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
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!!
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.
Really interesting topic and amazing explanation! I had never heard of Umbral Calculus up until now. I'm loving this summer of math exposition, I'm learning about so many new interesting things!
I already had a quick introduction to discrete calculus. This is wonderful next step. I see someone else already compared your approach to Grant's. So generous and clear, both of you.
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).
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?
(T-1) is the forward difference, -(1+T+T^2+T^3+...) is the taylor series of its inverse which just so happens to be the negative sum from 0 to infinity of f(x+k), since one is the inverse of the other this is what you're supposed to get: -(1+T+T^2+T^3+...) * (T-1) = 1 On one hand it makes sense for everything to cancel out because of the properties of infinity, on the other someone might read this as -T^inf+1 = 1 or T^inf=0 which seems like nonsense
@@КириллБезручко-ь6э I do agree that if "(T^n)f(x) -> 0 as n -> inf" the relation T^inf=0 holds, that is quite obvious. To me the problem is that this is supposed to be a relation that holds for any f(x). The idea of f(x)=f(x)-f(inf) may be actually quite interesting tho. For example you could think of -f(inf) as some sort of constant, and so you'd have an equality relation that looks like f(x)=f(x)+C, which would be a system in which translating a graph in the y direction does nothing to it. Also, I have tried extending these ideas to the hyperreal numbers and have noticed that with the definition of T[f(x)]=f(x+e), such that 0
For some reason I hate discrete calculus (or anything discrete for that matter), but the fact that I can transform every discrete problem into a continuous one makes me like it a little bit more :)
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
Wait, if phi of e^-x is 0, then phi isn't injective, so how can phi inverse exist? Or can we only write phi inverse when we restrict attention to just the polynomials (on which phi is injective)?
Phi has an inverse for all the "nice" functions mentioned in this video (except e^-x). Characterising which holomorphic functions phi is bijective over is an interesting problem I don't have an answer to yet
12:10 but doesn't the binomial theorem state that it is the sum from 0 to x and not infinity, I doubt this as a mistake but I'm not someone to talk with confidence here I suppose it is like a limit as x tends to infinity or something similar??
This is the generalised form of the binomial theorem that extends to negative and noninteger x; when x is a positive integer most of the terms end up just being 0. You can compute (x choose n) for any complex x using 11:03 :)
@@Supware Wow! Okay I googled it and now I totally get what you mean so that's what has been wrong in my mind throughout the video, I thought "why isn't he talking like with factorials and nPr's and stuff" but yeah that's just me with my highschool level because after all I'm just a math enthuthiast, but anyways thanks so much it is an enlightening observation for me :D
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!
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.
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
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.
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 :).
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
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?
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.
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!
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.
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.
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.
@@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 !
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
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.
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
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
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?
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.
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.
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' :)
@@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.
“By rearranging the question, we get the answer.” Imma use that
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
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!!
Wow haha thank you! Guess I've no choice but to keep it up :)
it's already there but not much on it. like 3 line paragraph style
At 13:40, why is there a phi before D^n? isn't (D^n f(0)) just a constant?
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.
What a legend only one ad in the beginning . Your so damn underrated
3:01
I've never seen that explanation for the fundamental theorem of calculus... it seems so simple now.
Well, unfortunately, the equation shown on screen is not actually what the fundamental theorem of calculus is or says.
Extremely clear, insightful and interesting exposition!
This video is so amazing, it blew my mind, please continue making these
Really interesting topic and amazing explanation! I had never heard of Umbral Calculus up until now.
I'm loving this summer of math exposition, I'm learning about so many new interesting things!
I already had a quick introduction to discrete calculus. This is wonderful next step. I see someone else already compared your approach to Grant's. So generous and clear, both of you.
Excellent video. Thanks for sharing.
Totally worth the wait!
Definitely looking forward to that followup!
Fantastic, you Sir did a Great Work!
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).
And at time 13:14, shouldn't there be just x instead of x-1 in the subscript?
