The Abstract World of Operational Calculus

[@akdn7660]

You managed in fourteen minutes to render about a quarter of my college math courses redundant. Subbed.

[@vnever9078]

Based channel, based comment. Liked.

[@MDNQ-ud1ty]

If you learned category theory you would realize life is redundant. If you learned about "capitalism"(economics, psychology, finance, evolution, and politics) you would realize why college is redundant.

[@TecknoVicking]

Yeah... as if what you previously learned has nothing to do with understanding this video...

[@alexweschler9470]

If that’s true, congratulations, you played yourself. Most ppl I know who went to college and had to take math courses ended up doing pretty well for themselves lol

[@ILSCDF]

Jaw dropping

[@brendanmiralles3415]

I started this video assuming this comment was hyperbolic... it was not

[@tomctutor]

The simple shift Theorems themselves are very useful, you can even apply these ideas to integration: D^(-1) ≡ ∫ D^(-n) ≡ ∭..nX or multivariate calculus; ⅅ_t f(x,t)= ∂f/∂t ⇒ e^(Tⅅ_t) f(x,t) = f(x, t+T) which is very useful when using periodic functions like trig. the list is endless and because we are dealing with linear operators, we are familiar with e^(DA) I = 1+A, where A^2=[0] the nul matrix. Yes totally agree a very powerful analytical technique if you deploy operational methods! Thankyou for a very well presented video, I appreciate the amount of work you put into making this, Mathologer would be proud!

[@logo2462]

Wow! This really cleared up why we can solve recurrence relations with “auxiliary polynomials”. My finite math course just had us plug and chug to solve these!

[@Supware]

This is something I really wanted to get right in particular :) I was wondering why auxilary polynomials work for differential equations, since I was similarly taught about them without explanation

[@pantoffelkrieger8418]

Another great way to derive these "auxiliary polynomials" is by looking at the generating function of the series. If you haven't heard of that, you should check it out; it's pretty cool.

[@Supware]

@@pantoffelkrieger8418 what that guy said :p if you're interested in this stuff and somehow haven't come across generating functions yet there are plenty of excellent videos on them here on yt

[@lexinwonderland5741]

I can't WAIT for the rest of this series! Both of your videos were extremely eye-opening even to a long-time maths student like me, and gave me that wonder of when I was first discovering a new field. Please please please keep it up, great job!

[@pandavroomvroom]

this channel is underrated

[@Supware]

I do need to make another video eventually haha, but thank you!

[@GeoffryGifari]

as a physicist, i imagine the shift operator working similarly to "ωt" expression in a wave ψ(x,t)=exp(ikx-ωt), so now we have a pattern that moves

[@eliavrad2845]

Yeah, quantum mechanics is mainly operational calculus (plus wave mechanics, probability, regular linear algebra...). The most famous exponentiated operator is the formal solution to Schrodinger equation exp(Ht/iℏ)|Ψ(0) ⟩ = |Ψ(t) ⟩ i.e. the time translation equation for the physical state |Ψ(0) ⟩ with propagator U= exp(Ht/iℏ) to the physical state at time t |Ψ(t) ⟩ . H, the Hamiltonian or energy, is at least a second order differentiation operator H=-(1/2m) ∂^2/∂x^2+U(x), with the kinetic energy -(1/2m) ∂^2/∂x^2 and the potential energy U(x) which is just a regular function. Especially in physics context, a lot of time the differential operator is shorted to ∂, rather than D, so expect a lot of ∂^2, ∂_x, ∂_t.

[@hyeonsseungsseungi]

Yeah! It's also amaging in quantum mechanics.

