The Abstract World of Operational Calculus
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- čas přidán 31. 05. 2024
- An introduction to the core concepts of operational calculus (requires some differential equations and Taylor series).
↓ Info and Timestamps ↓
Part 2 in my Abstract Calculus series!
Operational calculus has largely fallen into obscurity in favour of more modern techniques, but that far from makes it obsolete - it provides a framework for us to explore the amazing algebraic structure that makes calculus tick, without the need for any of the more advanced abstract algebra we’d otherways have to use to get here.
Huge thanks to TC159 for helping me structure the second half of this lesson! Expect some excellent videos on these topics on his channel: / @tc159
Discord Server: / discord
Patreon: / supware
0:00 Intro
1:02 Arithmetic
4:37 Differential Equations
5:46 Unit Shifts
8:56 Exponential Shifts
12:15 A Cliffhanger
13:05 Outro + Announcement
Music:
Chris Haugen - Northern Lights
artificial.music - Grandma’s Riding a Polar Bear Again
#some2 #operator #calculus
Corrections:
4:40 I should be calling this the derivative operator (of which differential operators are generalisations) - Hry
You managed in fourteen minutes to render about a quarter of my college math courses redundant. Subbed.
Based channel, based comment. Liked.
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.
Yeah... as if what you previously learned has nothing to do with understanding this video...
Jaw dropping
I started this video assuming this comment was hyperbolic... it was not
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!
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
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.
Yeah! It's also amaging
in quantum mechanics.
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!
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
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.
@@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
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!
I am now upset that they didnt teach us operational calculus upfront when i was learning quantum mechanics. Wtf, this clicked immediately
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
Really glad you were able to experience the video this way! :) this is pretty much what I went through while writing it
I've completely lost it when he divided by 1-D and expanded as a power series xD
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
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
Hey, I love your videos
@@ILSCDF thank you
Still make it. I'm still don't understand 100% of this video even after watching the umbral video n this one
@@juanaz1860 did you end your calculus 1?
@@alang.2054 I did college Calc 1,2,3, diff eq, linear algebra
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.
I understood the thumbnail just by reading it, yet I had never thought about it before. Just beautiful.
Keep it up man, you are making great videos.
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.
Why did computers render this topic redundant, and is there is a reason why it could make a comeback?
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.
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.
this channel is underrated
I do need to make another video eventually haha, but thank you!
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!
I'm looking forward to your next videos! These topics are so interesting
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.
I am really looking forward to seeing more of this series. These first two videos are great.
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?
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.
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.
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?
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
Soooo awesome! Simple and elegant, yet such non-trivial results!
This is the coolest math I have seen in a long time. Love it, thank you!!
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.
Theres "Guy Drinks Soda and then Turns Distorted Meme but it's an ADOFAI Custom Level" and theres this:
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.
these ideas are so beautifully explained
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
Thank you, this was sublime.
Duuude, great stuff, keep it coming
Umbral Calculus and Operation Calculus are a marvel in the math world
You make such great videos !
Thanks for the shoutout, great video!
Amazing video!! Thank you so much!
Great video! What an interesting way to think about things!
Thank you!
super cool, can't wait for the next one
great video, super fun but insightful.
My mind exploded seeing how Binet's Formula was so easily derived just by treating the translations in the recurrence relation as linear operators.
The moment you got phi to just pop out of nowhere I literally screamed! "No fucking way! Holy Shit!!!"
i read about functionals, which map functions to a number. is it right to say that operators and transforms map functions to other functions?
Good lecture video.
I've just found your channel and have subscribed.
As a physics and electrical engineering student this absolutely jaw dropping!
Never could I ever imagine that subtracting a number from a letter would get me a triangle
Looking forward to more videos like this one.
REALLY COOL STUFF!
Quality in form and content: some world-class video. My compliments and looking forward to the next video 🙂
Just superb
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?
Thank you so much!! 🙏🏻🙏🏻🙏🏻🙏🏻🙏🏻
Amazing! Thank you.
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?
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 :)
darn knowing abstract algebra seems very useful for stuff like this
man, absolutely amazing
Great stuff, thanks
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.
Not number theory per se but 3b1b has a couple videos (e.g. 'Euler's formula with introductory group theory' ) about these ideas :)
I was amazed by the fact that, it seems just so simple now the way you can solve for nth fib number
all of these sound real arcane. you mathematicians are real life wizards
Well the previous video on this channel was on _Umbral_ Calculus, which seems to have been named such because it looked like witchcraft.
How did you learn about all those techniques? Is there some book or vid series which you could recommend?
