did not have with my youngest brother. I prefer that my soon choose not to do math than the math choose him NOT as capable of it. There a lot of youtube videos (quality varies), there are some parents reading STEM toddler books (again quality varies). HTH | I think that's a good idea to an extent; if the kid's obviously uninterested and actively resisting, I would say perhaps to stop. Yeah, this. My mom put a ton of pressure on me to learn Math, I spent quite a bit of time learning to associate math with something to be gamed and just meaningless string manipulation bullshit. On the other hand, I built up a deep love for literature mainly through osmosis from the books lying at home, cause I was bored, not that I understood a lot of those books initially, but inductive style self driven exploration has a dramatic effect. Now much later, I love math and I sometimes wish my house had been filled with fun advanced math books where there was no pressure to learn it and I was pressured to instead write ten page essays on the meaning of a rose or something. | a 3 year old I was under the impression that kids aren't even sentient until about 4-6 and everything they do is just instincts and emotions until then. he's frustrated, which is often for a 3 year old Which is the expected result from this experiment. They're taking shots in the dark trying to gauge your reaction without context. This can develop into neurological issues later in life. Source: parents tried this with my brother and I. We're now technically smart, but emotionally stunted. | It's hard man. As the dad of a 3 year old we're trying to teach him all kinds of advanced shit as quick as we can. He knows the basic skeletal structure excluding the arms right now. Once he can tell me the bones in the arm I'm moving into basic muscles and which bones they connect to. We're trying to teach him basic math but getting the concepts across to someone that doesn't understand what plus means is tough. We've been trying a lot of different things to show the concept also, from playing a game where you add one thing to another to get a 3rd number, To fun shows about it, to "video games" based on adding. It's like, explaining what a color looks like to a blind person lmao I know one day soon it will click and make sense to him, but we ALSO have to teach him lol. How to use the potty all by himself, how to put on his own clothes, why we can't throw things, what emotions are, and how to healthily deal with them in yourself, how share, etc etc. So the balancing act for all of this, doing it in a fun and structured way that makes him WANT to learn, and keep trying, (even when he's frustrated, which is often for a 3 year old), is tricky. These are the foundational years of his life and reinforcing that being able to learn, and keep trying, and even if you keep fucking it up it's no big deal as long as you keep trying and improving, are things I REALLY REALLY wish my parents would have driven into me instead of, "you'll do great cause you're so smart and special", cause I I would have been able to deal with disappointment better, and it isn't a cushy soft world out there if you're on your own. Granted, I'll do everything in my power to help my kid out all his life to make sure he doesn't have it as tough as I did, but I won't be around forever and I want him to be okay when that time comes. Sorry, this got WAY away from me xD also forgot to include my wife and I are now more fluent than ever in spanish, weve been learning it as we go so we can teach it to him! |
[
"My Version of Completing the Square"
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"math"
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"exrrw6"
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"Removed - incorrect information"
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0.38
] | null | Step 4 is incorrect: if x is not 0, you get 1 = -2/x^2 - 3/x. Your method fails for any solution with |x| different from 1. | I don't believe that you get a choice on the bracket. Look at deciding each side by x as multiplying by 1/x , then you have (-2/x)(1/x)=-2/(x^2) | Your examples are missing solutions | For the first example, we can add the absolute value of -4 to -4 and get the other solution of zero. For the second one, the value we set the equation equal to, which is -2, is our other answer. | This is just some arbitrary coincidence? If we look at x -4x = 0, and do your steps, we get x =4. if we now add the absolute value of 4, we get 8, which is not a solution? |
[
"Does the \"Achilles and Tortoise\" paradox debunk the claim that fractals have an infinite perimeter?"
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0.66
] | null | No. It's a sum of infinitely many numbers. 1+1/2+1/4+1/8+... converges, but 1+1+1+... does not. When people say that some fractals have infinite perimeter, it's not just "oh, there's infinitely many pieces, the sum must be infinity!" They computed the sum and found that it actually does diverge. | But that's balanced out by the fact that more of them are added at each step. In a Koch snowflake the total permiter actually increases by factor of 4/3 at each step (so 3->4->16/3 etc). And in any case, just the fact that the things getting added are getting smaller at each step does NOT mean that the total sum is finite. | But don't pieces of fractals get infinitely smaller the more you zoom in? In a Koch snowflake for example, the triangles that are placed in every corner get smaller and become a fraction of the previous triangle | Ah that makes sense. Thanks for clearing that out! | So does the harmonic series |
[
"What is your favorite math/logic puzzle?"
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""
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0.99
] | Edit: Wow, thanks for all of the responses! I am no puzzle expert, but I love going through these, and now have a ton to keep me busy. | What causes the paradox? The computed mean is wrong, because the event space does not contain both 2x and x/2 with equal probability, but either A or B with 100% probability depending on which envelope I happen to hold. The expected value would be 25% greater only if the content of the other envelope was probabilistically independent from the content of the envelope I am currently holding. Really, we need to compute the of switching which is: E[y-x]= ∑ (y-x)p(y, x) = ∑ (y-x)p(y|x)p(x) E [y-x] = (B-A)p(y=B|x=A)p(x=A) + (A-A)p(y=A|x=A)p(x=A) + (B-B)p(y=B|x=B)p(x=B) +(A-B)p(y=A|x=B)p(x=B) = (B-A)∙1∙½ + 0∙0∙½ + 0∙0∙½ +(A-B)∙1∙½ = ½(B-A) + ½(A-B) = 0 | The Two Envelope Paradox: Let's start with a simple game: say I have two envelopes, A and B, and I let you pick one. You open it and see that it has $100 in it. Then I tell you that one envelope has twice as much money as the other one. So the other envelope has either $50 or $200, and I give you the option to switch. The expected value of switching is (50 + 200) / 2 = $125, so you switch. Now, consider the game again, but you don't get to open the envelope. You know that your envelope contains x dollars. The other envelope is either half the amount or twice the amount, so x/2 or 2x. The expected value of switching is then (x/2 + 2x) / 2 = 1.25 * x, so it makes sense to switch. You switch envelopes. Now consider that this new envelope has y dollars. The other envelope has y/2 or 2y dollars in it, so the expected value of switching back is 1.25 * y, but that would mean that it's always the right choice to switch, no matter which envelope you're holding. What causes the paradox? | Suppose you are hosting a party, and assume any two people at the party are either friends or strangers. What's the smallest number of people that need to be at the party in order to guarantee that there are either 3 mutual friends or 3 mutual strangers at the party? Solve that one? Okay, what about 4 mutual friends/strangers? Get that one too? That's pretty great. Now try 5 mutual friends/strangers. Get that one?? Okay, let's write a paper. | You might enjoy taking a look at this: https://projecteuler.net Have fun! | I think an easier proof is as follows: There are two envelopes, E[A]=x and E[B]=2x. You have equal probability of starting with either A or B. The expected gain when you switch is then P(A) E[A-B] = ½ (x-2x) = 0 |
[
"/r/answers/"
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0.33
] | null | Prove for me that the smallest spherical cap containing a fixed collection of points on a sphere is always unique if the points fit within the same hemisphere, but might not be unique if the points cannot fit within a single hemisphere. | What’s your background? | 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! | Igcse maths Olevel | https://en.wikipedia.org/wiki/Riemann_hypothesis |
[
"I need advice!"
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"math"
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"exj37b"
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0
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""
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] | null | Thank you for the response, I know my questions weren’t worded entirely correctly I appreciate your feedback. I am very anxious as you can probably tell. | Thank you for the response, I know my questions weren’t worded entirely correctly I appreciate your feedback. I am very anxious as you can probably tell. | chances are you'll need 1st year calculus and depending on program you may have options for which calc course you can take. At my school the lifesci is very easy essentially gr 12. You can find tutors at your school, most grad students offer tutoring services | my brain simply cannot do math There is a remote possibility this is true, but with very high likelihood it is not. In my personal experience people who say this usually have a kind of psychological block or phobia which prevents them from relaxing and focusing on the work or ideas in front of them (e.g. many are afraid of looking stupid in front of better prepared peers, and fixate on how other people are judging them to the extent they have no attention left for the math per se). When they are put into a context where that phobia can be avoided – e.g. where there is no pressure, nobody watching, no grade on the line, some help/support from a trusted teacher or mentor – then their supposed inability vanishes, they start to focus, they discover that they are entirely capable of “doing math”, and they even begin to enjoy themselves. Unfortunately many classrooms aren’t the most friendly or welcoming environments to help people through this. I second the recommendation to spend some time 1:1 with a private tutor. | Depends entirely on the school that you go to. You need to meet with your adviser there and ask them. Really depends on your background. Most of them tend to go through a lot of material though because some people in them go onto do higher math where all of that is required. Most colleges have free tutoring rooms and also office hours where you can go for help. Maybe see if you have a learning disability too, there are a few things like dyscalculia that are math specific that could be an issue for you. |
[
"How many combinations can exist in this pattern?"
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] | null | This is equal to the number of ways in which you can tile a 3x3 chessboards with dominos and monominos, so, according to the entry A028420 in OEIS, there are 131 such tilings. | You can see all the patterns (except the one with 9 single dots) here arranged in a 10 x 13 grid. | I thought this was a dot game. I’m too fucking high for your shit. | Aren't these Domino tilings ? | Sort of, yes. I just need to apply it to a 3 by 3 grid while using 1 by 1 tiles |
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"I want to get better at Math But I'm afraid"
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"math"
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"exipde"
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] | null | Don't know what to say to this but math will get you much farther than games. But not really enough information to properly respond to your post. Algebra? Calculus? Arithmetic? Complex variables? Where do you need to go? | There comes a time where you need to set your priorities. Doesn’t mean you have to abandon what you like forever, but the earlier you get your stuff together and figured out the better. | What do you feel you are lacking? Tiny mistakes? Logic? How to solve things? Try to ask yourself that, I remember starting with doing a lot of exercise about precision since my mistakes were always forgetting a sign, adding a ghost number, forgetting a step. Then you can start teying different things that you want to improve. | I have good grades and all, but I want to do math problems by myself. I'm a senior in high school and I feel I just don't know where to start. | How good do you think you are? You could start with olympic level problems. I would suggest looking for IMO problems and start with old contests and number 1's. These problems will appear in ascending order of difficulty. Also MIR books for olympiads were aleays good, you need to solve to move to the next one. If those are too hard, try looking for other countries' contests. Balkans are good and Iberoamericans also have decent level. You'll need to learn four basic things, numer theory, geometry, algebra and combinatorics. |
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"Educators: I need your help"
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] | null | what's 24 time 32, you may intuitively say 68. Anyone who says that has no intuition or number sense at all. First thoroughly understand addition and subtraction. Do many examples with three, four, and five digit numbers. Then understand the basic idea of multiplication: 4 x 2 means 4 rows of 2. Understand why 4 x 2 = 2 x 4. (Draw a rectangular picture, rotate it 90 degrees.) Understand why a(b + c) = ab + ac. (Again, a simple rectangular picture makes this obvious.) Study the base ten number system. What does 376 really mean? With all of that, it is obvious how to multiply 24 by 32. | > Anyone who says that has no intuition or number sense at all. Wow, than I definitely need your help. It might be the wrong subreddit though, someone to edit. | 24 × 32 = 768 24 + 32 = 56 7 plus 5 may force someone to pause, and in my case, count out five on my hand. This causes a typical 5 year old to count 5 on his/her hand, a perfectly natural and reasonable approach for someone who is just learning to count. With any guidance at all and just a tiny bit of practice most people can learn that 7 + 5 = 12. I would hope people get to this by age 7 or 8 at the latest. Unfortunately some classrooms aren’t the best at helping young students, in some cases because the teachers themselves have a math phobia. If this is giving trouble, try playing some simple counting games, play around with an abacus (or perhaps a rekenrek) a bit, try to answer some arithmetical questions for yourself as you go through your day (e.g. compute the number of windows in a large grid on a nearby building, estimate the number of people living in a city block, tally up the price of a basket of groceries in your head, ....) With consistent practice a typical person can improve at basic arithmetic skills pretty quickly. My mother recently bought my 3.5 year old this deck of counting cards. If you know any young children you could try regularly playing some of the suggested card games. https://tinypolkadot.com For suggestions about number sense more specifically, you could look at these books https://www.amazon.com/dp/1571107908 https://www.amazon.com/dp/1935099655 | This is kind of the purpose of the handout. I remember when someone was trying to teach me long division. I just gave up. Ever since then, I've never been very good with numbers, always limited to that surrender. Let me give you another example. 38 + 46. Someone without a good number sense may do 8 + 6 = 14, and 3 + 4 = 7, (pausing at each equation, mind you), than 7 + 1 = 8, 84. Someone with a good Number Sense knows 8 + 6, knows 3 + 4, and doesn't get stuck with the remainder. It's not out-of-reach for the ordinary person to do the entire equation through feeling rather than a step-by-step process. To put it simply, you don't have to break it down into individual equations; you can teach your mind to do it intuitively. Sometimes people just don't catch on, for whatever reason. I'm trying to design an analytical test to find where the problem is, and then a mechanism for rectifying it and how it relates to math at large. | 38 + 46 Someone with “good number sense” may simplify this to 40 + 44 = 84 or 34 + 50 = 84 or 90 – 6 = 84 or 35 + 3 + 45 + 1 = 84 or 100 – 12 – 4 = 84 or ... But there is also absolutely nothing wrong with 70 + 14 = 84 I think you may want to read/learn a bit more about the general idea of number sense and work on your own practice a bit before trying to produce a document for external consumption. Perhaps by reading some books or academic papers about it. I don’t think examining autistic savants is a useful guide for ordinary people. Their methods are not the same as the methods employed by people with “good number sense”. |
[
"Can someone explain how “100” “5” and “10x” are polynomials? I’m so confused."