[@mehulborad2400]

U know when you mentioned the factorisation of linear diff eq i paused the video and then tried to prove everyting rigorously and it was very beautiful how linearity can be exploited and i actually had then thought of the solutions to recursive relationals as well. At this point i was amazed and in awe at how abstractness is not only beautiful but very useful and guess what u go ahead and take the inverse of 1-d and the Fiinng geometric series to find the solution of a very famous diff eq in one step 🤣🤣🤯🤯🤯. I HAVE NO WORDS i am still jumping around like a mad man at how CRAZY this is. This has gotta be one of if not the most beautiful thing i know . Never expected differentiantion to work like this, it was always very tricky to find solutions, yet somehow magically hidden from me all this time it was secretly behaving like a real variable and polynomial. INSANE JUST INSANE

[@Supware]

Really glad you were able to experience the video this way! :) this is pretty much what I went through while writing it

[@MessedUpSystem]

I've completely lost it when he divided by 1-D and expanded as a power series xD

[@EpsilonDeltaMain]

Wow I was going to make a video on this topic eventually, and you did it so much better than what I would have done!! Congrats

[@ILSCDF]

Hey, I love your videos

[@EpsilonDeltaMain]

@@ILSCDF thank you

[@juanaz1860]

Still make it. I'm still don't understand 100% of this video even after watching the umbral video n this one

[@alang.2054]

@@juanaz1860 did you end your calculus 1?

[@juanaz1860]

@@alang.2054 I did college Calc 1,2,3, diff eq, linear algebra

[@hwendt]

Keep it up man, you are making great videos.

[@dmytrolevin738]

I was familiar with operational way to solve ODEs, but it have never come to my mind that this idea can be extended this far. This is amazing! Looking forward to the next video.

[@gustavoexel5569]

Almost all of these ideas we learn separately in college for example, within its own applications. What I found watching this video is that operational calculus makes these ideas so much closer, and interrelated among themselves, without the need for so much arbitration when deriving concepts and ideas. Really enlightening

[@alejrandom6592]

I understood the thumbnail just by reading it, yet I had never thought about it before. Just beautiful.

[@Orionhart]

The hard work you put in to these videos shows. I hope more folks see this video, and maybe some drop you some Patreon! Proud to be a patron.

[@KakoriGames]

Umbral Calculus didn't interest me that much, but Operational Calculus intrigued me that I went back and watched both videos. And boy, I don't regret doing that, awesome videos, can't wait for more.

[@netcat22]

I'm looking forward to your next videos! These topics are so interesting

[@ianrobinson8518]

This topic was first treated in great depth as far back as the mid-1800s. The types of general results that came out it are fascinating but all but forgotten. It is actually a sub-topic of became known as the calculus of finite differences. It was used a lot in empirical research areas and professions such as actuarial studies. With the advent of computers, the topic fell by the way side. Old treatises can still be found online and Schaum had an edition covering it thoroughly.

[@4grammaton]

Why did computers render this topic redundant, and is there is a reason why it could make a comeback?

[@ianrobinson8518]

The methods were used to provide numerical solutions to otherwise intractable big data problems in insurance and other professional fields. The old methods required simplifying assumptions, slide rules and log tables. Desktop calculators and mainframe computers went some of the way to easing the burden, but it was the advent of the modem desktop computer with almost unlimited computing power and ubiquitous tools such as spreadsheets which allowed us to dispense with approximations. I’ve no doubt that the finite calculus is used at a rudimentary level in some fields of work and research. However the subject matter was developed to a great depth with magical formulae and approaches somewhat akin to infinitesimal calculus’s. This is what has been “forgotten” and no longer taught.

[@geraltofrivia9424]

Great video. Please do some more. Their quality is just amazing.

[@Supware]

Thanks so much! I will 100% post another video *one day* (I have started on it already), though I've just about run out of strong ideas for this calculus series for now (and I've been busy with other projects and irl stuff the last year or so). The video I'm working on next is about rational points on conics and elliptics, if that's of any interest :)

[@pacotaco1246]

I am now upset that they didnt teach us operational calculus upfront when i was learning quantum mechanics. Wtf, this clicked immediately

[@yamansanghavi]

Wow, this was so good. Thanks a lot. A lot of things are something we know from quantum mechanics or differential equations but seeing them under one roof is absolutely amazing.

[@Sk8aterBoy132]

My mind exploded seeing how Binet's Formula was so easily derived just by treating the translations in the recurrence relation as linear operators.

[@hawkeyeplank]

Just finished the first class I have taken on abstract algebra and really enoyed it. It is amazing how analogous umbral/operational calculus are to some of the things we learned about this semester! Specifically seeing the conjugate pop out in the previous video, and seeing how operational calculus preserves calculus theorems while behaving like abstract algebra is so cool. I want to look into some of this more! Can't wait for the next video, and I am pissed at the youtube algorithm for not recommending your channel sooner!

