You explained the topics enough to understand what was going on and showed barely enough for us to be intrigued and interested in this without us getting really spoiled or you being tiresome. I am fully convinced to at least attempt and learn more from these fields eventually because of this video. I cannot help but praise you
Oooh, this seems like it could have lots of utility in digital audio processing, since you're regularly moving between the discrete and continuous domains.
There is so much mystique in this area. I feel like there is a mystery that is just lurking, waiting to be discovered. I see little tidbits of group theory conjugation, analytical combinatorics, probability density functions, so many paths begging to be traversed. From a personal point-of-view, so many potential application to physics
Absolutely fantastic video. The Newton difference formula derivation was simply amazing, i used it before but never knew where it came from and this was just the cherry on top. Can't wait for the follow up!
this is SO much better than the wiki page. It left me fascinated (even moreso than Dr. Michael Penn's talks on the matter), and honestly someone should REALLY add the homomorphism between discrete and infinitesimal calculus you described here to the wiki AT THE MINIMUM. thank you so much for the contribution to math education!!
In the sci-fi rouge like rpg "Caves of Qud" dark calculus is a forbidden field of mathematics, because it's study opens a path into a transcending layer of reality inhabited by an infinite ocean of psionic minds... After watching this i am impressed by how accurate to real life the devs made their lore.
Very nice! The example of umbral calculus on the Wiki page is pretty cool too, it relates to Bernoulli polynomials B_n(x) which satisfy the identity B_n(x) = (B + x)^n, i.e. B_n(x) = sum (n,k) B_k(x) x^(n-k). And you can actually simplify some proofs of identities involving the Bernoulli polynomials by doing "calculus" with this umbral notation.
I love umbral calculus and generating functions. Ive been reading George Boole’s book on the calculus of finite differences, and I really appreciate videos like these which make the ideas more accessible to the general public
Very high quality vid! I once read the Wikipedia page on discrete calculus, and the conclusion I came up with after reading for a bit was that it was dumb people calculus for dumb babies, and also that it was boring and dumb. But this was actually pretty interesting! The video & graphics quality here was great, loved the visualizations and I would've loved it if you had even more graphing and illustrations, especially in the later parts of the video. I'm looking forward to your next video!!
I’ve been shown another area of mathematics that peaks my interest, and has given me a decent view into the essence of it! Thank you, when it’s a drag it’s always better learning something new, and maybe finding some meaning within it.
Oh wow, this is really cool! I've played around with the umbral operator before without realizing what it was. I think the most recent time I used it is when I was converting a formula for factorial moments into a formula for non-central moments a few weeks ago.
Super interesting Stuff! I like how categorically you can see in this topic that calculus itself is a limit of this discrete version! The exposition was super easy to follow, love it.
Umbral calculus is truly a shadow of school calculus. I played around with umbral calculus and discovered that the sequence 2^n is its own difference. Therefore, 2^n is a shadow of e^x. Really cool. EDIT: If you apply newton's forward difference formula to 2^n, you get something that is disturbingly similar to the maclaurin series for e^x
One of the best SoMe vids yet! I literally just learnt about the binomial theorem and summation, intriguing to see it can also be expressed using discrete calculus
I like that you mentioned the prerequisites at the start of the video, and I also liked that you didn't explain what a complex number is like an average math channel
Thanks! The ideas there were that #1 lets the audience know they're watching the right video (or not), and #2 complex numbers aren't particularly necessary but can be used if you're familiar with them :)
9:03 I see what you did there: n_2 = n(n-1) = n^2 - n, n = n_1, n^2 = n_2 + n_1 and then phi inverse (n^2) = phi inverse (n_2 + n_1) = n^2 + n. Why didn't you explain this part? When I saw it for the first time I got confused a bit. UPD: I saw you are actually explaining it right after this example :).
12:25 it all looks nice but these formulas should only work for the natural x, right? Otherwise your n choose k should be transformed into something with the Gamma function, right? Basically, it should only apply for e^(ax), integer x. We should be able to pass complex numbers for a for sure, but x should stay a natural number for all of this to work. Or am I missing something?
