The Shadowy World of Umbral Calculus

[@nbspWhitespaceJS]

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

[@Supware]

Yep! This is a special case of the stuff I talk about at 12:00ish in the video (a=1) :)

[@krhino42]

“By rearranging the question, we get the answer.” Imma use that

[@Qreator06]

All of math in a nutshell

[@al.7744]

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

[@elkinmontoya9640]

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

[@marcoottina654]

maybe, about prime numbers ...

[@lexinwonderland5741]

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!!

[@Supware]

Wow haha thank you! Guess I've no choice but to keep it up :)

[@pauselab5569]

it's already there but not much on it. like 3 line paragraph style

[@fabiant.2485]

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.

[@jayst]

I could feel my Glimmer rise after watching this

[@cerulity32k]

dark and EVIL calculus 😨

[@魏寅生]

I've been struggling with abstract algebra and your video presents a perfect example for why isomorphisms are useful! Really appreciate it!

[@Supware]

Great to hear!!

[@vladimirmakarov9849]

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!

[@corbinsmith5015]

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

[@TheRationalPi]

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.

[@Supware]

Interesting, I'd love to see more practical applications of this thing

[@OdedSpectralDrori]

brilliant direction. time to see how these transforms would help analyze some filter and the fourier transform

[@Supware]

@@OdedSpectralDrori don’t quote me on this but it seems the Laplace Transform is VERY relevant here :p

[@OdedSpectralDrori]

@@Supware this will make a fine quote

[@Supware]

@@OdedSpectralDrori what have I done

[@johnchessant3012]

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.

[@Supware]

They’ll be appearing in the follow-up!

[@GalHorowitz]

At 13:40, why is there a phi before D^n? isn't (D^n f(0)) just a constant?

[@discreet_boson]

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

[@funnydog7817]

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.

[@Tom-u8q]

Just in case you didn't know, it's "pique" in the phrase to "pique one's interest"

[@lookupverazhou8599]

@@Tom-u8q Like Piqueachu.

[@MultiAblee]

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.

[@mcdudelydoo3116]

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!

[@rosettaroberts8053]

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.

[@Supware]

Oh nice! I'd love to know if it has a name or symbol, I somehow haven't come across either yet

[@diogeneslaertius3365]

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.

[@miklosbognar1449]

this is probably one of my favorite math videos on youtube, well done

[@therealjordiano]

The part at 7:35 when you introduced the main idea I near enough leapt out of my chair and yelled 'holy shit!' from the bottom of my lungs (not rly but still), amazing stuff and very nice explanation

[@MirMarksman]

Rad! The familiar-but-different feeling makes this feel almost like a math dream

[@Supware]

After working for a while on a second video I think this might actually be a major vibe I wanna aim for haha

[@alpers.2123]

Is this similarities related to group theorem?

[@Supware]

@@alpers.2123 phi is a homomorphism between discrete and classical calc

[@LebronJ0]

What a legend only one ad in the beginning . Your so damn underrated

[@SouthTC2]

your channel is a Godsend, you're welcoming me into the abstract and lovely world past what most people learn in their lives, you are so helpful, please don't stop making videos on forms of mathematics not usually talked about! You are gonna blow up bro, keep it up!

[@Supware]

Thanks so much man! I have been a little distracted by other projects and IRL stuff, but I do already have another video in the works that I'll get back to soon enough (conics and elliptics) :)

[@SouthTC2]

@@Supware Can't wait! I've been pretty busy so I can't wait to finish this video soon, same with your other one, I have both in my watch later playlist, so far I really like what you have, please continue making God blessed content :D, ppl need more math in our lives

[@harelrubin1432]

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

[@Supware]

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 :)

[@vnever9078]

Thank you Supware for introducing me to this beautiful world of Umbral Calculus!

[@proxagonal5954]

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!!

[@Supware]

Thanks! Illustrations are certainly gonna be an interesting challenge in the next one...

[@Briekout]

@@Supware what are you demonstrating with sir? MathCad ?

[@Supware]

@@Briekout Manim

[@alpers.2123]

It is calculus for engineers lol

[@proxagonal5954]

@@alpers.2123 Don't laugh at engineers man. They cool

[@dcterr1]

Wow, this is fascinating! I never learned much about discrete calculus before, but you've definitely whetted my appetite! Great job!

[@firefox7857]

3:01 I've never seen that explanation for the fundamental theorem of calculus... it seems so simple now.

[@angelmendez-rivera351]

Well, unfortunately, the equation shown on screen is not actually what the fundamental theorem of calculus is or says.

[@SirTravelMuffin]

Really solid pacing, and definitely leaves me with a lot of curiosity for the subject!

[@mbdtsmo]

Great video, crystal clear. Had never heard of umbra calculus before, but recently read about umbral/monstrous moonshine, so I was curious. Thanks

[@jmcsquared18]

That was one of the most entertaining things I've ever watched. Bravo, subscribed.

[@Supware]

Wow, thank you!

[@Fysiker]

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.

