Dope Tech Episode 2

<<@filakyle3663 says : Thouse Jesus Christ adds are making me sick. I know its irrelevant to this video. But thats actually the punchline.>> <<@dermasmid says : Thank you brilliant>> <<@sufthegoat says : Wow>> <<@surrealistidealist says : My cat jumped onto my lap and fell asleep, so I had to stop working through my Calc book. Good thing I had this video to watch! Btw, any favorite books on DiffEQ?>> <<@GarthWhoever says : Your brain is really awesome btw>> <<@natanprzybylko7227 says : Saw this video a couple weeks ago before I learned about logistic growth, now I'm coming back while I'm learning it lol>> <<@PmoOgFrNc says : This is the first video ive seen of Zach Star or Zach Star Himself where he is serious and the whole time ive been at edge waiting for him to turn something sarcastic and its just not happening fuck fuck fuck this must be worse than gooning>> <<@stokedfool says : Logistics growth explained with visuals. Great example involving the effects of hunting on deer populations.>> <<@hammerbg5816 says : i cant stop thinking god is going to call him any second now.>> <<@dcterr1 says : Very good, educational video! It seems I've always placed too much importance on solving differential equations - perhaps because they're like puzzles in a way and fun to solve. But now I see that it's not inherently necessary to do so. Instead, what's needed is a good intuitive grasp of what the solutions look like. In any case, if you do want precise solutions, all you need to do is to run a computer simulation using numerical methods, lile Runge-Kutta.>> <<@lazartomic5800 says : I was worried that Zach only filmed comedy.>> <<@futuresmkt says : Great description!!!!>> <<@mattias2576 says : One of the most fun courses i had in University was non linear dynamics which was essentially doing analysis of differential equations which are non linear (hard to solve usually) in a qualitative way. There is so much information you can infer about solutions without ever solving anything This video reminded me a lot about how we approached things there>> <<@Dr.Rajkumz says : 7:30 wow I was mind blown. Never thought of it like that.>> <<@SetTheorist9 says : Those hunters man, hm...when will they ever learn. Had they taken and stayed at least for the intro of Differential equations on population growth and predator prey models, they would've learnt a thing or too.>> <<@drmsanford says : You’re just a quick step from phase lines and bifurcation diagrams. I’d throw in that the solutions don’t cross because of the theorem of existence and uniqueness.>> <<@km4168 says : It would have been nice if you had spent another minute or two at 7:30 to expand this to Lyapunov stability. This intuition for a stable equilibrium is exactly the second condition for a lyapunov function. Nonlinear control (stability) relies on transforming (finding a lyapunov) your system (differential equation) into such a form that you can conclude stabillity similarly to what you show in your video.>> <<@pendragon7600 says : I mean you do learn this stuff in any intro ODEs course>> <<@lethargogpeterson4083 says : Love the shirt.>> <<@owdeezstrauz says : Your sense of humor has drifted far from my ability to understand. If this is the new comedy I just won't understand what the world has become.>> <<@wrathofainz says : 6:35 Israel _loves_ this equation.>> <<@fegolem says : the sky is pickles>> <<@sonnyward9857 says : Learned Logistic Growth today in pre-calc. After I took the test, and got a 98%, I wanted to relax and watch some YouTube. The first video I see is this. What are the odds?>> <<@RFmath_ says : Great video! Could you do one on hyperbolastic growth?>> <<@ersetzbar. says : Can you quantum tunnel the 0 line?>> <<@jormoria says : Lotka-Volterra?>> <<@Stacee-jx1yz says : 1) Calculus Foundations: Contradictory: Newtonian Fluxional Calculus dx/dt = lim(Δx/Δt) as Δt->0 This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale. Non-Contradictory: Leibnizian Infinitesimal Calculus dx = ɛ, where ɛ is an infinitesimal dx/dt = ɛ/dt Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities. 2) Foundations of Mathematics Contradictory Paradoxes: - Russell's Paradox, Burali-Forti Paradox - Banach-Tarski "Pea Paradox" - Other Set-Theoretic Pathologies Non-Contradictory Possibilities: Algebraic Homotopy ∞-Toposes a ≃ b ⇐⇒ ∃n, Path[a,b] in ∞Grpd(n) U: ∞Töpoi → ∞Grpds (univalent universes) Reconceiving mathematical foundations as homotopy toposes structured by identifications in ∞-groupoids could resolve contradictions in an intrinsically coherent theory of "motive-like" objects/relations. 3) The Unification of Physics Contradictory Barriers: - Clash between quantum/relativistic geometric premises - Infinities and non-renormalizability issues - Lack of quantum theory of gravity and spacetime microphysics Non-Contradictory Possibilities: Algebraic Quantum Gravity Rμν = k [ Tμν - (1/2)gμνT ] (monadic-valued sources) Tμν = Σab Γab,μν (relational algebras) Γab,μν = f(ma, ra, qa, ...) (catalytic charged mnds) Treating gravity/spacetime as collective phenomena emerging from catalytic combinatorial charge relation algebras Γab,μν between pluralistic relativistic monadic elements could unite QM/QFT/GR description. 