I gotta admit, this is one of my favorite videos of the SoME2 this year. This intrigued me so much and you explained it pretty straightforward even though I didn't completely understand everything on the first viewing. This year's SoME really gave us some banger math videos, can't wait for next year!
Wait, at 13:39, you performed the steps as if ϕ(fg) = (ϕf)(ϕg), which for me it isn't clear at all if it is true, or why it'd be true. Can someone explain to me how he distributed the ϕ operator in the summation?
Thank you for the video! I spent like 45 minutes going over the video, writing everything down, checking. It was a great experience. Please make more videos like this one, including the follow-up video on the Umbral Calculus.
It is the kind of video that I am looking since years ago. I found a formula for integrating analytic functions using series, more exactly summing derivatives of the function want to integrate. If have some interest let me know. Or at least could you recomend me some books for this fi function and this idea you are dealing with in this video. Best regards.
0:25 - 0:28 I am not sure if what you have on screen is supposed to be the fundamental theorem of calculus or not, but that is not the theorem. What you have on screen is just the definition of antiderivative. 1:05 - 1:10 The problem here is that the notation If(x) is just ill-defined. There are many unequal quantities that are nonetheless said to be equal to If(x), and this is just nonsensical. You cannot use notation like that. You have to choose a specific antiderivative of f, and call that If. 12:32 I think it is more enlightening, at this stage, to rewrite (a + 1)^x as exp(ln(1 + a)·x), so Φ[exp(a·x)] = exp(ln(1 + a)·x), and here, it is immediately clear that Φ's role in this particular context is to transform a -> ln(1 + a). This is foreshadowing for something you already plan to bring up in a future video, which is that D = ln(1 + Δ).
The bits in the intro are just meant as illustrations yeah :p and I like the I and Σ notations for indefinite stuff, I figure we're decluttering by removing the dx while we don't need it (and bringing it back when we do!) I really like this idea in your 3rd paragraph, I'll have to work it into the new video somehow. I didn't know about D = ln(1 + Δ) at all when I was working on this one and I'm still getting my head around it
Did anyone else get excited at 7:42 when they realized that he's drawing a commutative diagram? (with elements of the objects instead of the objects themselves, but still)
Amazing educational content! The only thing I would like to comment is that the delivery is somewhat stressed. There's hardly any breathing room in the video. Sanderson often gives you some slack to ponder after an information dump, where you can reflect a bit on what was presented and absorb the material.
@@Supware Well it is very easy on the ear.. And does you great credit. Apparently there is a Welsh Northern and English... creole. In any event your accent and proffessionalism of Narration is a great joy to endure. Much Thanks !
Amazing video! I really want to dig deeper into this but can't find anything online, where did you do your research for this video? Thanks in advance :)
The community has since found some promising resources! The books Gian-Carlo Rota: Finite Operator Calculus and Steve Roman: The Umbral Calculus, as well as Tom Copeland's blog 'Shadows of Simplicity' :)
There's a interesting relation between \Delta and D \Delta = 1 - e^D and S = 1/\Delta = 1/D (D/(1-e^D)) = 1/D + B_0 + B_1/1! D + B_2/2! D^2 + B_3/3! D^3 ... which is Euler-Maclaurin formula. The relation mentioned in this video is also interesting. thanks.
I've loved what I've seen of the video, I love calclulus, but I think I've fallen asleep both times I've tried to watch this, something about your voice and the pauses to think/read tell my body to sleep. I will return, you can look forward to the difference created by the sum of my discrete efforts to finish this delightful presentation.
@@Supware Thanks for the video, the comment is not a complaint about its content, it is more clear than other resources, this is more accessible, but still I have some struggles.
I don't understand what's happening at 13:30. Firstly, D^n f(0) is essentially a constant, and the operator phi cannot act on it. And if it can, then phi is not multiplicative, and therefore it cannot act on both x^n and D^n f(0). The final answer is correct, of course, but the approach is very strange
I feel cursed. The man plays isaac, and now speaks of umbral calculus. What dark abyss has he gazed upon to have an epiphany about "umbral" calculus. What dark sorcery is this
At 7:13 you define the falling power for negative n, with an image that says "n terms", yet the image shows n+1 terms... Should it start at x+1 or should it finish at x+n-1?
