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...

[@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!

[@ILSCDF]

Jaw dropping

[@brendanmiralles3415]

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

[@pandavroomvroom]

this channel is underrated

[@Supware]

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

[@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

[@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.

[@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!

[@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

[@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.

[@hwendt]

Keep it up man, you are making great videos.

[@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.

[@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.

[@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.

[@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.

[@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.

[@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.

[@inventorbrothers7053]

Just superb

[@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

[@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.

[@netcat22]

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

[@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.

[@citycrafter578]

man, absolutely amazing

[@pacotaco1246]

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

[@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?

[@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.

[@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

[@__-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

[@Duskull666]

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

[@TC159]

Thanks for the shoutout, great video!

[@GeoffryGifari]

darn knowing abstract algebra seems very useful for stuff like this

[@oblivion5683]

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

[@LukePalmer]

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

[@TheActurialRepository]

Thank you, this was sublime.

[@gabrieletrovato3939]

Thank you so much!! 🙏🏻🙏🏻🙏🏻🙏🏻🙏🏻

[@scottmiller2591]

Looking forward to more videos like this one.

[@OdedSpectralDrori]

great video, super fun but insightful.

[@logicprojects]

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

[@Supware]

Thank you!

[@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 :)

[@calvingakunju7580]

these ideas are so beautifully explained

[@toizh_x]

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

[@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

[@00000ghcbs]

Duuude, great stuff, keep it coming

[@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

[@cthoyt]

super cool, can't wait for the next one

[@fedorkochemasov4533]

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

[@alejrandom6592]

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

[@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.

[@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

[@bennyloodts5497]

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

[@Thejosiphas]

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

[@michaelriberdy475]

We need more supware

[@pacificll8762]

You make such great videos !

[@symbolspangaea]

Amazing video!! Thank you so much!

[@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

[@yash1152]

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

[@Mikey-mike]

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

[@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.

[@jkid1134]

Very hard to articulate how good this video is

[@wyboo2019]

i used something similar to this to derive the binet formula when i was just trying new things without concern for rigor usually you derive the binet formula using a generating function, but I actually imagined the (naturally-indexed) fibonacci numbers as the components of a vector in an infinite-dimensional vector space (ie f = 1e1+1e2+2e3+3e4+5e5+8e6+...) and then, i kind-of defined into existence a linear transformation that brought every basis vector to the next-indexed one, ie. s(e_i)=e_(i+1), pretended i had an inverse for this even though obviously one doesn't exist for e1, and it led to a polynomial in s applying to the fibonacci vector equaling the RHS, so the next problem was to find the inverse of this polynomial in s i got stuck there, until i realized i could factor the polynomial in s into two monomials and then just apply the inverse to each monomial separately, eventually bringing me to the Binet Formula as well as some very cool identities involving power series of the golden ratio i was unaware of its a very fun thing to work through i highly encourage, because ive never seen anyone else fiddle with a "generating vector" but essentially my approach seems to just be 'operational calculus' but translated to the language of linear algebrs

[@crueI]

Subbed immediately.

[@mrtfttkhv]

I wish I were thought solving DEs like this

[@GustavoOliveira-gp6nr]

This is way too cool

[@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

[@jens6076]

Amazing! Thank you.

[@PeterBarnes2]

Replaced by Laplace transforms!? This theory has the capacity of including the Laplace transform! It's the same as that (D+s)^-1 operator you showed, but written in the form of a definite integral of a dummy variable, rather than as taylor series!

[@wargreymon2024]

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

[@Supware]

The highest of compliments, thank you!

[@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 :)

[@Applied_Theory]

Great stuff, thanks

[@Henriiyy]

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

[@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

[@realcirno1750]

Great pacing

[@vnever9078]

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

[@michaelriberdy475]

Wonderful

[@TylerPerlman]

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

[@Supware]

Sounds like I have more googling to do...

[@ichigo_nyanko]

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

[@Nzargnalphabet]

Oooh more stuff from umbral calculus

[@minimath5882]

Amazing

[@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

[@ΒασίληςΖωγράφος-ρ7λ]

Wow, imagine defining a partial derivative operator that way. Also there is no need to define special notation for Fourier and Laplace transform because we have the D inverse operator. F f = D^-1 f * exp, L f = D^-1 f * exp(iω) or F f = I f*exp, L f = I f * exp(iω)

[@wargreymon2024]

To say it has been replaced by Laplace transform is just sad. It is more powerful and abstract techniques, just need more recognition 👏🏻👍🏻🔥🔥🔥🔥🔥

[@sergey1519]

I have very little idea how this magic works, but τ² = τ + 1(Fibonacci) (τ-1)τ = 1 Δτ = 1 τ = Σ or, from fundamental theorem, sum of F(n) to x = F(x+2) - 1

[@engelsteinberg593]

Recomended lecture?

[@mastershooter64]

This must be related to functional analysis and operator algebras

[@jaafars.mahdawi6911]

0:53 i think this one's gonna be fun.. Me (all along): it definitely is.

[@bentationfunkiloglio]

Mind blown

[@accountname1047]

Nice video

[@starkissed5795]

The goat 🐐

[@rajinfootonchuriquen]

Thanks you so much :)

[@fa-pm5dr]

lol i always had the question "what if i have a differential equation but my derivative depends on the function at a time before..." for example f'(x) = g(f(x-1)) well i never could solve it, and professor sent me to curiosity jail for asking stuff too hard to answer... now i think i may have the tools !

[@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 :)

[@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?

[@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 !

[@Wielorybkek]

operators are so cool :o

[@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?

[@malicksoumare370]

This video is very very coooolllll.....