Just found this, and your channel. This is the wildest math I have seen in a long time! I wonder if this has applications elsewhere.
I think "the wildest math I have seen in a long time" is what I'll be going for from now on hehe :) I think you'll enjoy the second video..!
@@Supware looking forward to it! :D
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?
Magic, beautiful content... but also explained equal!!! (I had some intermediate brain-meld-down, but: World-class video, my compliments!
what the fuck i just watched. There are so many new things that i learned from this video.
Awesome video!
Excellent, thank you very much 🙂
Best #some2 so far!
(T-1) is the forward difference, -(1+T+T^2+T^3+...) is the taylor series of its inverse which just so happens to be the negative sum from 0 to infinity of f(x+k), since one is the inverse of the other this is what you're supposed to get: -(1+T+T^2+T^3+...) * (T-1) = 1
On one hand it makes sense for everything to cancel out because of the properties of infinity, on the other someone might read this as -T^inf+1 = 1 or T^inf=0 which seems like nonsense
ну так это имеет смысл только если (T^n)f(x) -> 0 при n -> oo, это еще связано с нормой оператора
I have noticed that I made a simple mistake, maybe a typo, saying that -T^inf+1=0. It is now fixed
@@КириллБезручко-ь6э I do agree that if "(T^n)f(x) -> 0 as n -> inf" the relation T^inf=0 holds, that is quite obvious. To me the problem is that this is supposed to be a relation that holds for any f(x).
The idea of f(x)=f(x)-f(inf) may be actually quite interesting tho. For example you could think of -f(inf) as some sort of constant, and so you'd have an equality relation that looks like f(x)=f(x)+C, which would be a system in which translating a graph in the y direction does nothing to it.
Also, I have tried extending these ideas to the hyperreal numbers and have noticed that with the definition of T[f(x)]=f(x+e), such that 0
@@Fox0fNight удачи в исследованиях!
For some reason I hate discrete calculus (or anything discrete for that matter), but the fact that I can transform every discrete problem into a continuous one makes me like it a little bit more :)
I guess I'll endeavour to show you how beautiful finite stuff can be :D
Wonderful!
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
Please do more!
I found a new world here
This is the missing motivation from the semesters of calculus i took.
Taken a lot of inspiration of 3blue1brown I see. Even the way you speak is similar to the 3blue1brown chapter 1 Calculus series. Interesting. 😆
Nice video!
that was so cool!
Wait, if phi of e^-x is 0, then phi isn't injective, so how can phi inverse exist? Or can we only write phi inverse when we restrict attention to just the polynomials (on which phi is injective)?
Phi has an inverse for all the "nice" functions mentioned in this video (except e^-x). Characterising which holomorphic functions phi is bijective over is an interesting problem I don't have an answer to yet
12:10 but doesn't the binomial theorem state that it is the sum from 0 to x and not infinity, I doubt this as a mistake but I'm not someone to talk with confidence here
I suppose it is like a limit as x tends to infinity or something similar??
This is the generalised form of the binomial theorem that extends to negative and noninteger x; when x is a positive integer most of the terms end up just being 0. You can compute (x choose n) for any complex x using 11:03 :)
@@Supware Wow! Okay I googled it and now I totally get what you mean so that's what has been wrong in my mind throughout the video, I thought "why isn't he talking like with factorials and nPr's and stuff" but yeah that's just me with my highschool level because after all I'm just a math enthuthiast, but anyways thanks so much it is an enlightening observation for me :D
I have a feeling that Ramanujan would have liked umbral calculus.
Amazing
Volume was a little low...
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.
Interesting, I'd love to see more practical applications of this thing
brilliant direction. time to see how these transforms would help analyze some filter and the fourier transform
@@OdedSpectralDrori don’t quote me on this but it seems the Laplace Transform is VERY relevant here :p
@@Supware this will make a fine quote
@@OdedSpectralDrori what have I done
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!
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.
I could feel my Glimmer rise after watching this
dark and EVIL calculus 😨
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.