[@diana-pestana]

Soooo awesome! Simple and elegant, yet such non-trivial results!

[@DrJaneLuciferian]

I am really looking forward to seeing more of this series. These first two videos are great.

[@denki2558]

Fascinating. I used the thumbnail formula to derive the forward difference formula in just a few lines. With some rearranging, the backwards and central difference formula can be derived as well. It amazed me to see that the central difference formula has some connections to arcsinh. Our numerical methods prof didn't show derivations. I'm glad to learn that I could derive them on my own now.

[@inventorbrothers7053]

Just superb

[@oblivion5683]

The moment you got phi to just pop out of nowhere I literally screamed! "No fucking way! Holy Shit!!!"

[@defenestrated23]

11:38 - mind=BLOWN. This reminds me of dual numbers and how exp(a+bê) acts like a scale & translation, which means translation is like rotation around a point at infinity. It also kind of implies ê (epsilon, ê^2=0) IS the differential operator. You should also do a vid on dual quaternions!

[@andy_lamax]

Umbral Calculus and Operation Calculus are a marvel in the math world

[@rjfhpolito]

Thanks for the shoutout, great video!

[@toizh_x]

Theres "Guy Drinks Soda and then Turns Distorted Meme but it's an ADOFAI Custom Level" and theres this:

[@Bruno-el1jl]

This is completely insane! Amaaaazing video The shift in mental model for the e^(a+bi) to the D case was mind blowing Curious: where dos this fail? And why?

[@__-cx6lg]

bruh i started cracking up laughing when you expanded (1-D)^-1 as a geometric series 😆 And it actually works!! And then you did that thing with e^D.... I am flabbergasted This video is great

[@LukePalmer]

This is the coolest math I have seen in a long time. Love it, thank you!!

[@AshleyCog]

Using the first principles of differentiation you can right D in terms of T, h, and the "limit as h approaches 0" operator, D=L_(h -> 0)h^-1(T^h-1). Rearranging and replacing T with e^D, you can get a formula for this limit operator, L_(h -> 0) = hD(e^(hD)-1)^-1. Let h = 1 and replace e^D-1 with Delta to get L_(1 -> 0)=D(Delta)^-1 so the Bernoulli operator is the same as taking the limit as 1 approaches 0. The inverse of the forward difference is the sum so L_(1 -> 0)=D*Sigma is a cleaner form. This operator converts discrete problems into continuous. If you want to calculate the sum you can instead take the integral of the limit as 1->0 of the function. of if you want the forward difference you can instead take the limit as 1->0 of the derivative.

[@SapphFire]

Great video, both of them! Just as a small bit of feedback, it would help if the earlier steps were kept on screen so it'd be easier to follow (5:27 for example). You already did do that for most of them which is nice. I appreciate the slight pauses between the explanations. It gives time to think through what you said and makes it more relaxing to watch. Edit: To the other commenters, please don't increase the pressure to make videos. I know it's all meant well, but it can get overwhelming when so many people are expecting something from you.

[@Supware]

Really appreciate that, thank you! :) What (else) would you have preferred I keep on screen? I tried to keep stuff onscreen while relevant, except for in cases where it was meant as a refresher on assumed prior knowledge or where I could truncate it (like the mini textbox thing at 7:10) As for the edit, yes I probably would prefer less pressure, but it's lovely to know that so many people want to see more of these :p more WILL come *eventually* (just probably not calculus-based stuff for a while though, if only because I've pretty much said everything I wanted to about it in these two videos)

[@yongmrchen]

Another underrated math channel

[@TheActurialRepository]

Thank you, this was sublime.

[@00000ghcbs]

Duuude, great stuff, keep it coming

[@GeoffryGifari]

darn knowing abstract algebra seems very useful for stuff like this

[@Duskull666]

As a physics and electrical engineering student this absolutely jaw dropping!

[@braden4141]

7:05 in the video couldn't f(x) also be a multiplied with a periodic function with period 1 and still be a solution to the equation.

[@logicprojects]

Great video! What an interesting way to think about things!