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"math"
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"exihwg"
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"Removed - try /r/learnmath"
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0.15
] | null | Next time: r/learnmath instead of r/math | From the sidebar: Homework problems, practice problems, and similar questions should be directed to /r/learnmath , /r/homeworkhelp or /r/cheatatmathhomework . Do not this type of question in /r/math . | From the sidebar: Homework problems, practice problems, and similar questions should be directed to /r/learnmath , /r/homeworkhelp or /r/cheatatmathhomework . Do not this type of question in /r/math . | From the sidebar: Homework problems, practice problems, and similar questions should be directed to /r/learnmath , /r/homeworkhelp or /r/cheatatmathhomework . Do not this type of question in /r/math . | From the sidebar: Homework problems, practice problems, and similar questions should be directed to /r/learnmath , /r/homeworkhelp or /r/cheatatmathhomework . Do not this type of question in /r/math . |
[
"Noticed something odd about about sales totals at work"
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"math"
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"exf9u0"
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"Removed - incorrect information"
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] | null | Coincidence. | What are the odds? I should clarify it’s not just 5, 23, and 31. I see it all the time, including totals over 100 as well. Considering all the different combinations of items of differing costs, it seems like there’s got to be some kind of explanation. | It might be polling bias, it could be a plethora of things, but its also possible the tax rate mixed with .99 cent items in your town round to primes much more often. | That’s what I’m trying to figure out. It’s got something to do with combining .99 cent items plus the tax rate. But what would a number being prime have anything to do with totals rounding to $xx.00 Not polling bias by the way. I’ve made a point to mark down every time I see it and make sure the number I see is actually prime. | It could have to do with standard prices for items being like 19.99, like there are way more items at this price than any other, and it may round to 21 with your tax rate. I mean there's a ton of variables. I don't think this is a known science, but maybe number theory would have an answer |
[
"Amazon hanging cable problem"
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"math"
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"exc9ob"
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] | null | Hint... It's a job interview so they are not expecting the interviewed to spend a lot of time on one problem. | So people with less than graduate grasp of mathematics can’t discuss any problems? Seems kinda elitist | That’s not particularly helpful.... | Really, the cantonary problem is to simple? What kinda standard are you using to make the determination? | Hyperbolic functions But I think they asked another question, where, using the Pythagorean theorem, you could show that the question didn't have a solution. |
[
"I had a thought about time and I think this fits here."
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"math"
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"ex0c5t"
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0
] | [
"Removed - incorrect information"
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true
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false
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0.5
] | null | ...this is going on r/badmathematics . | arrogance of someone who's never learned a bit of mathematics in his life. learn some humility and accept your misgivings. we have , concrete definitions in mathematics and all properties come from those. a plane, and thus, a square, is two dimensional, because there are exactly two degrees of freedom on it. by definition. 0 is not "the absence" of something, it simply tells you that with a space with 0 points, there are 0 degrees of freedom. on a line, 1 degree of freedom. 1-dimensional. our 'primitive faulty mathematics' built this world and its technology. go dispute that. arrogant prick. | ...a point is 0-dimensional, a line 1-dimensional, and a square 2-dimensional. this obvious from the fact that you can construct a simple homeomorphism with R and the open square. | I'm a bot, , . Someone has linked to this thread from another place on reddit: /r/badmathematics A point is one-dimensional because 0 represents the absence of a point Info Contact | I think you’re making a mistake assuming humans are like line segments. First we are a collection of things that do change and are replaced. So In a very real sense our past and future selves don’t exist unlike it seems like you’re assuming. Also if your future self is simply just in the future, well then all movement may be a sham and nothing ever moves and is already there. That’s as likely as what you’re suggesting. Also many things in space cannot move themselves. I’m just going to stop reading now because I don’t want to be mean but it just seems like you’re assuming a million things about measurement systems. Because that’s what your talking about, measurement. Comparing things. So, as far as I can tell you aren’t making sense |
[
"Just some common knowledge"
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0.19
] | null | No. You actually have 100% chance to get a 6 or not a 6. | Plot twist, it’s a dodecahedral die with prime numbers, so you have a 0% chance to roll a 6 | Either way, it’s not a 50% chance | wording specifics, touché | Try again |
[
"Did you know that rectangles are bigger than squares?"
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"math"
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"ewxrnx"
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0.46
] | null | But squares are rectangles | How are you? | What is this even supposed to mean? | 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! | Edit: sorry not trying to be a dick |
[
"Is there another metric besides L2/Euclidean where pi is the usual value 3.14159...?"
] | [
"math"
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0.8
] | [deleted] | If I'm not mistaken, π as defined in any L metric will only take on values in [3.14159..., 4] with π = 4 for p = 1 (taxicab/Manhattan) and p = ∞ (Chebyshev/supremal/chessboard) and π = 3.14159... for p = 2. Of course, this is only a broken memory from an analysis class I had and not something I proved, so I would greatly appreciate help proving/refuting the claim. | according to this m.se qa , yes, in L the ratio is 4, it goes down to pi in L , and then back up to 4 for L . There's a long list of other values there. | Doesn't quite work. If you scale the circumference of your circle linearly, you also scale the radius an equal amount. The effects cancel out. | There's probably a slick dual argument: something like showing that it's monotonically decreasing over [1,2], and that with q such that 1/p + 1/q = 1, we have that the πs of L and L agree, then you're done. | I know I’m shouting to the wind here, but π doesn’t change its value in different norms on the plane. It is a constant which doesn’t depend on geometry or metrics or anything else. There are norms under which the unit circle (in that norm) has a different circumference than the circumference of the unit circle in the Euclidean norm. That’s nice and all, but it doesn’t change π. Really, it doesn’t. |
[
"Does math ever stop being intuitive?"
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] | [deleted] | Never stop drawing pictures! Even for things that can't really be drawn, a rough conceptual picture can be VERY useful. Leverage every bit of visual intuition that you have; it's a powerful tool. Just be aware of its limitations. | One of the reasons real analysis seems so intuitive is that it was invented in order to define the intuitive. | Haha go ahead. I apologize that you have to use my username for proper citation. | Yes. In my case, I got lost once we started talking about ℂℙ². Let alone ℂℙ³... | They’re the complex projective spaces of dimension 2 and 3 respectively. Essentially, they’re the sets of linear subspaces of C and C . |
[
"In Paris, a Mathematician Confronts the Political Odds"
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"math"
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] | null | Question for people in the know: Realistically, is Villani's work at all useful for the grand challenge each morning of transporting quantities of bread from bakeries to cafes. or is the NYT (possibly encouraged by Villani) reaching a bit to make it look like Villani's mathematical credentials are relevant to policy? I would presume that Villani's work would be far too abstract to be useful useful, but I've been surprised before. | Sure, but my question is whether his Fields-winning work on optimal transport (his tome is even mentioned by name) is relevant to his policy. The article certainly is painting it as if it is. | Yeah, let's just conveniently forget that riots in Paris have always been a common occurrence. Riots like this one have been a thing since before En Marche ! even existed. Let's blame it on Macron. | Yeah, let's just conveniently forget that riots in Paris have always been a common occurrence. Riots like this one have been a thing since before En Marche ! even existed. Let's blame it on Macron. | He does have some policy experience, mostly in the academical field. |
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"Bifurcation of Logistic Map"
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] | [
0.75
] | null | I just watched https://www.youtube.com/watch?v=ovJcsL7vyrk and was wondering how hard it might be to make one of those fancy bifurcation graphs he showed. Here is my own take on it. ```python from matplotlib import pyplot as plt from scipy.interpolate import make_interp_spline as mis import numpy as np rr = np.arange(2.5, 4.0, 0.015) it = 5000 for x in np.arange(0.1, 1.0, 0.1): xx = [] for r in rr: xn = x for _ in range(it): xn = r * xn * (1 - xn) xx.append(xn) xx = np.array(xx) xs = np.linspace(rr.min(), rr.max(), it) rs = mis(rr, xx, k=5)(xs) plt.plot(xs, rs, linewidth=0.5) plt.show() ``` | oh I see, you had the graphing software connect the points... I was so confused for a while lol | I kind of wonder if there is any relation between the logistic map and the https://en.wikipedia.org/wiki/Beta_function on some spiritual level or something similar to a Beta function, if only because of the same usage of terms. It would be neat to see a connection to the gamma function coming from the logistic map. EDIT: found this https://en.wikipedia.org/wiki/Beta_distribution on the logistic map wiki so I guess so. | Thanks for posting this. | This part of math is heavily intertwined in mysterious ways. Not only is the beta function/distribution connected, but also fractional integrals of noise (or generally fractional Gaussian/Brownian motion), turbulent flow and anomalous diffusion. I've been trying to untangle the parts we know about for months and still am nowhere...heck, it feels like I haven't even scratched the surface. |
[
"Three integers whose cubes sum to 42 have been found"
] | [
"math"
] | [
"exfn5z"
] | [
24
] | [
""
] | [
true
] | [
false
] | [
0.68
] | null | It's awesome but it's also old news. | Math journalism is hard. One reason is that many people want to know breaking news, and many mathematicians work off of preprints and personal communication. On the other hand, publications themselves come out through the process months/years later. This is an odd case. The result in the title is now relatively old by conference standards. In this standard, it's also very incomplete. The article indicates that 165 and others are unresolved as sums of three squares, but these have been found as well. For instance, (-385495523231271884) + 383344975542639445 + 98422560467622814 On the other hand, none of this work has actually been published anywhere yet. So on that scale, this is cutting edge. It's a challenging spectrum. | This is massively old news. | What's with this article? It's exactly the same as this article that was posted on the same website when this news was actually current (September 2019). Why was it reposted under a different URL and title with an updated date? | 42 was the question |
[
"Let's celebrate 2nd February!"
