Welcome to All Angles! We produce animations about mathematics, physics, engineering, and statistics. Our videos are meant for non-experts who want to get a good introduction to higher math, from group theory to Lie algebras. We love mathematics, and we can't wait to show you. Enjoy!
Wow, great video! I felt like as the video kept going it just gotten better and better (and I really liked the visualisation at the end) Btw I think a video that shows some possible usages of these concepts could be really cool
Grant cancelled #4 on October 16th, 2023. He wants to do it every other year. Too much work for him, I guess. Thanks for the video though. Expect me in your comment section asking dumb questions.
For my degree dissertation I wrote about Analytic Number Theory. I dedicated a whole chapter to Moebius function. I never got a deep intuition for the function, but this video changed that! Thank you
This highlights why referendums are important and should be more common and binding, since only being able to vote for D or R doesn't represent voter preferences.
I don’t think your example (at about 4:16) for an implication case where p is false but q is true is correct. The statement, if 2 + 2 = 5, then I’m the emperor of China, is logically false since neither “2 + 2 = 5” nor “I’m the emperor of China” is ever true and, therefore, there is never a time when “2 + 2 = 5” and “I’m the emperor of China” are both true. So the example you should use should be one that meets the following three criteria: 1) there are occasions where both p and q are true, 2) q is true for all occasions where p is true, and 3) there are occasions where q is true but p is false. So one example I can think of is . . . if a store is out of bread today, then I cannot buy bread from that store today. There can be other reasons why I can’t buy bread from that or any other store today (such as not having any money), therefore, the conclusion of the if-statement may still be true while the condition is false (the store has bread). If the example you gave is the same or very similar to the ones logic teachers use, then it’s no wonder why their students are very confused, and rightly so.
Here's another way to look at the mobius function from number theory. Similar to how we constructed the mobius function as the inverse of the dirichlet series having all ones, we can do the same thing with the power series having all ones, i.e. 1 + x + x^2 + ... = 1/(1-x) . The inverse of this is 1-x, the mobius function for the case of power series. One can think of mobius functions more generally for posets, and the one for power series is the simplest one, it is the infinite poset given by the natural numbers under the <= relation. If you look closely, the 1-x might remind you of the mobius function for dirichlet series. In fact, if you take a prime power, its divisibility poset will look like a chain, and the associated mobius function will have a 1 on 1, -1 on the prime, and zero everywhere else. For arbitrary numbers, the poset becomes a product of such chains. Using that the mobius function is multiplicative on products of posets, one recovers the mobius function for dirichlet series.
Sometimes things do not "slowly evolve" but just pop out when doing something. Sometimes one makes some definition because it is helpful and later on finds out where it fits. E.g., |μ| is something that someone would easily come up with as an "indicator" function for squares because indicator functions are very useful as they basically represent the idea of "has property" functionally which means they can be used as functions in calculus rather than trying to do everything as sets. This then lets one use the theory of functions to study things that may not really be that obvious as sets. It's obviously true then that |μ| = sgn(μ)μ which is equivalent to μ = sign(μ)|μ| then the question is "what could the sgn(μ) be"? While in some cases there might be many choices it is obvious that in this case a natural choice is quite limited since we are talking about the prime factorization and there is a limited number of choices with square free: p1^a1*p2^a2*...*pn^an. ak < 2 else the integer is not square free. So all we have in square free integers are a product of "singular primes". So all we can really do is count them(that is the most natural thing) and talk about the parity of the number. Hence sgn(μ) = (-1)^(# of prime factors in factorization of a square free integer). So the point here is that coming up with μ is really not that difficult and something likely that didn't take that long and many people came up with it or the variant |μ| and this likely happened even thousands of years ago when people were thinking of prime numbers and thinking about just multiplying singular primes(which then they are ultimately thinking about |μ|) and maybe wanting to distinguish between having even and odd number(which gives μ). What took time is to develop and see how μ showed up in a variety of places and how it could be used as a fundamental function in number theory. There are likely millions of functions people have defined easily/quickly using similar logic that have not yet been shown to be connected to any deep theory. Usually the simpler the function the more places it will show up. μ is quite simple. It's about as simple as one can get when talking about primes and trying to investigate their exponents. Primoridal primes[square free integers] come up with often when looking at integers because they sort of act as the prime building blocks "2nd stage". E.g., if one is doing sieves and such. Because any integer is either a Primoridal or a product of Primoridals. So I would say, at least in this case, that μ is not complex at all. It only seems that way at first because number theory is pretty unnatural for people. Almost no one things to investigate numbers in the way that number theory does. But once you realize numbers have all these intrinsic relationships and start thinking about them and how they interact you'll very likely to stumble upon |μ| sooner or later. In your video, as you explain, when you move multiplying and involving polynomials to that of Dirichlet series(which isn't a leap) then one naturally will seek out μ. These things can happen out of the blue when one studies such things. Of course we have hindsight but I think most people that have spent years doing math everyday can attest to how they figure out "new" things only to learn they already existed. It's sorta like once you get going down a road you're gonna start coming across the same things other people have went down that road before have saw. If you are the first to go down that road it might be a little slower and you are the first to see those things but sometimes it just all works out surprisingly well. Basically almost anything is going to be "foreign" to someone that doesn't spend time learning about something. The more time one spends on something the more "patterns" their brain will automatically come up with. Probably the biggest issue with people on the forefront of knowledge is that they are traveling down a dark road and don't know if it will yield anything of value. Usually the people that care less(and can care less) about where the road is taking them and only care about the sites they see are the ones that make the most progress going down the road(which is never ending).