[@Supware]

Thank you!

[@hemat8129]

As a first-year electrical engineering student, the D operation was a mystery for me. Thanks for making this mystery more mysterious.

[@Supware]

The D thing is just shorthand for d/dx haha, anything you want clarifying?

[@matiasbpg]

Great video! I always found interesting how these concepts are made rigorous and expanded in functional analysis and operator theory. Also extensively used in quantum physics

[@bennyloodts5497]

REALLY COOL STUFF! Quality in form and content: some world-class video. My compliments and looking forward to the next video 🙂

[@calvingakunju7580]

these ideas are so beautifully explained

[@cthoyt]

super cool, can't wait for the next one

[@galladeguy123]

This is super interesting! Where can I learn more about this? Are there any good textbooks on operational calculus and umbral calculus?

[@Supware]

I've been recommending Steve Roman's and Gian-Carlo Rota's books for general umbral stuff, but to learn more about specifically the things I talk about in these vids I'd urge you to join us in the Discord server

[@pacificll8762]

You make such great videos !

[@OdedSpectralDrori]

great video, super fun but insightful.

[@IIAOPSW]

this is simultaneously mind blowing yet unsurprising. like, at least in terms of the way I learned fourier analysis, it makes the most sense when you think about functions as infinite dimensional vectors where each adjacent entry is just [0, dx, 2dx, 3dx....]. So the meaning of sin and cos being orthogonal is the same as it would be if your vectors were ordinary finite dimensional. Showing that you can construct a delta function proves completeness of a basis for the same reason being able to construct the vectors with a single 1 in them is sufficient to span the space. In this context, obviously differentiation is just a matrix with 1 on the diagonal and -1 on the band one off from the off diagonal. All of which is to say, its not entirely crazy to see that calculus is linear algebra as dimensions approach infinity and linear algebra is calculus approximated with finite dimensions. But it is crazy to see how the results in one domain map so cleanly, even mechanically, to the other. Almost as if the infinitesimal is just an arbitrary choice of a real number precision no better or worse than 1, and everything can be done in discrete land if we want.

[@fedorkochemasov4533]

Never could I ever imagine that subtracting a number from a letter would get me a triangle

[@gabrieletrovato3939]

Thank you so much!! 🙏🏻🙏🏻🙏🏻🙏🏻🙏🏻

[@zuzaaa1998]

These ideas are also applied to partial differential equations where you can solve equations by using formal sums of laplacian operator. I remember that these ideas were really fascinating form me during my PDE classes but I haven't seen much of it since then. Do you have any books recommendations on the operational calculus?

[@Supware]

Not yet I’m afraid, but I think I’ll need to find some books before I continue this series! I’ve been recommending Rota - Finite Operator Calculus and Roman - The Umbral Calculus but those are more umbral than operational

[@symbolspangaea]

Amazing video!! Thank you so much!

[@scottmiller2591]

Looking forward to more videos like this one.

[@GeoffryGifari]

all of these sound real arcane. you mathematicians are real life wizards

[@angeldude101]

Well the previous video on this channel was on _Umbral_ Calculus, which seems to have been named such because it looked like witchcraft.

[@plucas2003]

omg where are the rest of the series??

[@Supware]

I'll make more when I have enough ideas 😅 it's a little frustrating but I really don't wanna make a third video that doesn't match the quality of the first two

[@DepozidoX]

Looking at 11:50, these can serve as transformations between the addition and multiplication worlds. I think that such transformations could be really useful to solve some hard number theory problems.

[@Supware]

Not number theory per se but 3b1b has a couple videos (e.g. 'Euler's formula with introductory group theory' ) about these ideas :)

[@yash1152]

5:47 > _"it's about time we introduce a new linear operator: the unit shift"_ i guess that's where my existing knowledge with operator calculus ends in this video. (except that some knowledge that i have is not covered here so far, maybe further in video) 8:57 > _"where right side ain't just zero"_ yeah, i guess this will cover the remaining part of my knowledge *Edit:* no! the aim/answer is same, but the method here is doing it from scratch

[@ZeDlinG67]

this is what Grant had in mind when started the #some

[@Supware]

This is certainly becoming a passion :p and I probably wouldn't have gotten started without the nudge from Grant

[@Henriiyy]

"Despite the lack of rigour..." As a physicist, this makes me comfortable xD

[@Mikey-mike]

Good lecture video. I've just found your channel and have subscribed.