] | [
"math"
] | [
"exdad0"
] | [
628
] | [
""
] | [
true
] | [
false
] | [
0.95
] | null | gonna be my best birthday ever | So I guess, Happy Palindrome Day 2020? :D | Can't wait to see someone do some travelling shenanigans to have 2 palindromic days next year (12th Feb. in a place that uses the British dating system, 2nd December in a place that uses the American one). | Happy birthday dude from Argentina! | Mine too!! |
[
"Complex cobordism questions"
] | [
"math"
] | [
"exsocu"
] | [
17
] | [
""
] | [
true
] | [
false
] | [
0.99
] | I have some questions about complex cobordism that I'd like to ask. I'll start with a vague question prompted by glancing at the Wikipedia article on "Complex Cobordism". In that article it says that as a generalized cohomology theory complex cobordism can be "hard to compute". It strikes me though that in contrast to some other hard-to-compute generalized cohomology theories such as stable homotopy theory, complex cobordism is at least easy-to-compute on the spheres, since its graded stable homotopy ring is simply the Lazard ring (concentrated in the even degrees). Is this perhaps a clue to the special importance of complex cobordism in the world of commutative ring spectrums in general? For example, is complex cobordism about as hard-to-compute as a generalized cohomology theory can get while remaining easy-to-compute on the spheres, and if so then is that related to the uses to which complex cobordism can be put? (I'm not sure to what extent the article intends "hard to compute" in a merely informal way vs in some more rigorous way, nor to what extent that might affect my question.) | What do you mean by "easy to compute on spheres"? While its true that it has been computed and the Lazard ring is polynomial, it is not obvious either that it is the Lazard ring or that the Lazard ring is polynomial. I think it is reasonable to say that the various types of cobordism are kind of intermediate between singular homology and stable homotopy and saying that complex cobordism is hard to compute is saying it lies closer to the stable homotopy end of the spectrum (ha), though I haven't actually heard that complex cobordism is particularly difficult to compute. | I feel like it is probably a good idea to study the specific details of complex cobordism before you try to think about it in this setting. | The proof of Lazard's theorem takes about 5 pages in these lectures: http://people.math.harvard.edu/~lurie/252xnotes/Lecture2.pdf http://people.math.harvard.edu/~lurie/252xnotes/Lecture3.pdf | According to rumors that I've heard, that exposition does an excellent job of demonstrating how obvious Lazard's theorem is. It should be helpful to me to try to make sure that I really understand the theorem, though, so I'll try to present for criticism my own exposition of it. This might be a slight digression from my current goal of learning the concrete details of complex cobordism, but probably a worthwhile such digression. My strategy here is first to cheat by stating "Lazard's theorem over k" for k an algebraically closed field k of characteristic zero, then sketching a proof of it, and then standing back and considering to what extent there are extra complications in working over the integers Z instead of over k. The theorem gives an explicit GL(1,k)-equivariant isomorphism between the affine k-schemes of "abelian group operations on the formal neighborhood of zero on the line, with zero as the identity element" on the one hand, and of "basepoint-fixing automorphisms of the formal neighborhood of zero on the line, " on the other hand. (Translating "GL(1,k)-equivariant isomorphism between affine k-schemes" from geometric to algebraic language gives "isomorphism of Z-graded commutative k-algebras", which is the usual way of thinking of Lazard's isomorphism.) Given a basepoint-fixing automorphism f of the formal neighborhood of zero on the line , we conjugate addition by f to obtain an abelian group operation for which zero is the identity element. In reverse, the formal exponential map of the abelian group operation can be construed as a basepoint-fixing automorphism of the formal neighborhood of zero, , by identifying the line equipped with its standard vector space structure with the Lie algebra of the abelian group operation. These processes are inverse to each other and manifestly GL(1,k)-equivariant. Then I hope that any extra complications due to using the integers Z in place of k are manageable, though I haven't worked out many details yet. Earlier I wrote that "the Lazard generators are (I think) more or less just the power-series coefficients of the infinitesimal generator of the formal group" but that was imprecise or perhaps wrong, as from the sketch above I'd say rather that the Lazard generators are the power-series coefficients of the formal exponential map of the formal group. As usual, no one else has any responsibility to find my mistakes here but I appreciate it when anyone points them out. [Corrected omissions above indicated by .] | My next question: I'm trying to get an intuitive sense of what complex cobordism is like as an E-infinity object in the (infinity,1)-category of spectrums, based on intuition about cobordisms and on the alleged fact that the graded stable homotopy ring of complex cobordism is the Lazard ring (freely generated as a commutative ring by one generator in each even degree). (Here I'm trying to think of "E-infinity object in the (infinity,1)-category of spectrums" as a sort of homotopification of "commutative ring", so I require E-infinity objects to have homotopy units just as I require commutative rings to have units. Also to keep things simpler maybe I should consider only connective spectrums in this context.) So the stable homotopy spectrum is the initial E-infinity object here, so we have an E-infinity homomorphism from stable homotopy theory to complex cobordism, and we can understand this E-infinity homomorphism pretty well by thinking of stable homotopy theory as "smooth cobordism with stably normal framing" while thinking of complex cobordism as "smooth cobordism with stably normal complex structure"; the homomorphism is essentially just the act of forgetting all of the stable normal structure of the framing except for the complex structure that it provides. But then I'm trying to see how the relative unruliness of the graded stable homotopy ring of stable homotopy theory maps to the relative ruliness of the Lazard ring. So somehow all of the higher homotopy groups of the stable homotopy spectrum (or equivalently all of the "torsion") are getting killed off. So a compact smooth manifold of dimension greater than 0 with stable normal framing is stably null-cobordant via a cobordism with stable normal complex structure? Meanwhile for each even degree d there should be a special d-dimensional smooth manifold with stable normal complex structure to correspond to the Lazard generators. As a wild guess might these be the complex projective spaces? I've placed question marks at the ends of some of my statements but really you should think of all my statements as having question marks at the end. Unless I allow myself to make lots of mistakes, the speed at which I can frame my questions is much too slow to permit any resonable discussion. |
[
"Are there any mathematicians that have designed a new ideal math education curriculum?"
] | [
"math"
] | [
"exr5xd"
] | [
19
] | [
""
] | [
true
] | [
false
] | [
0.84
] | We know the usual math education curriculum that runs something like arithmetic > prealgebra > algebra > geometry > some form of precalculus > calculus ... but I wonder if anyone has innovated a better system, whether or not they expect it would be implemented. I imagine the guiding principles would be that students learn the major milestone concepts and become ready to apply math in various careers. So, I'm most certainly not looking for people trying to define the problem and outline solutions or ideas, but that have actually created an outline or fully expressed curriculum. And I'm interested in a curriculum that begins from the beginning of learning math and at least approaches university level subjects. Who are the authors? Where are their papers or websites? My searches so far have yielded articles and sites about education, as regards talking about math but not math itself, to "elementary" math that's really advanced math and doesn't actually touch elementary math for kids. Thanks for your time. | Bill McCallum, who was a research mathematician at University of Arizona until recently, led the development of the Common Core Curriculum in math and has started a company to help implement it well ( there is probably no shortage of bad implementations). https://www.illustrativemathematics.org/ | The courses you've outlined are really only standard in the US as far as I know. Most of the world has math classes that integrate algebra and geometry throughout the years. Any set of courses can be done well or poorly, but I imagine it's a lot easier to put together a meaningful progressions of skills and concepts when they don't have to be broken up into one year of algebra, then one geometry, and then another algebra. You also need to define what you mean by curriculum. If you're looking for a set of standards defined by year the Common Core has done it pretty well, going from grade to tenth grade. It includes both traditional and integrated pathways and has a very well though out progression of skills. If you're looking for an actual progression of lessons there are also companies that provide those, such as Illustrative Mathematics and Great Minds. I don't believe that scripted curricula like those are productive, though, as they attempt to do the work that teachers really need to do themselves. | The details are foggy and I don't remember the source, but there was a proposition down here in Australia to split the end of high school maths into three more distinct areas: - Calculus, etc. for people who'll essentially study higher level maths in university (as part of engineering, finance, etc.) - Statistics, more for people who'll deal with it eventually e.g. psychology, biology, etc. - Practical everyday maths e.g. business, accounting, tax returns, for people who don't fall into the other two categories. Right now there isn't as clear of a distinction between the three maths subjects (Further, Methods, Specialist, where I am). I don't think this proposition will be implemented anytime soon, but it was very interesting to think about when I read about it. | Common Core is not good. | The Khan Academy page has Mathematic I, II, and III, which is integrated HS math based on CCCS. |
[
"Generally speaking, is Wikipedia reliable for math?"
] | [
"math"
] | [
"ex31vc"
] | [
455
] | [
""
] | [
true
] | [
false
] | [
0.98
] | null | If you're learning maths, you should attempt to understand the proofs anyway, so you're already checking it. Even if not, Wikipedia is built on nerds' need to correct each other, which is one of the strongest forces in the universe, so... | nerds' need to correct each other, which is one of the strongest forces in the universe ackchyually... | It's very accurate, but it's not always useful when learning an area of mathematics because the articles aren't designed as lessons for beginners. But it's a good reference to use while learning from another source. | As long as the subject isn't mixed with politics, Wikipedia should be reliable. However don't take it as an absolute source | Not only for beginners. If I want to learn topic #8 and I know prerequisites #1, #2, #3, #4, #5, #6 well, the Wikipedia article about #8 is likely to be difficult for me because I do not know #7. |
[
"The most important math concepts to add to your toolbox"
] | [
"math"
] | [
"exljgm"
] | [
20
] | [
""
] | [
true
] | [
false
] | [
0.88
] | I like to think about the math I learn as a toolbox, were each new concept or technique I learn forms a tool that I can then apply to solve different problems. That said, what are the most important and effective math tools that non-mathematicians must add to their toolbox? | Basic linear algebra and analysis imo. | Logic. It gets a shallow treatment in high school geometry and most non-mathematicians hardly see it again unless they take philosophy. | Hands down probability and statistics. No matter what your job is, the world is messy and uncertain and probability gives you a rigorous way to approach it. I guess not after you learn probability and realise that your EV is negative. I guess not after you understand that the law of large numbers doesn't work that way. I guess not once you realise that a sample size of one is meaningless. I guess not after you study statistics and watch their argument fall apart. I mean, these are all stupid examples but you get the point. There is too much uncertainty and incomplete information in our lives and going by gut feeling is dangerous. | Well you do machine learning right? It’s founded on probability theory which is basically an offshoot of analysis. The book on machine learning I read is pretty heavy on the analysis-like stuff. (Like for every e, exists a d etc etc) | This depends a lot on what you are doing. If you deal with anything physical, then linear algebra and real/complex analysis will pop up left and right. An astonishing amount of problems can be reduced to eigenvalue/vector problems. If you lean more towards computer science, then combinatorics and graph theory will be your bread and butter. If you do a lot of experiments or deal with people, than statistics will be your friend (or your love-hate). Though, if we broaden up a bit and look beyond the borders of math, then the most important toolbox in Joe and Jane Average life should be questioning their believes. Most people hear something and once they start thinking it's true, they cannot be convinced otherwise anymore. Not even if the situation changes completely. Being able to reevaluate one's believes and biases is something that is getting more and more important in today's fake news society. |
[
"How do you read (maths) journal articles? Computer screen? Tablet? Good ol' paper?"
] | [
"math"
] | [
"exc4gk"
] | [
21
] | [
""
] | [
true
] | [
false
] | [
0.87
] | I am looking for an alternative to printing the articles for reading. I find uncomfortable to read on the computer, and I tried with a tablet but the note taking is a problem. Usually, when I read I take hand notes in the margins (for all that remarkable proofs, you know), and so far nothing beats pencil and paper for that. What do you use? There must be a good alternative! | What are you currently using to read pdfs? I suggest sumatrapdf to everyone and I believe it could make or break your experience reading article (and even textbooks for that matter) | I use my tablet - in my case it’s an elderly iPad Pro with an Apple Pencil. I paid out the nose in all honesty, but I think the good quality screen and stylus does make a world of difference. | What are the advantages of sumatra? | Paper, go out side and read. | It feels really optimized for reading. I've never had to consider alternatives, but from what I remember with having textbooks as pdfs on chrome it was a lot more disorganized, searching by pages and by words was a little less evolved than Sumatra, and now instead of having to keep track of chrome tabs I can just Alt+tab for my textbooks whenever I'm in chrome. Adobe's interface is just a mess , especially if you don't plan on modifying documents (as I myself don't) you're losing a lot of valuable real estate and popups and long load times just makes it a pain. |
[
"Is mathematics as intrinsic as language?"