I wonder if there’s any way to use normal or umbral calculus to find an exact functional way to do that, I believe you could very much just use factorials or something
Well done for the video, it is amazing. Question: In the Newton's forward Differences formula at time 13:43 in the right hand of the equation inside the sum we replace x^n/n! -> x_n/n! But why there is a φ in front of D^n(f)(0) and since D^n(f)(0) is a number, what does φD^n(f)(0) means ?
φD^n(f)(0) is the phi of the nth derivative of f, evaluated at 0 (constants are φ-invariant too!) The derivation is my own; if I've done something dubious without realising I can start an errata :p
@@Supware I think the only dubious part I spotted was that pulling 𝜑 into the summation technically swaps two limits so somewhere here some dominated convergence / uniform convergence assumtions are neccesary since not every C^oo function is real analytic, but surely 𝜑 has the correct properties for this to work out. (I assume here that 𝜑 :IR--->Z but maybe it's smarter to define 𝜑: IR[x] ---> Z[x] ^^' not sure tho)
@@MultiAblee yes this has been a big question for me throughout haha, my analysis skills are abysmal but I think a friend of mine recently managed to nail down what domain to use to keep 𝜑 bijective etc. If we make much more progress with 𝜑 itself I'll put it in another video :)
@@Supware Maybe we can use the fact that [𝜑^-1g](0) = g(0) to show D^n f(0) = [𝜑^-1 Δ^n 𝜑 f](0) = [Δ^n 𝜑 f](0) ? Then there is no need to keep 𝜑 on the right hand side of the equation at 13:41 in the first place
this umbral calculus is the quantum mechanics of mathematics where the wave function is applied to the derivative of the delta operator and the result is a function amplitude which, if squared, gives you the probability that you have the correct answer in terms of the sigma of the exponential.
14:48 more advanced discrete calculus? : yes more combinatorics stuff? : umh, maybe no. but it depends. but please do at least noise cancellation of your audio in the post processing. i am not saying to get a new mic or whatever.
newton series is such a nice analogue to taylor series and a special case of the interpolation polynomials when written in newton form. Tried to use this on a test and the teacher just said "there is no such thing as a 0 in the indexing set" and didn't even bother looking at the rest. like just change the index if you don't like it? it's much uglier with index 1 than 0 because it doesn't ressemble classical calculus anymore. it'd be quite unnatural though still technically correct if taylor series started at 1 instead of 0.
It is, if you want to think of it that way! Phi and its inverse can be expressed as matrices with Stirling number coefficients, and the monomials / falling powers are the different bases
Very cool topic and great presentation! 9:00 phi takes a regular power to a falling power, so I take it that inverse phi takes a falling power to a regular power; so how does inverse phi work on a regular power, n^2? The only way I can see to make this work out like it did in the video is that phi n^k = n(n-1)...(n-k+1) = n!/(n-k)! phi inverse n^k = n(n+1)...(n+k-1) = (n+k-1)!/(n-1)!
It is so sad that the real good content creators don't get enough attention and need to stop. And we get stuck with so many overrated sh1tty fake content creators.
To the correction regarding time 7:13. I think it would make more sense if the denominator was just (x+1)(x+2)...(x+n), giving us an empty product for the special case n=-n=0. The previous correction with extra (x+n+1) in the denominator would give us (x+1) in the denominator for this n=-n=0 case, which is not compatible with the formula shown at time 6:46 at the top (where by setting n=1 we obtain x₁=x₀⋅x and from that for x=1 we get 1₁=1₀⋅1 and therefore 1₀=1, whereas using the previous correction we get 1₀=1/2, which does not make any sense to me).
Just…wow. Super concise, accurate, insightful, intuitively explained… definitely a video worth seeing by every modern mathematician. I’ll certainly look a little deeper on umbral calculus research thanks to you. My dearest congratulations. Thanks for sharing! 🧮
Obviously there are plenty of limitations (that the Discord server folks are still figuring out!), but yes for e.g. all sums of polynomials, trig and exp functions :)
That φ smell's a lot like a isomorphism, that makes we wonder if you can define some sort of abstract "umbral space" of which calculus and umbral calculus are only 2 examples.