[@alejrandom6592]

I was amazed by the fact that, it seems just so simple now the way you can solve for nth fib number

[@citycrafter578]

man, absolutely amazing

[@dj_laundry_list]

l am so insanely mad that I wasn't taught calculus, or at least DiffEq this way. Learning the algebra of any kind of operators (or mathematical objects in general) should be considered essential

[@Supware]

Agreed!

[@jinjunliu2401]

For linear operators that'd be something you might see in a linear algebra course :)

[@Fru1tyy]

These are some novel concepts that I've not seen before, interesting stuff

[@Supware]

no idea why I didn't give this a heart earlier :D

[@GeoffryGifari]

i read about functionals, which map functions to a number. is it right to say that operators and transforms map functions to other functions?

[@PavanKumar-xv1hg]

wait how did the last part of solving the differential equation come like the so called complementary function ? at 10:53

[@Supware]

y = (1-D)^-1(0) ⇔ (1-D)y = 0 ⇔ y = Dy

[@PavanKumar-xv1hg]

@@Supware ok thanks that cleared things up for me !

[@wargreymon2024]

It's informational and inspirational, even better than 3B1B

[@Supware]

The highest of compliments, thank you!

[@GustavoOliveira-gp6nr]

This is way too cool

[@jens6076]

Amazing! Thank you.

[@Applied_Theory]

Great stuff, thanks

[@realcirno1750]

Great pacing

[@crueI]

Subbed immediately.

[@asthmen]

These are really fun topics! One question about your DE example, (D + 3)(D + 2) f = 0. Is it not possible for (D+2) f to land in the kernel of (D+3) without f itself being in that kernel? Obviously (D+2) f = 0 means f is in Ker(D+2), so... let g = (D+2) f. Then (D+3) g = 0 implies g \in Ker(D+3), so g = c exp(-3x). Then (D+2) f = g = c exp(-3x) means that f = c (D+2)^-1 exp(-3x) + h, h \in Ker(D+2) is there something in the commutativity properties of (D+2) and (D+3) that says that (D+2)^-1 g has to stay in Ker(D+3)?

[@Supware]

There are people smarter than me in the Discord server who can answer questions like this effectively :p

[@Thejosiphas]

fire. i wish they taught us this in odes!!!! i hate analysis & love operator algebras

[@michaelriberdy475]

We need more supware

[@nihilumaeternum6555]

Incredible stuff. When's the rest of the series coming up??

[@Supware]

Eventually lol, when I have more ideas for topics (and understand them well enough myself)...! I don't want to post another video until I have something matching the quality of the first two

[@nihilumaeternum6555]

@@Supware Can't wait for the material about iterated functions that you alluded to in this video

[@yoavboaz1078]

5:29 is it right to use the word "or" here? For the general solution neither of these 2 equations are satisfied. so you can't really say you need one or the other to be true since sometimes neither are true

[@Supware]

You're right, it's a bit clumsy and technically incorrect; just a concise way to show going from the auxiliary equation to the general solution

[@Fox0fNight]

@@Supware It reminded me somewhat of zero divisors, though I don't have enough knowledge to see how this could be useful

[@srather]

Is it possible to write f(1-x) with a linear operator?

[@Supware]

Sure! Define N as the linear operator that maps f(x) to f(-x), then what you're looking for is N(T^-1)

[@Supware]

(Which you can even write as N/T since they commute!)

[@srather]

@@Supware But something like f(x+1)*f(x) wouldnt be linear anymore?

[@Supware]

@@srather I don't think so

[@solarfridge]

At 10:44 when you solve the y-y'=x^3 differental equation by generating the series expansion for (1-D)^-1=(1+D+D^2...) and then apply these to x^3 and get the solution, then what happens when we use it for something like e^x where no matter how much we derivate it stays the same: (1+D+D^2+...)*e^x=(e^x+e^x.....)=n*e^x (where n->inf), implying that y-y'=e^x does not exsist, but it does. Is there an answer tho why does this method fail when we use functions outside of polynomials (or any functions that eventually reach 0 when derivated enough times), or I did something wrong and it actually works with e^x?