They’ll be appearing in the follow-up!
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!!
Thanks! Illustrations are certainly gonna be an interesting challenge in the next one...
@@Supware what are you demonstrating with sir? MathCad ?
@@Briekout Manim
It is calculus for engineers lol
@@alpers.2123 Don't laugh at engineers man. They cool
I've been struggling with abstract algebra and your video presents a perfect example for why isomorphisms are useful! Really appreciate it!
Great to hear!!
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.
Just in case you didn't know, it's "pique" in the phrase to "pique one's interest"
@@Tom-u8q Like Piqueachu.
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.
Oh nice! I'd love to know if it has a name or symbol, I somehow haven't come across either yet
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
Yep! This is a special case of the stuff I talk about at 12:00ish in the video (a=1) :)
Rad! The familiar-but-different feeling makes this feel almost like a math dream
After working for a while on a second video I think this might actually be a major vibe I wanna aim for haha
Is this similarities related to group theorem?
@@alpers.2123 phi is a homomorphism between discrete and classical calc
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.
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 :).
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
Umbral Calculus is just the best when you're deep in some special functions, like Bessel, Laguerre, and so on.
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?
If newton used this, was discrete calculus developed earlier than the real number one?
Not sure but I'd imagine the formula was discovered before the formalisation of limits, yeah :p
and here i thought umbral calculus is the mathematical machinery needed to manipulate the shadow realm
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.
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!
What if we replaced the falling powers with gamma function to make *continuous discrete umbral calculus*
This is so cooool. Which books or other texts could you recommend about this topic?
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.
That was one of the most entertaining things I've ever watched. Bravo, subscribed.
Wow, thank you!
5:53 - 5:59 PAUSE
Have you ever read the chapter on umbral calculus of steven roman’s book?
No but I've seen it mentioned quite a few times, I really gotta check it out
Thank you Supware for introducing me to this beautiful world of Umbral Calculus!
wow, great video, thanks!!
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)
More coming! I got some bad bois in the follow-up whose objects aren't even labelled ;)
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.
Absolutely amazing video! Subscribed.
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 believe that the word "umbral" comes from the idea that the superscript n casts a shadow down to the ground of subscript n.
I like to think it means one calculus is the shadow of another; we're solving problems by looking only at the shadows they cast :p
Very pleasant regional accent you have. Is is a variant of Welsh ? My wife is from Swansea.
Just a generic Yorkshire accent I suppose :p you can hear it come through a little stronger on "definitely" at 13:17 haha
@@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 !
this is probably one of my favorite math videos on youtube, well done
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
At 13:25 you write phi(e^-x)=0, but that's not true, because phi^-1(0)=0!=e^-x
In actuality phi(e^-x)=0^x, with 0^0=1
9:30 How do we know phi is linear? (Also, it's more like function application and not multiplication really.)
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.
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
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
5:22 Where did the minus disappear?
The whole expression looks a bit different because the sum is going all the way up to x rather than just x-1
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?
This video changed my (math) life. I can't think of anything else anymore.Thanks
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.
I like this video a lot
Hey thanks! I installed a de-esser for the next video, hopefully that'll do it :)
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.
Last time I watched this I was very confused, is thinking of these things as linear operators on a vector space of functions valid?
Wow, that is pne of THE BEST videos I've seen! I am impressed! This is magic in real life!
Wow, i definitly want more of this
Wow, this is fascinating! I never learned much about discrete calculus before, but you've definitely whetted my appetite! Great job!
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.
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 :)
Mostly Wikipedia haha, I'm afraid I'm in the same boat!
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' :)
fantastic video
The best videos use the worst microphones. I really liked your video though! :)
This is a gem!
I don't know how many times I have to see this video to understand it 😆
I'm open to suggestions if you'd like anything explaining in more detail :)
@@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 am getting it but its moving so fast we need more deeper videos on this
Awesome video 😍
Does it mean that umbral calculus is something more suited to be used for algebraic geometry? Or is it already?