] | [
"math"
] | [
"ewzi5c"
] | [
5
] | [
""
] | [
true
] | [
false
] | [
0.73
] | Almost every society/group or people in human history has developed some sort of spoken language. If they haven't, they found another method of communication which may include simply pointing at things. But so many civilizations also independently discovered mathematics. The biggest example I can think of is the Mayan empire which independently discovered/invented (whichever you believe) math like a counting system and the Mayan calendar. I believe more civilizations developed language than mathematics but that's not a fair judgement because language is necessary to communicate math. So, is math as intrinsic as communication? | I believe that mathematics is what our minds do when we're both thinking and interacting with the world, and that it is therefore actually a generalization of language. And that makes sense, since linguistics has taught us that human languages have identifiable structures, rule systems, and ways of dealing with novel situations. Languages are used to encode information and then convey that information (whether through speech, writing, or otherwise) to other people with the expectation that they will be able to faithfully decode your message and understand the point you were trying to get across. The human brain just does math. It does it every moment of our lives, even if we don't recognize its information processing as "mathematics" in a conventional sense. | There's no place where they don't use language, but there are places where they don't have more than a few numbers. According to the World Atlas of Language Structures : A number of languages of the world have numeral systems that extend only as far as 3 (e.g. Mangarrayi (isolate; Northern Territory, Australia)), while others show slightly higher but nonetheless heavily restricted upper limits, such as 5 (Yidiny (Pama-Nyungan; Queensland, Australia)). I don't think you could argue that people using these languages are doing math when using numbers. Of course math isn't all about numbers, but I think it's pretty easy to imagine a society where nobody does geometry or graph theory. It may not be as easy to imagine a society where people don't count as such, but it turns out those people are out there. So I doubt the evidence exists to decide whether math is an instinctive human behavior like language, but it's rational to accept the possibility that it's not. | Languages are used to encode information and then convey that information (whether through speech, writing, or otherwise) to other people As an extension to this, you could replace "other people" by "other people, or yourself". This seems to be an important part of why math/language is useful. | Mine is a very unsophisticated thought, but I've pondered it over the years: virtually all people can run integers forward and backward with no issue. Far fewer (offered with no proof, but including me) can run their language's alphabet backward. I realize there is zero social value in being able to run the alphabet backward, save as a parlor trick, and that lack of utility drives its lack of observation. But it's been on the tip of my mind to think this says something about math's inherenticity. (That is not a word.) But in my right mind, I realize this is a dumb observation :) | > I don't think you could argue that people using these languages are doing math when using numbers. Why not? They're just using a heavily restricted number system which is sufficient for their purposes. Imo they're still doing math, just very simple math. |
[
"How to calculate angle?"
] | [
"math"
] | [
"tk73bu"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
0.4
] | null | Beware though that the tan function is periodic with period 180°. That means that e.g. tan(30°) = tan(30°+180°). Assuming that the direction can be in any of the 360° angle, you need a way to check whether to add 180° or not. One way to do this is to check the x-velocity, call it vx. If vx<0, the angle should be betelween 90° and 270°. Example: You have velocities vx=-1, vy=-1. Hence you calculate the angle atan((-1)/(-1)) and get the value 45°. However, since vx<0, we know that the angle should be between 90° and 270°. Since 45° isn't I this range, we need to add 180°. Your angle should then be 45°+180° = 225°. Here's an example of pseudocode you could use. input vx, vy angle = atan(vx/vy) if vx<0 and (angle <90 or angle>270): angle = angle + 180 return angle Some libraries, such as NumPy for Python, has this already builtin. For NumPy you would use the function numpy.atan2. This also works for vx=0. | Use inverse trig functions. Usually you want to do atan(y/x) (because the tangent of an angle is the length of the opposite side of a right-angled triangle divided by the adjacent side) | Thanks a lot! | If you want to do this quickly in a game, you can create a table of value for fast retrieval. Also the atan method fails when Vx = 0. | Most programming languages will provide an atan2(y, x) function that takes two arguments, avoiding the problem when x=0 and providing a full 360 degree (or 2pi radians) range. |
[
"International student surviving in a Russian university: how Latex saved my life"
] | [
"math"
] | [
"exjpfo"
] | [
888
] | [
""
] | [
true
] | [
false
] | [
0.98
] | Hi guys! I'm from Argentina, studying at the Moscow Institute of Physics and Technology (MIPT). Came here with practically no knowledge of the language, studied Russian for a year, and then started my bachelor's degree (Physics and Mathematics) completely on Russian. I remember my first few weeks very clearly. I quickly realized I was way behind, since the preparation people here get during high school is insane. It doesn't help I had to spend half my time translating before I could get to actually studying. Here everything is proof based. For the final exam, you are expected to know over 60 theorems and their proofs. Just understanding those things is hard... how was I supposed to get all of that in my head? All of that for one subject: mathematical analysis. Physics was also very much a struggle, but this was a whole different level. One thing that helped me while learning Russian was Anki, spaced repetition software. I thought, well, if I want to learn a proof, I should try to write it down several times until I got it right. Each time I'd have to look less and less at the textbook. Now, what if I had an algorithm help me, so that I was 'reviewing' more often the parts where I struggled the most? I had little time, so I had to be effective. This is exactly what anki or other SRS does. It's supposed to make you review a card right before you forget it, this way you train yourself to recall. That's when I decided to try and make some flashcards with Latex. I started out small, with a few theorems that were giving me a hard time, and I realized that it was working. Of course, this is no shortcut for understanding a proof, that's a step prior to this, and without understanding, you have nothing. My idea was to train myself to remember the logic behind a proof, so that I could remember all the steps. Guess what? It worked! I was able to study practically all those theorems and proofs, and passed the semester! I wanted to ask, is this a popular thing? I know there is sort of a controvery because, of course, you can't memorize a proof, but I think this is just an optimized way to do what we were already doing, isn't it? I wrote a guide, in case anyone is interested: | Wow! Alot of people consider physics and math to be difficult subjects. I can't imagine having to learn them in a foreign language. As a math major who is learning Russian too this post interests me greatly. Do you find your Russian vocabulary includes alot of very technical terms you don't encounter in daily life? | Yeah, actually, its funny how sometimes I realize I know weird technical terms, but can't remember some basic everyday vocabulary. It's also frustrating when I know how to say technical stuff in Russian but not in English or my native language. | Sounds like Anki saved your life more than latex though. | Curious why you went to Russia to study university? It is popular for people from Argentina to go to Russia to study? | I think it’s popular for people from Argentina to leave Argentina. Don’t get me wrong, it’s a wonderful place with wonderful people, but the political and economic scenes haven’t been helping very much. |
[
"If you could have any function, what would it be and why?"
] | [
"math"
] | [
"tk3kp0"
] | [
0
] | [
"Removed - low effort image/video post"
] | [
true
] | [
false
] | [
0.17
] | null | Have? Like… as a pet? I’m not sure I follow | I’d like to have bowel function. Why? I’ll die. | A function Ans : (Set of Questions) --> (Set of answers) which takes questions as input and gives its answer as output. Do I need to tell why? 😂 | 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'm pretty sure that's not a function, at least not in ZFC. Gotta be some sort of paradox buried in there :D |
[
"what books are good for studying discrete mathematics"
] | [
"math"
] | [
"tjz7gd"
] | [
7
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
] | [
1
] | null | I like "Discrete mathematics with Applications" by Susanna S. Epp. Kenneth Rosen's one is also pretty good. | Most of the materials for UC Berkeley’s discrete math and probability theory class for computer science are available for free at https://www.eecs70.org . There’s no class textbook afaik, and the only thing really missing is the lecture recordings, but the slides are posted and the notes are really good as like a mix between a lecture and a textbook. I’ve been loosely following along with some friends who are actually enrolled and really enjoyed it so far. The homework problems can be very difficult, and there’s quite a few I had to give up on and check the solutions, but the problems are all very interesting and fun to attempt to solve. I highly recommend it if you want a challenge. | Beginner friendly : •Discrete mathematical structures by kolman busby and ross •Discrete mathematics and it's applications by Susanna Epp, A little advanced: •Discrete Mathematics by VK Balakrishnan | Epp is classic. Definitely recommend. | Concrete Mathematics by Graham, Knuth, Patashnik is easily my favorite. Though it's arguably a bit more advanced. |
[
"Any proof on 1! + 3! + 5! + 7! + ... sequence will/will not always give a prime?"
] | [
"math"
] | [
"tk3tg5"
] | [
2
] | [
""
] | [
true
] | [
false
] | [
0.75
] | null | 1! isn’t prime. After that the next non-prime is 1!+3!+5!+7!+9!+11! It doesn’t seem surprising that the first few of these numbers are prime, but I don’t know how you’d prove they’re “more likely” to be prime than other numbers of a similar size formally. | Nice ty | i= 0 ; 1 i= 1 ; 7 i= 2 ; 127 i= 3 ; 5167 i= 4 ; 368047 i= 5 ; 307 * 131221 i= 6 ; 1361 * 4604927 i= 7 ; 201413 * 6523619 i= 8 ; 661 * 8231 * 65616917 i= 9 ; 23 * 269 * 1861 * 10595972521 i= 10 ; 1291 * 2153 * 87011 * 211755799 i= 11 ; 503 * 3456107 * 14900428290907 i= 12 ; 31 * 52169713 * 9607053087589249 i= 13 ; 10024387 * 6375434257 * 170621768533 i= 14 ; 103 * 12457 * 259691 * 26568467889745343027 i= 15 ; 79 * 167 * 857 * 32904013 * 22126658040650014219 i= 16 ; 487 * 2797 * 4248689 * 1501830183738718869343757 i= 17 ; 431 * 154417 * 352438054566919 * 440902557389180519 i= 18 ; 41 * 233437 * 1439161464961198736099202193704034691 i= 19 ; 36477586459 * 622565279381 * 898808668823682643861193 i= 20 ; 83 * 111439 * 10276495349 * 352154715305853297531044490702319 i= 21 ; 62431070602437079834607 * 968247627636926649219253990721 i= 22 ; 79 * 2570369 * 589398059688931909980928710756563118943148344097 i= 23 ; 313 * 8785210429997286127 * 94096176344289335558698165301267629897 i= 24 ; 521 * 11933 * 2356862323 * 6197317339 * 1685779681186373 * 3975240748352325566759 i= 25 ; 242971 * 2140199 * 9775298775180527 * 305264822670597616742574910673548667509 | It fails if you go up to 11 (source: Wolfram Alpha). | You're welcome :) |
[
"Calculating how many towers you can build such that a layer can't have more blocks than the last one"
] | [
"math"
] | [
"tkd7su"
] | [
6
] | [
""
] | [
true
] | [
false
] | [
0.88
] | null | The answer to this problem is even better than you think. If you take j_l=i_l+l-1, your problem transforms into "how many possible combinations for 1≤ j_1<j_2<...<j_k≤ n+k-1". So, as you noticed in the beginning, the answer is C(n+k-1,k). The other way to see that the answer is C(n+k-1,k) is to do what you did and then simplify the answer. ∑(C(n,i)*C(k-1,i-1))=∑(C(n,i)*C(k-1, k-i). (Since C(a,b)=C(a,a-b).) And ∑(C(n,i)*C(k-1, k-i)=C(n+k-1,k). (In general ∑(C(a,i)*C(b,k-i))=C(a+b,k) since the right side is ways to pick k objects out of a+b objects, and the left side counts ways to pick some (i) objects from a red objects and pick remaining (k-i) objects out of b blue objects.) | Another easy solution: note that there is an easy bijection between this question and "how many paths are there from (1,k) to (n,0) on a square lattice, moving only rightwards or downwards?"; the region to the left of this path is always such a building and all such buildings can be represented like this. The latter is well-known to be C(n+k-1,k), since you're choosing k downwards steps out of (n+k-1) steps. | The proposed way to solve it should in principle also be applicable to the new problem. But now it's a huge pain in the ass because you have a tower with two parameters. The fun with summation is left as an exercise to the reader. | More difficult version of this problem for you to ponder. If you have n blocks in total, how many different towers can you build? (The rules are the same. No empty layers, no larger layers above smaller ones, but same size is allowed to stack.) | So I'm assuming that equally many blocks is also forbidden or else the number is always unbounded. n=0 is an empty tower. n=1 has only one version. n=2 can be 2-1-0 or 2-0 3 can connect to any of the previous towers ->4 (3-0, 3-1-0, 3-2-0, 3-2-1) 4 can connect to any of thr previous towers ->8. You see where this is going. This is like Fibonacci but you sum all previous terms. So you get 2 . |
[
"How can a math person best contribute to climate solutions?"