[@kkski4817]

I like this video a lot

[@Supware]

Hey thanks! I installed a de-esser for the next video, hopefully that'll do it :)

[@riccardofiori828]

This is one of the best math videos i've ever seen! Ideas are presented so neatly and however so mind-blowing. You earn a subscriber, I'm hoping for more videos like this!

[@NoNTr1v1aL]

Absolutely amazing video! Subscribed.

[@EC-pf1tr]

This was awesome. I feel like I finally have a window into why I leaned everything that I did and how everything is connected from combinatorics to sums and differences and the discrete, to the continuous, to linear algebra, to complex numbers

[@whitestonejazz]

this video was a trip. crazy to think this was never mentioned in any calculus classes. Very cool, thanks!

[@madeline-capi]

Great video about a topic I didn't really know anything about. Surprised of seeing stirling numbers too, when I learned about them they were shed by a completely different light and these kind of connections are what make math so interesting. Looking forward for a follow-up! And I hope this video gets blessed by the algorithm just like other SOME2 videos have been. Thanks and have a great day.

[@qazxwecvr]

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)

[@Supware]

More coming! I got some bad bois in the follow-up whose objects aren't even labelled ;)

[@willcollings5681]

Just want to say thank you. I'm working on translations between the Laplace transform and the Z transform (it's discrete counterpart) and this definitely sheds some extra light on the topic

[@striga314]

This is so cooool. Which books or other texts could you recommend about this topic?

[@orirrr8]

Extremely clear, insightful and interesting exposition!

[@netcat22]

This was beautiful, I had never heard of this branch of calculus before but I'm absolutely in love with it. Your sub count honestly shocked me, this was such high quality.

[@ficakcblack1090]

This video is so amazing, it blew my mind, please continue making these

[@kiiometric]

Oh my god my brain is tickling, this is beautiful

[@hotterfrenchfry]

I don't usually comment on videos but such quality from a channel with 254 subscribers amazes me. This video's topic is really well chosen as it is understandable and complex and your explanations make it a great time !

[@Supware]

Thanks, means a lot!

[@luiz00estilo]

Just leaving a comment to help with the algorithm. This video was _enlightening_ . Great work man!

[@arongil]

Agreed, the algorithm needs to know this video is top quality! Leaving this comment for the same reason :)

[@georgecote]

awesome quality! I'll be very happy to see more of that :) good luck!!

[@MCLooyverse]

Absolutely wonderful. Something about conjugation (q * x * q^(-1)) makes me happy every time it comes up (it comes up a lot).

[@utof]

13:23 this is my favorite part. Makes me so excited to learn and explore this branch of math!

[@Supware]

The bottom one amounts to a logarithmic spiral being the "ϕ of a circle", and as far as I know we're still figuring out the top one (for which the equation is incorrect but still interesting :p)! Both courtesy of the guys in the Discord

[@utof]

@@Supware woahhh awesome thank youu!!!

[@tricanico]

Wow, that was so cool. Thanks for posting this.

[@joshuaiosevich3727]

Man, this stuff is really my wheelhouse, I messed around with umbral calculus a lot in high school and early in college. Here’s some fun stuff I noticed. First thing I found was a formula for the n-th difference of a sequence, and then a formula to recover a sequence from those differences. What was striking, is that the “sum” of a sequence depended on the same difference coefficients as the original sequence. Meaning that if a sequences differences terminate, or obey a discernible pattern, computing a closed form for its summation is trivial. This lead me to a “better” version of fahlbauers formula that also works for any polynomial. Another fun thing I noticed is that this umbral calculus lets you count permutations that fix different numbers of elements. Take the group Sn, and let P(k,n) denote the number of permutations in Sn that fix k elements. Summing over all k gives you the size of the group which is n! . Note the P(k,n) can also be written as (n choose k)P(0,n-k), there are (n choose k) to pick the elements you fix, and those that remain, are not fixed at all. At this point in the argument, if you know umbral calculus, you’re all done! You see that P(0,n-k) is the k-th difference of n! ! Once you figure out permutations that fix 0, you get all the other ones too. This all amounts to a curiosity though, not good for much.

[@Aditya_196]

😵‍💫 i am getting it but its moving so fast we need more deeper videos on this

[@jaafars.mahdawi6911]

Simply fabulous! More of these, please, sir!

[@simonflavioibanez7715]

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! 🧮

[@Supware]

What a wonderful thing to read! Thank you so much!

[@kuppersrocky6834]

wow, great video, thanks!!

[@Malk007]

This video changed my (math) life. I can't think of anything else anymore.Thanks

[@Celastrous]

Please keep making these!!

[@KoHaN7]

This was an incredible video. The way in which you merged everything together was mesmerising! I can't wait to se the follow up and some more exceptional quality of work and educational content. Bravo!

[@HaliPuppeh]

This was really interesting. Thank you for explaining things slow enough that I could keep up. So many videos on advanced topics go too fast and I get really lost.

[@timkohl9831]

This is something I've been curious about for a long time, thank you. This is very well made.

[@PamSesheta]

This has reminded me of something from a calculus class near the end. This was amazing, first encounter with phi operator and now I wanna know more because this seems really handy for dsp

[@ILSCDF]

Wow, i definitly want more of this

[@okoyoso]

Out of all of the videos from SOME2, this one was the most eye-opening. Looking forward to SOME3!