[@danieldias3192]

I don't understand how the complementary function added at the end of the geometric series expansion solution works. How does (1 - D)^-1 * 0 equal ce^x? Where can I find more info on this?

[@Supware]

y = (1-D)^-1(0) ⇔ (1-D)y = 0 ⇔ y = Dy I'm afraid I haven't found any info for this kinda thing yet; I'll post about resources both in the comments and on the Discord server as I come across them :)

[@瑠ちゃん]

Where can I learn more about this stuff-umbral calculus, the shift operator, etc? It's all so cool and interesting I'm amazed I was never taught any of this before! It looks like it has some really cool applications as well. It doesn't have to be books, videos, anything is okay. Telling me what the subject is called would go a long way! Is operational calculus part of abstract calculus or are they separate things? The same with umbral calculus, is that part of abstract calculus? Where did you learn this stuff? I also always annoyed at people factoring differential equations but being completely unable to explain why that is okay.

[@Supware]

It seems operational and umbral calculus are just different names for different approaches to this stuff. 'Functional calculus' is another keyword, and I've been recommending Roman's and Rota's books on the subject. Most of my personal "research" so far has just been translating Wikipedia I'm afraid lol "Abstract Calculus" doesn't mean anything canonically as far as I know, it's just the name I gave to this series

[@angelmendez-rivera351]

@@Supware I think abstract calculus probably refers to calculus in arbitrary topological spaces, generalizing to the maximum.

[@mastershooter64]

I bet it's a part of functional analysis and operator algebras

[@yash1152]

Wow, there's also a new section of corrections in youtube. wowwww!!

[@michaelriberdy475]

Wonderful

[@starkissed5795]

The goat 🐐

[@minimath5882]

Amazing

[@yash1152]

0:17 ha, no I - cards for me, no links in description either :)

[@vnever9078]

THIS IS HOW YOU MAKE A MATH VIDEO.......

[@mrtfttkhv]

I wish I were thought solving DEs like this

[@brunojani7968]

Shame you stopped, great videos

[@Supware]

Thank you! I will be back eventually, though I'm pretty much out of good ideas for this calculus series in particular for now (next video idea is conics/elliptics) and other stuff has been distracting me this last year or so

[@bentationfunkiloglio]

Mind blown

[@jkid1134]

Very hard to articulate how good this video is

[@firefox7857]

10:53 How does that last part work? Where does the eˣ come from? I get lost here every time I watch this.

[@Supware]

y = (1-D)^-1(0) ⇔ (1-D)y = 0 ⇔ y = Dy

[@yoavboaz1078]

Hi I've got one more question. In 10:55 what do you mean by the complementary function? Anyway i just wanted to tell you how great this video is. Ive been watching it all day pausing all the time to play with matrices since I've wanted to see what i can do with what youre showing. Do you plan on making any more videos?

[@Supware]

Complementary function is just the name of the part of a general solution that's not the particular integral. Nice, glad you liked it :) what've you been doing with matrices? I do plan on making more eventually, still just waiting for the right inspiration to strike though (don't wanna make another video that's not up to the quality of the first two)!

[@yoavboaz1078]

@@Supware you can associate polynomials with column vectors (ie. 3x^2+7x+2 is the vector [2, 7, 3, 0, 0, 0...]) and linear operators like the derivative with matrices. Using this video i was able to derive the matrices for Δ and T and look at the relations between them. For example T is indeed e^D and D is (T^h-1)/h as h approaches 0

[@Supware]

Ohh yeah, nice nice! phi and phi inverse themselves can be rendered as matrices quite easily, as in edit1 here: math.stackexchange.com/questions/4530940/trying-to-characterise-an-umbral-shift I made a phi calculator for polynomials in Python using this approach :)

[@engelsteinberg593]

Recomended lecture?

[@accountname1047]

Nice video

[@TylerPerlman]

Reminds me of the use of annihilators to solve inhomogeneous linear ODEs

[@Supware]

Sounds like I have more googling to do...