Also, we can use the Euler product of the zeta function: ζ(s) = prod (1 + p^-s + p^-2s + ...) = prod (1 - p^-s)^-1 so 1/ζ(s) = prod (1 - p^-s), and expanding it out, we see the coefficient on n^-s matches the definition for μ(n)
I'm at @15m44s and can't help noticing that this is describing exactly what the entirety of simple linear regression does in statistics. I also can't help noticing that the μ and σ have very particular meanings and that any statistical estimate with a single degree of freedom will decompose into an estimate of the value of that function which we call μ that has an error estimator ɛ. The measurement of the spread of that error estimate is measured in units of standard deviation that just so happen to be labeled σ. I also know that with each additional variable we use to try to make a better estimate of that original function are estimates of a coefficients vector β of each additional parameter in a vector x with the initial β₀ canonically set to 1 (because it is already the estimate μ. Each βᵢ coefficient marginally has a corresponding error estimator ɛᵢ that measures a component of the total error that exists between the points being estimated and the βxᵢ parameterization of that component of the vector. For components that have no explanatory power the β fails to meaningfully project the x component from 0 and the error ɛ remains not meaningfully changed from its value prior to that component being added. The component's individual contribution is measured in squared distance from a "sum of squares" model for the geometric fit. A component may also have non-orthogonal components that must be rectified with all other component vectors and these form a covariance matrix which is measured by the cross product of their error estimators... so help me if this doesn't very quickly start discussing statistics and the minimization of error via a sum of squares and covariance in the next i dunno 45 sec imma gonna lose it
It's really weird, when you think about it. multiplication = rotation. The complex numbers are hiding in the space between -1 and 1. Thank you for illustrating this so clearly for us.
When does one start ignoring a pattern? The 0^0=0 people deduce the pattern from positive integers. But we note the pattern doesnt work for negative integers. So that suggests the 0^0=0 reasoning is biased. 0^0=1 is the hero.
💛 I may add to the final note: although a definition kinda presents itself, there are infinitely many possible constructions in maths that can be done, but only some of them get selected because they’re cleaner to work with (and for example define!) and they are more connected to other things (and often applications). From this standpoint, one can ask why had μ as it is made it into mainstream theory, if it has a definition that seems a bit clunky, but you show how it’s also simply related to the 1̅ sequence and also it can be noted from Hasse diagrams that in the end the definition itself is quite graspable too: alternating ±1 parity thing happens very often in other math, and adding a 0 clause is only too useful if we want to deal with square-nonfree numbers at all (and we do!). Videos that touch this process of working with constructions-and even more, choosing one over another (which is undoubtedly harder to show, so I’m okay with that being presented rarer in popularization/explainers-I get how much work it is to make a good clean thing)-are so important to make people consider trying doing more math of their own and experimenting and not just staring at some text without an idea that it can be visualized and can be understood by themselves. Thanks!
Thank you for the kind comment. We have made another video that follows the same idea: we show that there is only a single correct formula for 2D multiplication that satisfies all the required properties. It turns out to be the complex multiplication. You can find it here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-di5QKO9xg2I.html It's often very instructive to show why a definition is "inevitable" once you have agreed on its important properties.
Not all functions are formulated from a dirichlet series. Dirichlet series and functions are also two different things but if you regard the dirichlet series as a function, then they only classify as a small subset of all functions as they follow a specific form as seen in the video. If you are given a dirichlet series and want to see how it s composed, however, that’s a different story
Great video. But I couldn't understand after around 3:40. I've heard about a related concept, Möbius transformations from a popular book (VCA) by Tristan Needham. I still don't really see the connection yet other than they are both connected to group theory. Perhaps I need to rewatch the video a few times and reread VCA a few times to understand it. I'm glad that some4 is happening, and not some3.5 as I've previously heard.