Mainly just scouring sites like Wolfram and Wikipedia for keywords to search and rinse and repeat so far. I'm increasingly on the lookout for better resources; I think I'll be using Gian Carlo Rota's 'Finite Operator Calculus' for future videos in this series
this is what Grant had in mind when started the #some
This is certainly becoming a passion :p and I probably wouldn't have gotten started without the nudge from Grant
Hi, I was wondering if you could recommend any resources or books to learn more about it? It seems a rather obscure topic
The two books Rota - 'Finite Operator Calculus' and Roman - 'The Umbral Calculus', and Tom Copeland's blog 'Shadows of Simpilicity' :) and of course our Discord server!
Subbed immediately.
Great pacing
10:53 How does that last part work? Where does the eˣ come from? I get lost here every time I watch this.
y = (1-D)^-1(0) ⇔ (1-D)y = 0 ⇔ y = Dy
Nice video
"Despite the lack of rigour..." As a physicist, this makes me comfortable xD
This is way too cool
wait how did the last part of solving the differential equation come like the so called complementary function ? at 10:53
y = (1-D)^-1(0) ⇔ (1-D)y = 0 ⇔ y = Dy
@@Supware ok thanks that cleared things up for me !
We need more supware
Thanks you so much :)
fire. i wish they taught us this in odes!!!! i hate analysis & love operator algebras
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)?
There are people smarter than me in the Discord server who can answer questions like this effectively :p
Great video as the other one. What books do you recommend to an engineer in order to study this field?
I'm not much help here I'm afraid :< I've been struggling to find anything at all myself; I've heard good things about Rota's and Roman's books but they look fairly advanced and pure
Wonderful
Amazing
Wow, there's also a new section of corrections in youtube. wowwww!!
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.
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
@@Supware I think abstract calculus probably refers to calculus in arbitrary topological spaces, generalizing to the maximum.
I bet it's a part of functional analysis and operator algebras
Is it possible to write f(1-x) with a linear operator?
Sure! Define N as the linear operator that maps f(x) to f(-x), then what you're looking for is N(T^-1)
(Which you can even write as N/T since they commute!)
@@Supware But something like f(x+1)*f(x) wouldnt be linear anymore?
@@srather I don't think so
I wish I were thought solving DEs like this
These are some novel concepts that I've not seen before, interesting stuff
no idea why I didn't give this a heart earlier :D
Very hard to articulate how good this video is
I need more, function iteration pls
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
This video is very very coooolllll.....
The goat 🐐
WOW! Intuitive!
So glad to hear!
0:53 i think this one's gonna be fun..
Me (all along): it definitely is.
Mind blown
you can do 1/(c + operator)? is that always legal?
You can, but you have to be wary of commutativity. E.g., dividing by (c + operator) on the left is usually not the same as dividing on the right
If (c + operator) commutes with everything then you're free to use 1/ unambiguously
No it is not always legal. You have to ensure that -c does not lie in the spectrum of the operator.
@@strikeemblem2886 spectrum of the eigenvalues?
@@GeoffryGifari No, spectrum of the operator. This is a set that contains more than just eigenvalues, e.g. continuous spectrum, residual spectrum...
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
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
Agreed!
For linear operators that'd be something you might see in a linear algebra course :)
When you derived a formula for the Fibonacci numbers, I immediately recognised Binet’s formula, who was the one to discover it after Euler. Now I can’t stop thinking if they also used this way of deriving the formula, or if they used different tools. If somebody knows, can they please tell me?
Interesting question! I don't know, but this is a pretty natural way to come up with the formula and Binet was active around the time this stuff was a thing
Reminds me of the use of annihilators to solve inhomogeneous linear ODEs
Sounds like I have more googling to do...
Great!
Sorry, but to me, stuff just happens without reason at some point. How do we expect that we could use mysterious "super position" finding two different conditions separated by 'or' involved in final step of tackling the differential equation? Then the idea spread somehow further reaching the equation of the recurrence in Fibonacci numbers. I felt left without a clue despite the "Super position" actually making *some* sense and I don't see why it could be so obvious that it was completely omitted while taking next steps. It may be so that I'm a little bitter now because it was left without explanation oftentimes in every course i took at my uni pertaining to alike problem-solving techniques. But I'd really like to know how to answer these questions
I don't think this is obvious at all, it's only omitted because it's included in the diffeqs prereqs.
Essentially we've found every expression we could plug into f to make the equation hold, so the entire general solution will be a linear combination of all those.
But yeah I don't remember this being adequately explained in any of my school/uni classes/textbooks either...? Bit strange lol, maybe it is meant to be obvious 😅 I think the best way to convince yourself is to just verify some general solutions by plugging them back into their equation
@@Supware omg right ok well now i get it thanks
@@Supware An assurance that it is the general solution is that since it's a second-order differential equation, the solution space is two-dimensional, so if you have two independent solutions, then you have the most general solution. No idea how to prove the dimensionality of the solution space though.
@@flyingpenandpaper6119 love this idea!
It's informational and inspirational, even better than 3B1B
The highest of compliments, thank you!
Oooh more stuff from umbral calculus
0:17 ha, no I - cards for me, no links in description either :)
So cool