[@akio-the-lazzycatto]

Wow, that is pne of THE BEST videos I've seen! I am impressed! This is magic in real life!

[@GelidGanef]

This hit the sweet spot for me in that it's perfectly intuitive that this should work, but how it works and why blows my mind.

[@КириллБезручко-ь6э]

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

[@joda7697]

Umbral Calculus is just the best when you're deep in some special functions, like Bessel, Laguerre, and so on.

[@symbolspangaea]

This is a gem!

[@pr4_kp]

Incredible! I hope you do more on this topic - really got me thinking :)

[@jonipaliares5475]

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!

[@paulkohl9267]

Loved it! Please, by all means, more on this topic.

[@george_miller_1089]

Brilliant calculus video!

[@robharwood3538]

Great video! I had been exposed to Umbral Calculus a wee bit, but not much stuck with me; and I had also learned a little bit about doing finite sums (a la Newton) using falling and rising powers. But the two different parts of my brain didn't make the connection between the two topics until this video. Thanks! This new (for me) connection has already illuminated a lot of things I had been confused/stumped/stuck about before, and I can't wait to put this into practice!

[@djamondaxuzm4712]

Thank you for the intro, really professional and keeps everyone on the same page

[@sejr8053]

Awesome video 😍

[@dyllanusher1379]

Definitely looking forward to another video, thank you for sharing these mind blowing ideas!

[@diegohcsantos]

I really want to see more advanced discrete calculus stuuf!!! Thank you so much for the high quality video!!

[@Supware]

I'm planning some shorter videos on summation by parts and summation using complex numbers :)

[@kodirovsshik]

This was just WOW I always had the idea and the basics of the discrete calculus calmly sit somewhere in the back of my mind with me knowing that is it like "somewhat" related to the canon calculus, but now that i see this video, i realize that OH MAN is it something completely different!! I would surely love to see more content on some more advanced stuff on this topic, top interesting stuff

[@pedroth3]

Great explanation of this concept! Congrats and thank you

[@mitch9224]

Incredible work - this has to be one of my favourite math videos. It really does leave you wanting to learn more about the subject which is such a good feeling. Fabulously presented as well :) My only question is concerning how at the end you mention that the two series are the same up to an isomorphism. What exactly is meant by this?

[@Supware]

Wow, thank you! By "isomorphic" here I just meant that one is a direct analogue of the other: the underlying structure of what's going on is the same, but in one case we're in the differentiation world and in the other we're in the differences world

[@ColeCoug]

Great explanations and animations for a topic that I have never heard of before!! This video makes me want to learn more about it

[@Orionhart]

Totally worth the wait!

[@orktv4673]

The way that forward difference operator is defined strongly reminds me of Dirac's derivation of the gamma matrices.

[@victor1978100]

5:22 Where did the minus disappear?

[@Supware]

The whole expression looks a bit different because the sum is going all the way up to x rather than just x-1

[@randomz5890]

Amazing work here. Grant will definitely take notice to this! I generally love videos explaining more niche areas of maths, so I unfortunately have to be a bit selfish here and ask that you keep making more videos like this because they're amazing!

[@Supware]

Working on it!

[@randomz5890]

@@Supware awesome, I should expect you to be two steps ahead 😄!

[@SnapThority]

The maths animations are really good, I appreciate the effort you put into them!

[@Supware]

Thanks, and sorry for only just noticing this comment haha! The animations were made possible by 3B1B's Python library. Great software but takes some setting up

[@notjerrett]

Definitely looking forward to that followup!

[@theseusswore]

3b1b's manim is an incredible tool for stuff like this. calculus and its practicality go hand in hand, so being able to visualise stuff in calculus is incredibly important to understanding it

[@Supware]

It's such a great tool! Even for simple things like manipulating equations, it makes everything so much faster and easier to follow If I were using paper or a whiteboard instead this video would be 2 or 3 times as long for sure

[@alejrandom6592]

I can't believe you only have 321 subscribers, you deserve like a ton more

[@Supware]

Hah thanks man! Guess I need to make more of these :p

[@tomkerruish2982]

One day later, it's 583 (including me now).

[@minimath5882]

This was great, I look forward to seeing more in the future!

[@michelangelowebb4362]

I hope you continue with this videos. It it amazing work. Thanky you man !

[@gustavoexel5569]

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?

[@atrophysicist]

That's true i dont understand it either!

[@SeanGhaeli]

fantastic video

[@gianpierocea]

Incredibly well done, this is exactly my kind of thing. Thank you :)!

[@eduardoabreu78]

Fantastic, you Sir did a Great Work!

[@TheActurialRepository]

Hi, love your videos. Just wanted to point our the upper limit of the summation in the binomial theorem is "x" and not infinity. 0:42

[@Supware]

Thanks man! I believe 0:42 is correct, but I should definitely have reiterated that I'm talking about the general version

[@meccamiles7816]

Excellent video. Thanks for sharing.

[@pawebielinski4903]

Excellent video. I'd love to see more on the topic!

[@danieljulian4676]

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.