] | [
"math"
] | [
"tkdpz0"
] | [
13
] | [
""
] | [
true
] | [
false
] | [
0.68
] | I have a background in physics and mathematics, and I've been spending a lot of time researching the different paths I could take to maximize my positive impact on Earth's environment. The scale and complexity of modern environmental issues makes it difficult to get a sense for what to focus on, so I wanted to crowdsource some thoughts on this and get a discussion going. Besides the title question, I also specifically wanted to hear some thoughts on these (related) questions: It seems like there is a wave of people with questions to the tune of "how can I be a part of the solution?", so this is both me selfishly asking for career advice and me hoping to add to the growing pile of Internet advice for people who want to dedicate their careers to solving global problems, but have no idea where to start. Also, let me know if I should cross-post this anywhere else which is better suited for career-y questions! | Are you still in university? If so, the best way to get started is going to be to take a class on climate modeling. In terms of mathematical background, you should be very familiar with pdes and numerical analysis. As for your more specific questions: If you want reading, I'll suggest Randall's and and Lauritzen et al's . | I would say that our theoretical understanding of climate is fairly good, but there is still work to be down. However, the work isn't necessarily mathematical in nature, and a lot of it is down with general circulation models. With regards to climate modeling, there are several factors at play: I am not aware of any online courses; this sort of stuff is usually taught in graduate level classes. As for actual solutions to reduce emissions, that stuff is usually more commonly found in engineering departments. Climate scientists themselves do not work on these types of things, just the modeling and prediction. | There’s a mix of technological and policy oriented ways one could try to get involved with. With a math background, you could try to get involved with work related to computational work on climate modeling, simulations of materials that could be useful for clean energy or reducing CO2 emissions, or things like process modeling to study scaling up new technologies. Many of these would rely on knowledge from physics, materials science, and chemistry. There’s also policy hurdles associated with things like prioritizing funding for new technology and disincentivizing the continued use of CO2 emitting technology (among other things). It’s less likely studying math would prepare you for these, but there are people who go from STEM to work related to STEM policy. | Simplest way would be to acquire data science/ML skills and get a job at a non-profit or government in their environment sector. They always need high-skill scientists/engineers but cannot compete with the MAANG companies in terms of compensation. | I graduated recently, so I'd have to re-enroll as a non-degree seeking student to take a class there. I figure there are some online courses out there which could be a decent substitute, though. And thanks for the book recommendations! So we have a solid understanding of climate on a theoretical level, but modeling requires solving computational challenges? Is it an issue of not having enough data/enough precision in the data, or of loss of accuracy in numerical schemes (or both)? and to clarify my second question: should there be more focus on developing particular technologies and strategies (e.g. improving carbon dioxide removal technologies) or on improving our understanding of the systems themselves (e..g how mass implementation of CDR technology would actually pan out) ? |
[
"Question about infinity"
] | [
"math"
] | [
"tkgu8i"
] | [
4
] | [
""
] | [
true
] | [
false
] | [
0.58
] | Hi, I am not a math expert by any means (i am currently attending my first year of Computer Science in a italian university so excuse me also for my english) but today i was wondering about the concept of infinity. My question is the following: If the universe were finite would the concept of mathematical infinities and consequently of all sets (real,integer,complex...) become useless? Without the existence of infinity in the real world woudn't, for example, limits, indefinite integrals and all math become "wrong" in a way? Thanks in advance for any response and sorry if i'm off topic :) | This is an interesting question that brushes up against some ideas in the philosophy of mathematics. I'll respond to just one: Math has no relationship with, or any dependency on, the physical world. Every time a distance, a velocity, an amplitude, or anything else is described using mathematical language, it's just that: a , almost a metaphor. The usefulness of mathematical ideas is that they ! Because they aren't physical, they don't have any surprising properties. They only have the properties they are defined by. This is very useful because it means we can reason about their behaviors based on those properties without any unnecessary details getting in the way. When we say 2 + 3 = 5, we only need to think about 2 and 3 and 5, and what + and = mean. We don't need to know what we have 2 of, or what we might have 5 of, or anything like that. We just know the properties of 2 and 3, and how addition works, so we can learn the result 2+3=5. If we go to apply that idea to the physical world, we need to find things in the world that behave like 2 and 3 and addition, to be able to apply our mathematical learning. For example: Maybe we have a 2 liter bucket and a 3 liter bucket. If we fill both of them, and then pour them both into an empty tank, we can apply our 2+3=5 result to expect 5 liters of water in the tank. But we can only do that because we know that combining liters of water works in a way like addition. Counterexample: We have a 2 liter bucket of salt and a 3 liter bucket of water. If we pour those into the tank, we do not get 5 liters of salty water. This is because the salt dissolves. That doesn't mean that 2+3=5 is wrong, it just means that we were applying it to the wrong thing. So, if the universe is somehow finite, the concept of infinity is not wrong. It would just mean that describing the universe as infinite would be wrong. Infinity is defined by its properties and behaviors, so things that don't have those properties and don't behave like infinity just... aren't it. To sum up, and to respond to the version you posed in a comment below: But since in a way math is the science of quantities and space wouldn't it be wrong? Math is not the science of quantities and space. It is often taught that way to students, but that is not how mathematicians think about it, and does not really describe most of mathematical practice. | Math doesn't depend on the existence of any physical thing | If the universe were finite would the concept of mathematical infinities and consequently of all sets (real,integer,complex...) become useless? For what it's worth, modern cosmology is not totally sure whether our own universe is finite or not. So as far as we can tell, we might as well be in the very hypothetical you are asking about. Have we found various infinite sets to be useless? No. Sets can be used for lots of things besides measuring how big the universe is. | You need to be more specific with your question. Mathematics is not the "science of quantities and space" at all, at least from my perspective. It is, broadly, an exercise in formalized logic and communication - just distilling down our intentions to symbolic representations as best we possibly can, so as to communicate more precisely with one another. There are plenty of contexts where infinite actions (typically, iterative) end with finite results. Take the tortoise and the hare from GEB for example - that's a famous example, where an infinite process (continuously travelling half the distance) ends in a pretty logical conclusion, the hare obviously catches up and passes the tortoise. So I guess in the end while I'm not terribly sure what you're asking here, if my estimation is close to correct, the answer is no. A finite universe does not imply the "incorrectness" of infinite mathematics. | No. The mathematical applications of infinity would still be useful in determining convergence of finite mathematical principals that could be applied to a finite universe. There would still exist infinite possibilities for arrangement of matter in the finite universe. |
[
"What part of math makes you excited about it?"
] | [
"math"
] | [
"tkdf5d"
] | [
30
] | [
""
] | [
true
] | [
false
] | [
0.85
] | I am a programmer, I do work with some form of expressing the logic in the online systems, and this is something that from time to time can be exciting if my task resolves an important problem, or I am working with a modern tools. Otherwise it may be just another day in a job. I have noticed that mathematician's brains tends to find the rush if they "proof" something For me the proof I am looking for is a "ding" of the microwave, it proves that my food is ready. I will not die from starving. Okay, a meme content with math is also exciting: | This is going to sound weird, but being lost (and then unlost): the best moments for me are when I go from having no idea how to even begin on something to having some kind of clue on how I might go about doing it. The bits after that (including the proof and everything) tend to feel more like tidying up after that initial realisation. | Intellectual masochism. :-) | I’m only a high schooler who happens to like math, but for me, the main thing that makes math fascinating to me is its “purity” (couldn’t come up with a better word lol). Sure, math has numerous practical and important applications, but for me the fun part is knowing that, at least in theory, all you really need to do math is your own mind. It’s not tethered to external phenomena like the sciences are and it’s not restrained by limits on time or space like computer science is. People often find pure math arcane and largely useless because much of the math being done today doesn’t apply directly to any particular application, but I think there’s something uniquely beautiful about it that almost makes it feel like creating art out of logic and numbers. | The art of being stuck!! Yessss | Engineering student here. What makes me excited is how math can be used to build models of how a system will behave. More importantly however, you can also go in reverse and tell the model how you want the system to behave and it will tell you characteristics of the system. Want to know how much a beam will deflect with a given load? Easy enough. Want your beam to only deflect by a certain amount under that load? Your model (with some restrictions in place) can tell you the beam needs to be x cm thick. Naturally this means I am drawn towards calculus, with differential equations being by far my favorite subject. |
[
"Is 163 the \"best\" integer for Ramanujan's constant?"
] | [
"math"
] | [
"tkhxv2"
] | [
65
] | [
""
] | [
true
] | [
false
] | [
0.93
] | Hi all, I was thinking today about Ramanujan's constant, e^pi*sqrt(163), which is remarkably close to an integer (has fractional part 0.99999999999925... ), for reasons relating to modular forms and the j-function. I roughly understand why this occurs, and also understand that 163 is the largest "special case" (that is, it is the largest Heegner number) for which we have an explanation for why the result is so close to an integer. I have always wondered, however, whether this is the best we can do. For a real number, denote [x] as the distance from x to the nearest integer. The question I ask is, Does there exist an integer k > 163 such that [ e^pi*sqrt(k) ] < [ e^pi*sqrt(163) ] ? I have not been able to find anything on this online, and I'm not sure how one would answer this in the negative. An exhaustive search might lead to a positive answer, but the exponential nature seems to make this difficult. I would be surprised if typical computing power was able to produce such an example, yet it had not been made widely known. Yet on the other hand, it seems almost absurd to think that the distance to the nearest integer for this function is uniformly bounded by the case k=163 over all integer inputs. Does anyone know of any results for this problem? Thanks! | not for up to k = 100000 https://math.stackexchange.com/questions/1092248/e-pi-sqrt-n-is-very-close-to-an-integer-for-some-smallish-ns-what-about | This doesn't directly answer your question, but I think you will find it interesting nonetheless. There are several values of k which are Heegner numbers, but for which exp(pi * sqrt(k)) is still very close to an integer. The largest "fundamental" such k (which is not a Heegner number) is k = 58, for which exp(pi * sqrt(58)) = 24591257751.9999998222... The explanation for k = 58 is similar to the explanation for k = 163, except that instead of the modular curve X(1), you use the modular curve X_0(2)/w_2, and instead of the j-function, you use the Hauptmodul on X_0(2)/w_2. Here w_2 denotes the Atkin-Lehner involution on X_0(2). There are also many non-fundamental discriminants k for which exp(pi * sqrt(k)) is close to an integer. For example exp(pi*sqrt(652)) = exp(pi * sqrt(163)) is within 10 of an integer. You might think that squaring something close to an integer will "obviously" yield something close to an integer, but this is not true -- for example exp(pi * sqrt(163)) + 10 is still fairly close to an integer, but squaring this value doesn't yield something close to an integer. The correct answer here is to look at the non-fundamental quadratic order Z[sqrt(-163)] ... but I'll let you work out the magic for yourself. | This has been extended to 10 . See sequence A069014 on OEIS. | Nice, thank you for the link! I suppose if I make an algorithm I can start there, haha. | Oh that’s a lovely argument! That’s wonderfully clean. It’s fun to see also that the constant e shows up quite directly. I suppose the next step would be to implement an algorithm to see if such an example could be found! I might do this in the coming days as a fun little project. |
[
"What to do about math books?"
