Statements like these always brings me to memory the opening line of Hamming's Numerical Methods book: The purpose of computing is insight, not numbers. It is very easy to get carried away and forget that - in particular today when processing power grows exponentially. Even more when we know there are a myriad of problems that are uncomputable, literally, and human common sense and intuition (insight) are as relevant now as ever.
Indeed you can use symmetry, but it feels more like a mathematical hack, and the fact that it agrees with reality could be a coincidence. You can state that, and there is a lot of evidence for, that nature follows some basic geometrical rules. Applying that through a Lie theory framework on a symplectic manifold to see how charges behave differentially will eventually get you to Maxwell equations because of how those Lie algebras operate. However for me the real revelation was just using the Lienard–Wiechert approach to calculate how charged particles should behave in a relativistic field, which is as simple as it gets, and then see that you can build the full electromagnetic theory on top of that, with the bonus that the formulation is already relativistic. The same resulting symmetry in a corresponding Lie group is consequence of that (nicely captured by Hodge's equation), and invariance or operator rules don't need to be forced.
In the "opposite" direction, you might discover quantum mechanical "spin" from the Maxwell equation. Suggesting that coincidence is a kind of historical artifact :)
Thanks for the postclassical angle on this, I missed that in the comment below, which was only "charge"
Not sure what you mean by Hodge equation, care to elaborate?
I assume (for the lay physicist) it's the Hodge decomposition mentioned in here (pp6-8)
Correct, the famous d*F=J differential form formulation with one of the versions of the Hodge operator, which I have seen named in several ways. Also depending on your definition of the star operator and current density, you often see this as two equations with Hodge duals, like dF=0 plus d*F=*J. The tensor equivalent can be stated as a single equation or as a set, too.
To be fair and looking back at history, the discovery of Maxwell equations, relativity and quantum theory are so intertwined with the discovery, invention and application of new Mathematical ideas, in particular emanating from the work of Hamilton, Grassmann and then Lie, Levi-Civita, Cartan, etc. that is difficult to separate at what extent those concepts influenced over each other in their attempt to explain and describe reality. The ability to express Maxwell equations in a compact form with quaternions before vector calculus was even a thing provides some evidence. One can argue that the classical formulation for electromagnetism could be expressed that way because Hamilton was trying to find the proper framework that could capture his ideas about physics. Fast forward some 60 years and you also have a similar thing happening with Pauli matrices in quantum theory, and the work of Noether in modern physics.
Mixed Hodge structure en.wiki could use better (undergrad physics) examples if it wants to be the quaternion of our time, thank you for that rabbit hole :)
What I always miss from this introductory abridged explanations, and what makes the connection between Lie groups and algebras ('infinitesimal' groups) really useful, is that the exponential process is a universal mechanism, and provides a natural way to find representations and operators (eg Lie commutator, the BCH formula) where the group elements can be transformed through algebraic manipulations and vice-versa. That discovery offers a unified treatment of concepts in number theory, differential geometry, operator theory, quantum theory and beyond.
Which for some of us, Spanish speakers, was on occasion amusing or lewd, depending on the context and culture. The Spanish equivalent is 'tronco' which is very similar and it is slang for a couple of things.
They also mainly continued to be loyal to the Spanish crown after Argentina and Chile went through their independence, and carried out the final pacification of the Mapuche territories in the 19th century. By then only a very small part of the population had not mingled with Europeans.
You would have thought they learned from their mistakes implementing VST2, but they doubled down going even further basing VST3 on the Windows Component Object Model. I guess it was a decision to avoid reinventing the wheel, but you can quickly realize it is a very bad model for real time audio plugins and audio host support. The API just exploded in complexity, and testing was a nightmare. In contrast you can tell the U-He developers have all the experience from the trenches.
> COM is just 3 predefined calls in the virtual table.
COM can be as simple as that implementation side, at least if your platforms vtable ABI matches COM's perfectly, but it also allows far more complicated implementations where every implemented interface queried will allocate a new distinct object, etc.
I.E. even if you know for sure that the object is implemented in c++, and your platforms' vtable ABI matches COM's perfectly, and you know exactly what interfaces the object you have implements, you cannot legally use dynamic_cast, as there is no requirement that one class inherits from both interfaces. The conceptual "COM object" could instead be implemented as one class per interface, each likely containing a pointer to some shared data class.
This is also why you need to do the ref counting with respect to each distinct interface, since while it is legal from an implementation side to just share one ref count for it all, that is in no way required.
Note that VST3 doesn't implement the COM vtable layout, their COM-like FUnknown really is just 3 virtual methods and a bunch of GUIDs. They rely on the platform's C++ ABI not breaking.
You're right that QueryInterface can return a different object, but that doesn't make it significantly more complicated, assuming you're not managing the ref-counts manually.
Today I drove past a new Tesla Model Y, still with a temporary paper tag. As I was passing by I immediately noticed some pretty heavy water condensation inside one of the lateral tail braking lights. I just rolled my eyes.
Ahh... Brings me memories. Back in the 80s I did something similar after reading Dawkin's The Blind Watchmaker and his Biomorph evolution app. I wanted to recreate it but doing something more fun. So I hacked an Atari Logo space invaders game that used genetic programming on the invaders that survived most of the hits based on their shape (some pixels worked as shields) and motion. After 5 levels the game was almost impossible to beat.
To be fair, any multivariable regulator or filter (estimator) that has a quadratic component (LQR/LQE) will naturally yield a solution similar to backpropagation when an iterative algorithm is used to optimize its cost or error function through a differentiable tangent space.
I believe the reason it works in nonlinear cases is that the derivative is “naturally linear” (to calculate the derivative, you are considering ever smaller regions where the cost function is approximately linear - exactly “how nonlinear” the cost function is elsewhere doesn’t play a role).
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