4) Formal Limitations and Undecidability Contradictory Results: - Halting Problem for Turing Machines - Gödel's Incompleteness Theorems - Chaitin's Computational Irreducibility Non-Contradictory Possibilities: Infinitary Realizability Logics |A> = Pi0 |ti> (truth of A by realizability over infinitesimal paths) ∀A, |A>∨|¬A> ∈ Lölc (constructively locally omniscient completeness) Representing computability/provability over infinitary realizability monads rather than recursive arithmetic metatheories could circumvent diagonalization paradoxes. 5) Computational Complexity Contradictory: Halting Problem for Turing Machines There is no general algorithm to decide if an arbitrary program will halt or run forever on a given input. This leads to the unsolvable Turing degree at the heart of computational complexity theory. Non-Contradictory: Infinitary Lambda Calculus λx.t ≝ {x→a | a ∈ monadic realizability domain of t} Representing computations via the interaction of infinitesimal monads and non-standard realizers allows non-Church/Turing computational models avoiding the halting problem paradox.>> <<@xdzzz0 says : 6:35 - * Bill Gates drooling at the mouth *>> <<@jamesjohn2537 says : Thanks, for clarity. Visual interpretation of big idea logistics ODE telling us! Cheers and happy Easter>> <<@supernovaxs9480 says : It’s so weird seeing Zach go back into serious STEM mode from his second channel>> <<@_abdul says : I love to see people saying "Oh...he's the sketch guy" on math videos like this, There used to be comments like "Oh... he's that Math guy" when he started the himself channel. We've come a long way.>> <<@Ishugod1212 says : Bro how tf does George Washington know so much math>> <<@devilsadvocate3364 says : Honestly, if you could add in humorous examples, i feel like that would be both easy to engage with, as well as help with my small-brain understand a bit better 😅>> <<@vigyanumtube9154 says : I just realized that he makes sketchs too>> <<@ShinDMitsuki says : I was waiting for the punchline and then remembered I originally subbed to you because you are an engineer who made math videos 😂>> <<@attila0323 says : A reminder that this is the same guy who makes video series about Washington and Lincoln time traveling, and other funny sketches.>> <<@passengerplanetearth says : Nearly tricked me into learning something, nice try dude!>> <<@o_s-24 says : Calc TWO?? We only did improper integrals and power series in calc 2>> <<@ekandrot says : I remember a few years ago on the news, everyone was explaining virus spread using exponential growth rather than logistic growth. At that time and still, I believe logistic is what should have been used and your video is a great explanation as to why and what the correct curves should have been.>> <<@happyeevee4955 says : u should explain the lotka–volterra equations next>> <<@kennethhicks2113 says : Your FUNNIEST vidy yet!>> <<@void2258 says : I never knew you did videos like this. I had only seen the comedy stuff.>> <<@Meatloaf_TV says : I have heard calc 2 is pretty hard and im taking it ts summer for 6 weeks im starting to get nervous>> <<@no-bk4zx says : The timing is impeccable. I was just taught population modelling this semester but never really understood exactly what the differential equation meant. I knew how to solve it and how to model the curves but not what it meant. Thanks a lot this was genuinely insightful.>> <<@eguineldo says : I understand why they might not talk about logistic growth in calc 2 in depth as there's a couple other important topics that are better suited for an ODE course.>> <<@northernlight1000 says : Great video Zach! and yes often times in math, the higher level you learn certain fundamental topics a lot of the time people really stop looking for actual explicit solutions to things for the most part and focus much more on trying to find all the possible qualitative information they can about that certain problem>> <<@tobybartels8426 says : This kind of thing is often covered in an introductory Differential Equations course, and you might even get the higher-order version of this (where a non-linear autonomous equation or system of equations usually can't be solved exactly, but you can still analyse its qualitative features). But we could certainly put this in Calculus 2, with maybe one extra day on the subject.>> <<@JYF921 says : wait, this is NOT a comedy channel? XD>> <<@JxH says : BEWARE !! In the real world, it should (sometimes) be possible to cross the equilibrium lines. For example, if the logistics equation includes an equilibrium line at 4.5, but (for example) human babies arrive in integer units. So the curve might presently be at 4, and then this example could 'tunnel' straight over to 5, leaping over the equilibrium line at 4.5. Imagine if the 'litter' was a larger number, such as six puppies or piglets. It could overshoot the line. What happens then depends on the rest of it. ...Sorry, I always see the exceptions. It's in my nature.>> <<@isojapo4219 says : bald peanuts>>