] | [
"math"
] | [
"tkdgwv"
] | [
123
] | [
""
] | [
true
] | [
false
] | [
0.99
] | I am a math phd and over time I have collected a few math books. These books are close to my heart but at the same time due to practical constraints I have to get rid of some of these books. I could donate them locally or sell them to some local used book stores but these books have a niche audience and I feel an aspiring mathematician or a starting graduate student may find much better use of a book. I am wondering if: - there are other avenues aside from local books stores or donation? - is setting up a book swap/book market on this sub (akin to ) an option? - How do people feel about this anyway? ------------------------------------------------------------- Update: I am very happy for the overwhelming response I received. I will come up with a list of books in the next few days and post it here. I will reach out to everyone I promised to reach out. I apologize for prematurely putting up this post before compiling a list of books. | Can you find PDFs of the books via sci-hub, z-library, pirate bay, or similar? If not, please scan the books and put the files online in such free collections so others can find them all over the world. After that, the books are really just for collectors, so maybe ebay? | I unfortunately love old textbooks. Send me a message and let's see if we can work something out! | So much this. If you have books not already on libgen, please do scan and upload them! It takes a little time---I just pop on headphones and zone out---but it is a major help to the world, and to math. It sounds like you want these books to go where they'll be appreciated; this would guarantee that. | Hey, I'm poor and have some old textbooks (1 semester). Four robux? Please? | its amazing how quickly you start to lose those sentimental feelings about books once you've have to carry them all in a move once or twice. |
[
"Stuck in math. Or lost in math? How to understand better?"
] | [
"math"
] | [
"tk2ubl"
] | [
38
] | [
""
] | [
true
] | [
false
] | [
0.88
] | I'm in my fourth year as an undergraduate, on a part time course load. On average I took 3 math classes a semester. I've been having a problem for some time where I'm just too slow in understanding (reaching the point where I can attempt the homework problems) that I would spend way too much time, and eventually fall behind. Describing the problem is pretty hard. For real analysis, algebra and fields I was ok, though the last bit of Galois theory got a bit gnarly for me. I did eventually revisit and understand, though I've forgotten it. After that I attempted representation theory and algebraic number theory. Rep theory I dropped. It became clear in class one day that I didn't understand another student's answer and he couldn't even begin to explain to me. ANT, my prof encouraged me to just finish it, and he passed me. He said I would always go on the right path before just stopping, instead of completing the proof. Actually, I simply did not have enough time to do it. Finally I took Algebraic geometry and algebraic topology last semester and it was a disasterclass to say the least. AG was like.... even the notation confused me and I would just read a sentence and be unable to understand what it meant. Algtop was ok until we hit chain complexes. I still do not know what they are and cochains.... there was a property in the chains and it worked in the cochains. I spent ages trying to figure out why (for an assignment) and I did not succeed. Now I'm taking probability theory, analysis 2 (lebesgue measure, exterior measure, will soon talk about hilbert spaces) and category theory (apparently this is the easiest special topics class ever). Probablity and analysis are a relief to me. Cat theory is a bloody mess. It's never clear to me why I need to prove something, I cannot understand why I need to prove why certain diagrams commute. I will always just think it's by definition. And when I do have to prove I cannot. I'm not sure what to do. Next semester will be my last. It was my dream to do some independent study with a professor but it never worked out. My main goal is to understand all of this. Otherwise I will have come here for nothing. I occasionally do a stock take of what I learned. I learned far less than the classes on my transcript. It's not just about having difficulty proving. I don't know where to start or why I need to prove it. I hope I've done a good job explaining. My request is: how can I improve? I don't mind scrapping what I have and starting over. I want to review ANT, Rep theory, Alg Top and AG when I have the time. The problem is, I don't know where to begin. The majority of this stuff happened online. And I do not enjoy the mode of study but I can't do anything about it. | You know there is an old saying: In mathematics you don't understand things, you just get used to them. It sounds to me that you are being too hard on yourself. 3-4 math classes in a semester is a lot, especially as you get into higher level and proof based stuff. Its going to be hard to get a deep understanding of complicated stuff when you are spread thin across one semester. I also personally think that math has a pedagogy problem, which is that a lot of the proofs we do rely on common patterns of thought and proof strategies, but no one teaches them to you. Probably because most people aren't aware of it. So you have to grind through and discover them yourself. I recommend really studying and trying to reproduce existing proofs, for example those in the chapters of a math book. Eventually the thought processes will get into you. | Maybe you're just more of an analysis person than an algebra person. | Sounds like your brain is overloaded from trying to go too fast. Slow down, keep grinding. Mathematical maturity takes time to develop and you need to walk in circles for a bit to get used to the path before being able to go forward. | on a part time course load. On average I took 3 math classes a semester. I would not call 3 math course a semester a part-time course load, especially if they are upper-level undergraduate math courses or graduate math courses. It takes a lot of time for people to digest the information and to pace themselves so they would be kept motivated about whatever they are learning. AG was like.... even the notation confused me and I would just read a sentence and be unable to understand what it meant. Well, that's normal. But sounds like you are lacking a little bit of the basics presumed in your lectures/textbooks. Algtop was ok until we hit chain complexes. I still do not know what they are and cochains.... there was a property in the chains and it worked in the cochains. I spent ages trying to figure out why (for an assignment) and I did not succeed. Many times I am in the same position. I find it helpful to tell myself to stop thinking about whatever problem I am solving as soon as I feel that I am spend too much time on it or I am not on the full efficiency anymore. Brain chemistry is real. It also shouldn't be a shame to go back to the problem later, or ask for help from your professor or your classmates. It is still solid progress from having tried to solve the problem and learning from a little collaboration. Does your school offer you an undergraduate thesis option? If so, I would recommend you try to find a topic of your interest and do an expository/research thesis in your last year or semester. It would be an entirely different experience from taking classes, and it - more or less - gives you a perspective of what skills are more important from being someone actively learning/doing mathematics. | I felt similarly quite a lot in my math undergrad too. Especially about where to start or what needs proving in my later analysis classes, even though I was never taking more than two higher level math classes. Personally I think my problem was a weak foundation in proof writing/strategies and also a long time to understand things. I've found that building visualizations in my mind by explaining things to my cat or partner has been enormously helpful at tackling topics that were just insane to even read. Do your professors (or at least category theory) have office hours? If so, I think you should be going to every single one. If they're not good at explaining the class content in a way that clicks with you, I think you should ask them if *you* can try explaining a concept that's confusing to you, and have them point out what you're missing. I know it's really intimidating and I had a lot of anxiety about office hours as an undergrad, but know that they would probably love for you to be coming in to do this. The anxiety is just your own pride getting in the way of learning. My last suggestion would be to expose yourself to the ideas of the topic as early as possible. Know what you're going to take next semester? As soon as you have time, watch anyone's videos on the topic, the textbook beforehand, etc. Get the core ideas/themes percolating in the back of your head for as long as possible before class. I also think you're being too hard on yourself. Killing your self esteem at tackling complicated subjects will 100% decrease your productivity. Personally if you can, I think you should try taking less and you'll build your confidence back up (because you are very smart and very capable, getting in your head about this is definitely causing you so much distress that you probably wouldn't experience otherwise). |
[
"Math Related Tattoos?"
] | [
"math"
] | [
"tjrqmk"
] | [
3
] | [
""
] | [
true
] | [
false
] | [
0.57
] | The title says it all. I’m a math PhD student and I’m jealous of all my peers who have math tattoos (epsilon>0, pi, a straight line and a theta in the crook of the elbow, etc.). Tell me about your math tattoos, ideas, and wishlists and maybe I’ll get one :) | My partner is in the tattoo community and they have the best criteria for choosing a tattoo you'll love showing off to everyone: (1) Choose a concept that is meaningful to you. (In this case, a theorem/proof, a piece of equipment or tool, etc.) (2) Think of WHY it makes you excited. (Is it a sentimental tool you've used since learning to like math or did work with in undergrad? Is it related to your thesis? Do you love talking about it?) (3) Gather visuals of this concept. (Textbook covers, blueprint designs, illustrations from super old papers, teaching visualizations, etc.) (4) Find a reputable artist (not just some shop you find, check their Instagram) and go in to meet them. Talk out your ideas, bring your references, and explain the emotions and feelings you experience so they can get it across. Don't be afraid of revisions, it's part of the job. (5) If they do an amazing job, tip $10 per hour. I did this and, from my work in nuclear physics, got a killer optical illusion quantum tattoo. Even if people don't understand it, the most fun tattoos are ones where people still love to enjoy the art. Getting digits of pi can be an alright tattoo, but not forgetting the art is an important way to be proud of this body modification. Now, instead of getting just some text, it will be something everyone wants to ask about. | A nice embedding of the Peterson graph | Someone should get a tattoo where they add a tally mark every time this thread gets posted. If they scroll back far enough I'm sure they'll have a whole sleeve. | You'll want to go through all this too https://www.reddit.com/r/math/search/?q=tattoo&restrict_sr=1&sr_nsfw=&sort=top | stokes theorem |
[
"Help needed for proofreading a blog article on computational homology."
] | [
"math"
] | [
"tjox4t"
] | [
16
] | [
""
] | [
true
] | [
false
] | [
0.92
] | I am working on a blog article about computational homology, where I show how to write a Python program that computes the homology of abstract simplicial complexes. The fact is that I'm not a mathematician, just a computer dude who enjoys mathematics. So before I publish this article on my blog and post a link here, I'd like some help with the proofreading. If anyone is interested, start a chat conversation with me and I'll provide a pdf export! | I can probably help with this. | You may also find an article i wrote that does the same thing useful: https://jeremykun.com/2013/04/10/computing-homology/ | You have an excellent blog! | Man, I came across your blog years ago and thought it was awesome (although I could not understand much of it at the time)! It even motivated me at some point to learn more maths. As time passed by, I completely lost track of it though, glad to see you're still at it. I'll read your article more in detail, but I can tell we used quite different approaches for the computation. That's interesting. | might be worth mentioning the Smith normal form and structure theorem for fg modules over a PID, if people need to consider torsion and/or compute homology with coefficients |
[
"What Math Trick you've learned that you deemn useful and use it every time?"
] | [
"math"
] | [
"tjt4ip"
] | [
247
] | [
""
] | [
true
] | [
false
] | [
0.97
] | [deleted] | Add 0 in a clever way | A related trick is multiplying by 1 in a clever way. | You can differentiate by separately differentiating each occurrence of a variable, keeping all other occurrences constant, then substituting the variable again and summing the results. E.g.: ( x )' = (( a )' |a=x + ( x )' |a=x) = (( a log a )|a=x + ( a x ) |a=x) = x (log x + 1). This is essentially smartly evaluating the directional derivative of a generalization of the original function. In this example: f(x) = x = g(x, x), where g(x, y) = x . Edit: corrected the example. Edit 2: I should like to add that this result is pretty strong and general, as it generalizes both the summation and product rules for differentiation. This result, the chain rule and the partial derivatives of function w.r.t. to all of their parameters are all that are needed to evaluate every possible derivative. Edit 3: To show how stupidly general this result is: you can derive the Leibniz integral rule from this by considering there to be three occurrences of x: two for the integration domain a(x) and b(x), and one in the integrand f(x,y). | If the problem is too difficult, try solving a slightly easier version first and see if this might give me something to attack the more difficult problem. | Indirect proof like proof by contradiction and contrapositive. |
[
"[suggestion] Learning about math proofs using lean"
] | [
"math"
] | [
"tk4y1a"
] | [
236
] | [
""
] | [
true
] | [
false
] | [
0.97
] | I always have an interest in learning about how to write mathematical proofs. But even after reading many books and watching videos about the topic, I am still that dumb guy who sees mathematical proofs as alien language. Despite being a mathematical simpleton; I love programming and able to understand the logic behind it. I am also able to relearn many mathematical concepts which I thought was tough by writing programs or by reading math books written by programmers for programmers. I recently came across lean program which helps to validate theorems. Can some one suggest an material which teaches proof through lean programming from scratch? Whats your view about using lean for teaching math proofs from basics to advance level? | I LOVE LEAN!!!! | 🟪🟪🟪🥤🥤 | Perhaps start with the Natural Number Game ? | I thought you were talking about a different kind of lean :) | Before reading the post I was thinking “man, troll posts don’t usually get upvotes on r/math , this must be really well written” |
[
"Which mathematics books are suitable for self-study?"