While they're named after the same person, the moebius function here is otherwise *completely unrelated* to moebius transformations as you would see in Complex Analysis. So it's not weird that you don't see a connection: there isn't one.
Basically what @diribigal said. That Möbius guy must have been very busy, because he has his name on multiple unrelated concepts. By the way: I love VCA, it's an amazing book. But, indeed, unrelated to this video.
For me sets are just predicates. Like x in S just means that S(x) is true. In that sense, sets from set theory are just a bunch of 'or's, that's why the 'comma' is commutative and idempotent, because 'or' is.
It is not the inverse of the neutral element 1,0,0,0... is the neutral element 1,1,1,1... is the element taken in consideration The function of the video is the inverse of the 1,1,1,1... sequence
Yeah! Anyway, anytime you find that a neutral element has a different inverse from itself, it signals that either it wasn’t the neutral element, or it’s not the operation under which it’s neutral (compared to the operation wrt its inverse is a different thing), or maybe even they are indeed the same modulo some natural equivalence we forgot about, or something like that. And small errors like those do happen all the time, so it shouldn’t be any worry-despite math allows exact proofs of things by its very nature and it isn’t technically a natural science where you _absolutely need_ more varied evidence from the universe to be more certain about your theories, math still enjoys more evidence for one to become more certain in precisely the same way, so checking things from different angles (or considering particular cases and generalizations) is always a good idea even if you’re sure you have everything already proved by that point. And moreso if you’re not certain because the more insight and intuition one has, the better the road, and the more abundant are opportunities.
I can't watch it entirely now but I don't want the algorithm to punish the video. I'll keep going later. I didn't know the SOME 4 was already on going. Great news!
Thanks for the video. Am I understanding correctly that any function can be written as a linear combination of 1,x,x^2.... (or the terms in the Fourier series)? If so, is there a proof for this?
Thanks for the question. The answer is no: there are many exceptions. Plenty of functions don't have a Taylor series, and plenty more have a Taylor series that doesn't converge for all real inputs.
Know the video is already 3 months ago. But Find it facinating. Looking deeper I discovery symple typed lambda-calculus, that is exactly it. But here has a different notation.
I've had this discussion as well about the accuracy of the political compass before compared to other tests, and what's the "correct" number of axes. I said more axes explains the data better since there is less compression. I had the same conclusion that if you want to state your political identity with perfect accuracy, you have to state your stance for each possible issue so there's no compression at all and no nuance is lost. I came with the thought of doing Factor Analysis on everyone's stance for each issue. There will be a factor explaining most of the variance given that people's opinions on different issues can be correlated. In the video as well, you said that there are less people than the possible combination of stances given just 33 issues. That already suggests a lower bound to the number of Factors, or dimensions we can reduce the potentially infinite number of issues people disagrees on. That's because for issues #34 and above will necessarily correlate to issues #1 - #33 since the space of opinions is constrained by the finite number of individuals.
That's such a great idea. I wonder if it has already been done. I know that the psychologist Jonathan Haidt has made a kind factor analysis of political temperament, but that's not quite the same thing.
Nice, I used a similar visualization for the SVD in my video on the matrix transpose here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-wjYpzkQoyD8.html
Thank you for the well explained video. I wonder how this could be applied to financial modelling and risk analysis - my first thought is to run a Monte Carlo analysis with as many variables as possible and record all variable values with the output (for example profit or IRR). Then “just” do the “elipse thing” to figure out what variables are the most impactful?
This is a very nice presentation. I like the tie-in to data science and the covariance matrix. One small thing: at 27:22 you say you can compute V similarly to U. But there is a hazard: the eigenvectors of V are dependent on the choices made for U (even Gil Strang ran into this issue). It's best to substitute U back into the original decomposition definition and solve for V (the remaining unknown). I'm enjoying this series.
Regarding the uniqueness of the neutral element in your example, what is the difference between: make 1 blue ; do_nothing And make 1 blue ; make 1 blue
Good question. The 2 programs in your example have the same effect in the end, but they do very different things. In the second program, the second "make 1 blue" is not a neutral element. Just apply it to a situation where cell 1 is yellow, and you will see that "make 1 blue" has a very clear effect. The reason it *seems* like a neutral element is just because cell 1 *happens* to be blue.