] | [
"math"
] | [
"tjv86c"
] | [
68
] | [
""
] | [
true
] | [
false
] | [
0.93
] | Mathematics books have a bad name for being terse and abstract. Although being abstract is not bad. But most math books are written for people who already know something about the topic of discussion. There are exceptions of course and that is what this post is about. Which math books do you know about that a reader can work through on their own to gain an understanding of the topic i.e. suitable for self-study? This is applicable mostly for foundational subjects. That is when a student mostly knows nothing to very little about the topic at hand. Of course, the writing has to be extraordinary as if the author is teaching you individually like in an old apprenticeship system. Knows nothing about the topic -----------> Awesome book --------------------> Has some idea about the topic tldr; easy to read, "easy" to understand, delivers value | Books’ prefaces are usually very helpful in that regard. Look for the keywords “elementary,” “first course,” “self-contained,” “introductory,” “undergraduate,” etc.. and read the entire preface to see if it seems suitable- they’re usually a couple pages, at most. Amazon reviews with a high number of reviewers can be a decent gauge, too. They also sometimes provide a few pages’ worth previews. | I don’t know why you make out self studying math to be this huge, monumental, borderline impossible task. It’s not, and saying stuff like this just discourages people unnecessarily. | Elementary Analysis: The Theory of Calculus, Kenneth A. Ross | They just wanted a book recommendation, not a huge reason to give up on studying math | why my post are being banned with the same topic as yours 'self study' and this is not?. |
[
"22f with Zero math knowledge"
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"math"
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"weapi5"
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4
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] | null | Khan Academy has the entire US mathematics curriculum from PreK to College. It's the best place to start. | I'm not from the US. I'm from South Africa. What's prek? | Pre-kindergarten. Like, the very first class you would ever go to for anything. | Yes you can. just read elementary math and understand it well. once u get started it won't be very difficult. If u get stuck somewhere, do learn & repeat | One thing that I am very grateful for that my math teacher in secondary school made us do is around 20 calculations in our head everyday. He would give us a paper each week with 7 Sets of questions. This was simple exercises like multiplications,divisions, additions. I'm not sure how far behind on maths you are but this will help you immensely in anything in the future. Being able to do quick calculations all the time is a blessing and I'm very glad our teacher was strict about this. |
[
"Looking for help in Newton Raphson Method"
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"math"
] | [
"we9rmd"
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"Removed - ask in Quick Questions thread"
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0.86
] | null | Isn't NR the one that looks like x_(n+1) = x_n - f(x_n)/f'(x_n)? Because if I recall, that one finds roots | You can find the local extrema of functions using Newton's method on the derivatives of a function, but its purpose is to find the roots (or an approximation of the roots) of a function. Finding extrema is a side benefit, but not original objective of Newton's method. | Approximation to roots. I think the higher the derivative, the closest result of approximation to the root. | I used NR in maximum likelihood estimation (using gradient vector and Hessian matrix) in order to find (lokal) maxima... So I think you're right, it finds f(x) = 0. | I have a bunch of mathematics lectures on my YouTube channel, ThatMathThing , and a playlist for Numerical Analysis. This video is a bit older now, but it is a full lecture on Newton's method. https://studio.youtube.com/video/hvXtKtkjIUk/edit And I code it up in MATLAB in this video here: https://studio.youtube.com/video/5Ua0DAA7sEg/edit |
[
"What is the longest/complex math proof and definition?"
] | [
"math"
] | [
"tjz9t2"
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192
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""
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true
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false
] | [
0.9
] | In terms how much words/formulas/symbols you need to write it? | The classification of finite simple groups is a good candidate for the longest / most complex proof. | I would guess it depends on how much you are willing to "unroll" definitions. A Kähler manifold is a complex symplectic Riemannian manifold. A complex manifold is a manifold with holomorphic transition maps, a symplectic manifold is a manifold with a closed non-degenerate differential 2-form, and a Riemannian manifold is one with a symmetric positive definite rank-2 tensor. But even those definitions sound very complex without a first semester on manifold topology. | At over 10,000 pages, spread across 500 or so journal articles, by over 100 different authors Divine Copulating Bovines! That would make some very thick book! | I'll wait the movie. | The computer-assisted proof of Keller's conjecture in seven dimensions. It takes 200GB to store the complete proof, which would easily stretch to tens of millions of pages if printed out. |
[
"Why is the infinite monkey theorem a big deel? Is it just fun to imagine, or can we actually pull some knowledge out of it?"
] | [
"math"
] | [
"we9ggk"
] | [
28
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"Removed - ask in Quick Questions thread"
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true
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false
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0.75
] | null | It's mostly an introduction to the "weirdness" inherent in studying very large numbers. | Not exactly the monkeys situation but there’s a similar thought experiments to point out the “properties” of infinities. My favourite is the following: A group of librarians decide to write a dictionary with every “word” possible. By “word” I don’t mean English words, but any combination of letters. For example, AA would be a two letter word, although it has no meaning. Words can have from 1 to infinite letters, so it’s a hard task. They decide to work in alphabetical order. So they write a book containing every single one of the infinite words that have A as their first letter. The words would look like: A, AA, AB, AC, AD, …, AAA, AAB, …, ABA, ABB, ABC, … This takes a lot (indeed infinite) of time. But they do it. Thing is, they’re too tired to do it again with all the words that start with B. Not to mention the remaining 26 letters of the alphabet. So one of them has an idea. They take the “A” Volume and delete the first letter of every word. The remaining book would look like (I’m gonna use underscore to mark the removed A) _, _A, _B, _C, …, _AA, …, _AB, _AC, …, _BA, _BB, _BC, … Look at what we obtained. The first bunch of words before the dots are all the one letter words, the second bunch are all the two letter words where A is the first letter, the next bunch are all the two letter words that start with B, and so on. Deleting letters from one of the 26 Volumes, we obtained a Volume with all the possible words, starting by any letter. So, in a sense, one of the Volumes contains the totality of the Volumes. EDIT: As two kind redditors pointed out in the comments, words can’t actually be infinitely long. All words in the Volume are finite, but there is always a longer word. | What's particularly interesting about it to me is that it's often stated in a way that misses that weirdness. Most often, I hear: "An infinite amount of monkeys on an infinite amount of typewriters will eventually produce the entire works of Shakespeare" To me, that understates the vastness of infinite numbers. If there's an infinite number of monkeys, there's no reason to say "eventually". Instead, I favour something along the lines of: "An infinite amount of monkeys on an infinite amount of typewriters will produce an infinite number of copies of the entire works of Shakespeare in as short a time frame as possible for a monkey to produce one page... as well as an infinite number of copies with exactly one typo" It's maybe not as quick and easy to say, but I like to emphasize that you're limited only by what's possible and not by what's likely. | It's maybe not as quick and easy to say, but I like to emphasize that you're limited only by what's possible and not by what's likely. This is a great way to put it! | I think there might be a problem with allowing the longest words to be infinite length... |
[
"I've got a question for you"
] | [
"math"
] | [
"we8h2r"
] | [
0
] | [
"Removed - ask in Quick Questions thread"
] | [
true
] | [
false
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0.13
] | null | it’s just ambiguous notation. | To answer your specific question, multiplication like 2(3) is sometimes called multiplication by juxtaposition and I think (but don't know for sure) some places/disciplines have a different precedence for this compared to using the multiplication symbol. The question which you and your friend are arguing about is a silly one designed to provoke arguments - the nuances of PEMDAS and multiplication by juxtaposition and whatnot are never actually relevant to mathematicians, who would in this situation use parentheses or a fraction bar for clarity. I'd consider this expression a little ambiguous and would ask for clarification / look for context clues if I encountered it in the wild. | It's just convention matter, nothing to do with math | read the sidebar. | It doesn't fucking matter. It's about notation and convention. Noone uses the ÷ symbol because vertical fraction notation is much better and many people, myself included, treat everything that is joined by juxtaposition as a block. If you wanted x/yz to mean xz/y then you could have directly written xz/y, it is much more intuitive to understand x/(yz) |
[
"Can you name some Number sequences?"
] | [
"math"
] | [
"we3u9d"
] | [
4
] | [
"Removed - ask in Quick Questions thread"
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true
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false
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0.83
] | null | "The On-Line Encyclopedia of Integer Sequences® (OEIS®)" https://oeis.org | Fibonacci. There should be a lot written about it, if he also likes to read about the sequences. If he's into pi, look at phi, and research what that's about. If he wants to learn about famous numbers, look at chemistry and biology and physics, for avocados number, etc. Edit: a v o g a d r o ' s number | He was 3 a few weeks ago. And can play piano a bit as we put numbers on the keys and he memorises the sequence to play it 😂 | Thanks for the edit I was about to ask if that’s what u meant lol auto correct strikes again | Besides looking at sources online and in books in libraries, please be sure to use archive.org which has digital versions of lots of older books you can check online. I'm hoping you're getting guidance from social workers or other therapists. My one suggestion is to let him follow his interests, and don't artificially prohibit him from exploring certain areas. I've seen the negative consequences such limits can have on a child. Is he old enough to consider learning an instrument? He may find parallel stimulation from figuring out how to play musical pieces. Just a thought. |
[
"When I am asked for the exact value, does that mean decimal?"
] | [
"math"
] | [
"wdxx9o"
] | [
1
] | [
"Removed - ask in Quick Questions thread"
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true
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false
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0.6
] | null | Write it as 2*sqrt(3)/3 as any decimal answer is a rounded/truncated approximation | (2√3)/3 has an infinite number of (nonrepeating) digits. Can you write down of those infinitely many digits in your answer? If you try to write the decimal expansion of that number but you don't include of its decimal digits, then you're not being exact. | When you’re asked for an exact answer, it’s because they don’t want you to use a calculator. | Short answer: no, don't give the decimal. Most decimals are only approximations, not exact. Specifically, a decimal with only finitely many places is exact, like how 1/4 = 0.25. But pi is only close to 3.14 or 3.14159 or any other finite decimal. On the other hand, if you just say "pi", well, that's exactly pi. Expressions that involve pi or sqrt(whatever) or fractions sometimes feel less concrete to some people, but they are exact. | 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! |
[
"How hard is linear algebra?"
] | [
"math"
] | [
"we1ym4"
] | [
40
] | [
"Removed - ask in Quick Questions thread"
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true
] | [
false
] | [
0.91
] | null | High school linear algebra will mostly focus on matrix computations. You’ll probably start out with using matrices to solve a system of linear equations and then expanding into operations available for matrices. It shouldn’t be too hard, although it might feel boring just multiplying matrices or keep solving systems of linear equations might make linear algebra seem boring. Just know that there is a LOT more to linear algebra. What you’ll cover in a course like this doesn’t account for all the linear algebra theory there is. I realise the following does nothing to help you, but when I read the following definition of linear algebra, it made me very excited to learn about these things. So, hopefully this gives you something to look forward to in the future if you wish to learn linear algebra properly. More formally, linear algebra is the study of linear maps on finite dimensional vector spaces. Systems of linear equations are a special case of linear maps. Solving systems of linear equations play an important role in most computations done in linear algebra. Anytime you try to compute something, you’ll find that solving a system of linear equations is always one option among others. You will see this as you learn some proper linear algebra. But all in all, it shouldn’t be that difficult. These types of introductory courses are pretty accessible. | College linear algebra can mean either linear algebra with a focus on matrices or it could mean linear algebra with a focus on linear maps and vector spaces. It really depends which one of these it is. For the matrix one, it is a bit new and that could be confusing. For the linear maps one, it can certainly be made difficult and a class like this will mostly focus on proofs. So if you’ve never written a proof before you might struggle with this course. It really depends on which course it is. | Linear algebra is not too hard. Yes, it is heavily based on matrix algebra. You will likely spend most of your time on row operations. I think it would be manageable. | A professor of mine once said that linear algebra is the only domain of math that we truly understand in its own right. Every other domain is simply translated into linear algebra, solved there, and translated back. Ever since then I've been astounded by how close this is to the truth | You're going to hearing mixed results because everyone's experience will differ greatly depending on the professor they had. There are two extremes here with (widely) varying levels of compromise: With professors who emphasize application, there will be heavy focus on computation, especially in terms of (duh) applications. In this case "theoretical" side of things is sort of brushed over in favor of emphasis of learning to apply the various techniques (mainly decomposition methods) on a number of exercises with some nicely behaved applied examples. The difficulty here is learning how to apply the techniques appropriately - in particular, learning how to "linearize" commonly encountered real-world systems (traffic flow, force distributions, decision models, chemical balancing, etc.). On the other hand, if you have a professor which emphasizes the theoretical side of things (i.e. focusing on proofs of abstract results), the difficulty then comes from acclimating to one of the first exposures to (albeit relatively simple) proof-writing and reasoning. Computation with this case is largely done in the context of hoping to ascertain some general result from whatever you're dealing with. One common point between any interpretation of the above is the ease of computation and calculation. In general, if one establishes a good foundation with the early chapters (typically on matrix arithmetic), then it quickly becomes apparent that the computations involved in linear algebra are not difficult at all. It's the conceptual side of things (be it theoretical or application) in which many students have difficulty adjusting. |
[
"Post-quantum encryption contender is taken out by single-core PC and 1 hour: Leave it to mathematicians to muck up what looked like an impressive new algorithm"
] | [
"math"
] | [
"weecoq"
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760
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""
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true
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false
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0.98
] | null | It’s a good article. Explains the broader context, gives enough details to explain the situation but should be accessible to most people, but also provides links to papers for those who want the gory details. | It was impressive, and very elegant mathematically. I'm surprised that the article gives vibes of "mathematician shows the world how it's done" - in reality this is mathematicians improving work of other mathematicians, and landing a suprisingly powerful attack in the process. Both the cipher and the attack were a great contribution to science. See also https://old.reddit.com/r/math/comments/wc4gkx/supersingular_isogeny_diffiehellman_broken/ for an older discussion on this sub. | Well, it wasn’t quite as impressive as it seemed if that happened :P more like “mathematician shows how good a cryptosystem actually is”. Also, long live code-based crypto!!! :P McEliece for the win babyyyy! | Agreed. Just a horrible subtitle imo. | Slightly different than what this article is about. "Quantum supremacy" is about claiming that your rudimentary quantum computer can solve a specific problem faster than any classical computer. (And yes your right, google's claims of this are a little exadurated). This is about finding a long term encryption scheme that neither quantum computers nor classical computers can crack. The tech for quantum computers will eventually get there. Its just not right around the corner like some people want you to think. |
[
"How to know the path to take?"
] | [
"math"
] | [
"wdx8cv"
] | [
24
] | [
""
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true
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false
] | [
0.81
] | Something I'm struggling with now is I don't really know how to improve at math outside of my university's course structure. For some context, I'm a discrete math additional major (unofiicially as of now) and the most recent courses I've taken are abstract algebra (but like 99% group theory) and next semester I'm taking category theory. The problem I have is I see all this cool math that I want to be doing, but I'm not sure how to get there. Like it would be great if you could just open a book and get to work, but a lot of them require prerequisite knowledge for understanding concepts, or at the very least motivation. For example, I'm taking a Homotopy Type Theory Course right now, and sure, I can follow the material, but because of my inexperience I have no intuition for the motivation and frequently they mention that a concept is similar to <big scary math word> that I've never seen before. From what I can tell, I think Category Theory will help a lot with understanding the HoTT material, but in general, I feel like its hard to learn math without a dependency chart of sorts. Does anyone have any advice? | I'm surprised they even teach Category theory and Homotopy type theory as undergraduate classes. Both of these requires motivations from many different fields of math otherwise they wouldn't make sense. Technically, they do not logically depends on any other classes, but without other knowledge you're basically learning them dry with no motivations, no intuitions and almost no examples. Really, the instructor should have at least mention what kind of math the students should be recommended to know before learning the class. I think these are the worst topics to teach to undergraduate. | Ok, that makes more sense, even just a bit. Category theory is still quite unmotivated though unless you have a lot more algebra, and also algebraic geometry and topology. Homotopy type theory, you probably should know intuitionistic logic, algebraic topology and probably some type theory (through computer science or logic). Even the name itself suggested it as such: "homotopy" was originally from algebraic topology, and "type" of course come from type theory. To a certain extent, you can use category theory to motivate homotopy type theory....but category theory itself require more motivations, so you're back at looking at algebraic topology or something like it. | The HoTT course is online just for fun. The category theory course though is through my university. There are no official prerequisites but they recommend logic and algebra | Its just research I guess ? In France, a phd is 3 years and you dont attend class; you only go research (with a little teaching) and it is not rare to spend the first 6 months or even the first year of your phd to just read book in order to understand the subject. If your first steps in research are driven by your advisor, it is for a reason. | Ask for advice from people on prerequisites. It would help if you gave them a rough idea of what you’re trying to build up to. For example, if you wanted to do stochastic analysis someday, I could give a guide on how to go from calculus all the way to stochastic analysis. Other subjects will need a different guide of course. |
[
"If I want to learn about the mathematical basis behind Gaussian Mixed Models, which topics should I look for?"
] | [
"math"
] | [
"yu9hkv"
] | [
6
] | [
"Removed - ask in Quick Questions thread"
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true
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false
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0.88
] | null | Do you have any background in Calculus? I would recommend picking up a book on probability theory / statistical modelling. You could also check out some free materials from places like edX or OpenCourseWare ect... | This or something like it would be good if you want to learn about probability and statistics from the ground up. I'd try to find a course aimed at 1st year university students but this is dependant on your mathematical maturity. Since you said you have a biology background maybe it would be better to find a book or lecture series aimed at biologists/scientists. If you are less interested in learning everything from the ground up and instead just want to learn more about the kinds of models you have seen, prehaps a course like this would be a better fit. Good luck to you. | 🙈 | Mainly multivariable calculus, undergraduate probability and then the mixture models themselves. | I did study calculus as undergrad, I could probably catch up on the missing gaps. Any suggestion for specific courses/books? |
[
"What are some useful or cool techniques?"
] | [
"math"
] | [
"we78h9"
] | [
169
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""
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false
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0.95
] | What are some useful or cool techniques like strong induction or that proof technique in graph theory where u try to find contradiction on a minimal or maximal graph? | FullSimplify[] on Mathematica. It will change ur life. | One class of techniques that I particularly like is generalizing your problem. Counterintuitively, sometimes looking at a more general case than what you’re actually interested in helps remove some noise of the special case and give meaningful insights. A few examples: I’m sure there are other nice examples I haven’t thought about. | The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of the prescribed kind is strictly greater than zero. Although the proof uses probability, the final conclusion is determined for certain, without any possible error. Quoting Wikipedia | Cute example: paint a sphere 90% red and 10% blue. It can be any arbitrary crazy swirl of red and blue areas, tiny dots, etc. Place a cube inside the sphere so that all corners are touching. Can you rotate the cube so that no corner is touching blue? | Knowing when to exit a rabbithole — still working on that |
[
"Why is mochizuki still a professional mathematician?"
] | [
"math"
] | [
"we241o"
] | [
0
] | [
""
] | [
true
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false
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0.3
] | In other fields of the natural sciences, when people intentionally post verifiable false results and are caught, they are usually reprimanded, charged with scientific dishonesty and stripped of all their accomplishments. Think of who lost everything because her STAP cells were fake. If Scholze and Stix have verified that mochizukis proof has errors, yet mochizuki claims the proofs to be correct, how has he not been charged with academic fraud to some extent? | Proofs are wrong all the time. I don’t see charging anyone with academic fraud for that is required. What I think is that he should try to communicate his work in a better way. From what I have read, he uses very different and weird notation in his papers and is not open to communicate his work properly. I think I read somewhere that he thinks the only way for someone to learn all of his work is to come study under him for years. | I had heard that Scholze and Stix had constructed an explicit counterexample to Corollary 3.12, but I can't seem to find it. In any case, for the sake of argument, let's assume that to be the case (so we make the argument as unfavourable against Mochizuki as we can), and list what Mochizuki's done that you might be taking issue with: Written a paper with a serious error in it that invalidates the result. This isn't grounds for firing by any means. Errors infrequently do come up in mathematical papers and usually these are merely caught by peers and retracted/corrected. Refused to clarify that part of the paper towards Scholze and Stix. Also not grounds for firing, he's not obliged to do so. Managed to convince his coworkers that his flawed argument is correct. Also not grounds for firing; whether his coworkers agree with him should not be his responsibility. Published a revised version of the paper (still with the same error in it) in a journal. Again, this isn't grounds for firing. Published that revision in a journal for which he's the chief editor. While this might seem like academic dishonesty of some sort, this is apparently common in some areas of mathematics as long as the authors recuse themselves from the peer-review process. Mochizuki claims to have done so, and even if it were the case that the reviewers felt coerced to accept it simply because he's their boss, there's plausible deniability here. Not retracted his paper. Not grounds for firing, he's not obliged to do so if he genuinely believes his argument is sound, and the journal also isn't obliged to retract it for him if the reviewers believe his argument is sound too. Acted unprofessionally towards Scholze and Stix throughout this whole ordeal, as well as in a followup paper. Unprofessional conduct like this is sometimes grounds for firing and sometimes not (and it depends on the university). Probably a bunch of other things that I'm not aware of, but which also aren't grounds for firing because they're all similar to the above. In the specific case of Obokata, the main point isn't that her results were "verifiably false" (in the sense of "not reproducible"), it's that she outright fabricated the data and plagiarised the text, so she was acting in bad faith. Mochizuki, for all his faults, has made a genuine attempt at solving the conjecture, even if his proof is fallacious. To draw a parallel, to do what Obokata did, Mochizuki would have to outright plagiarise other people's papers, cite certain non-theorems he was using as being in some source that didn't actually contain them, and falsely claim that his proof had been computer-verified without having performed any such verification. Alternatively, to do what Mochizuki did, Obokata would merely have to have made an error when performing her experiment and/or have incorrectly analyzed her data (thus making her conclusion untenable), and then have been a dick about it to anyone who pointed it out. Obokata's actions were far worse than Mochizuki's; it simply doesn't even come close. In conclusion: Is Mochizuki's paper wrong? Very likely. Has Mochizuki behaved like an unprofessional dick? Absolutely. Has he ? Unless he secretly had a hand in the peer-review process for his own paper, absolutely not. Should he be fired for academic fraud? No, unless it can be proven that he did in fact coerce his reviewers. Should he be fired at all? Probably depends on your view on whether people should be fired for being unprofessional dicks. | That’s called a disagreement | Does anyone claim that Mochizuki intentionally published results he knew to be incorrect? Do you have evidence supporting such an allegation? If not then this is not a case of scientific dishonesty. | But Mochizuki is a notable mathematician himself. Should we say that Scholze and Stix, whose rebuttal of his claims he disagreed with, are frauds? After all, a notable mathematician disagreed with them, right? |
[
"Formula to convert meters in feets and inches"
] | [
"math"
] | [
"ytw21q"
] | [
0
] | [
"Removed - see sidebar"
] | [
true
] | [
false
] | [
0.5
] | null | Unfortunately, your submission has been removed for the following reason(s): /r/askmath /r/theydidthemath If you have any questions, please feel free to message the mods . Thank you! | EDIT: Thanks guys ! Because of you i was able to finish my little script :D Have a nice day ! | You need to do a bit of a remainders calculation. First multiply the number of meters (x) by the number of feet in a meter. Take the whole number component of this and set it aside, this is your number of feet. Then, take the remainder and multiple the number of inches in a foot by it. That gives you the inches. | The problem is your converting 2 units (feet , inches) in to a single unit meters, and vice versa. You will want to convert meters to feet. Say we get 3.21 feet. Use the floor function to separate the decimal off ie: Feet = floor(3.21) = 3 Inches_unconverted = 3.21-floor(3.21) = 0.21 Then convert 0.21 feet to inches (using the ratio) Then you can output the feet and inches. | You just compute how many feet are included, and then in what remain you see how many inches are. A ft is 0.3048m an inch is 0.0254m So if you have x meters, the number of feet is f = floor(x/0.3048). What remains is (x-0.3048 * f)/0.0254 inches Alternatively, you can compute the total of inches t = x / 0.0254 then the number of feet will be equal to f = floor(t/12) because there are 12 inches in a feet, and the number of inches will be equal to i = t - 12 * f. Note that funny thing can happen working with floating point numbers. I don't know the details of Python. This post will be deleted by mods because it is against the rules if this subreddit. You should ask on r/learnmath Even better, considering that the math needed for this is extremely elementary, you should ask on a programming subreddit. |
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