howling
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Same. I don't even use LLM normally as I found polars' syntax to be very intuitive. I just searched my ChatGPT history and the only times I used it are when I'm dealing with list and struct columns that were not in pandas.
The paper's introduction says:
we first establish algebraic product formulas for the direct computation of the Clifford product inverses of multivectors in Clifford algebras Cl(p, q), n = p + q \le 5, excluding the case of divisors of zero.
Sorry I forgot to answer your question: To rotate a bivector of the form v ^ w, just do R v ^ R w where R is the rotation matrix. This is a linear map so you can extend the operation linearly to arbitrary bivector.
There is no equivalent notion of rotor in EA. I would say its most important use is that you need it to define differential form in which we can define exterior derivative and integeration in arbitrary differentiable manifolds. It also enables to define determinant elegantly.
> In 2D and 3D GA, every non-zero k-vector and versor has an inverse.
Of course by definition every versor has an inverse. The invertibility of k-vector gets hairier for higher dimensions though. Even in 3D GA, some mixed-grade elements are not invertible.
> As to which is more fundamental, I don't think it matters.
It doesn't matter mathematically but it matters pedagogically. GA enthusiasts seem to advocate teaching GA to anyone that has learnt linear algebra. I believe it is more appropriate to stick to teaching tensor algebra and its quotient exterior algebra. Then it is up to you to learn Clifford algebra as a generalization of exterior algebra; especially if you are a game dev, a physicist, or a topological K-theorist.
You have to read the first paragraph as well.
> I have given a lot of reasons why I think GA is problematic: the Geometric Product is a bad operation for most purposes. It really implements operator composition and is not a very fundamental or intuitive thing. Using a Clifford Algebra to implement geometry is an implementation detail, appropriate for some problems but not for general understandings of vector algebra and all of geometry. Giving it first-class status and then bizarrely acting like that is not weird is weird and alienating to people who can see through this trick.
If I understand him correctly, he means Clifford algebra is "appropriate for some problems" but we should "move away" from "giving it first-class status" as it is not more fundamental and often does not help students understand geometry better. I also readily admitted that it has some use cases in game physics in my comment.
> but the biggest giveaway is how he says we should move away from quaternions
I'm sorry I must have missed that part. Can you point me to where did he say this?
First of all it is only invertible for some non-zero elements, especially if `a` is a linear combination of multivectors or we work in PGA that explicitly adds a basis vector of norm 0. Yes sometimes it is useful but that doesn't automatically makes it more fundamental than the inner product and exterior product.
All of these stuff can be done in normal linear algebra. Some (not all) of the operations can be done more efficiently with GA in low dimensions. It is neither more concise nor more intuitive to understand than normal linear algebra.
> Can you formulate a lot of stuff much more efficiently and concisely, and does it help gain new perspective on some things? yes, absolutely. It provides a wonderful framework for expressing geometric ideas.
Can you elaborate on what stuff does it help to formulate much more efficiently and concisely?
> Yes, sometimes you just need the dot product, and sometimes you just need the exterior product. If you are coding, or giving the final form of some formula, you don't have to always put both of them in your code or paper.
In my experience 99% of the time you just want the dot product or the exterior product. Even when you want both it is rare that you want to combine them linearly except in some niche physics/mathematics.
> But neither the dot product nor the wedge product are investable by themselves. Having an investable product on vectors is endlessly useful while you are deriving the formulas.
Do you mean invertible? Why is invertibility is so useful?
An article that argues against GA:
https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...TLDR:
- GA tends to attract a lot of crackpot. In fact most mathematicians avoid the name Geoemtric algebra and call it Clifford algebra to disassociate with them.
- Most of the usefulness of GA comes from just exterior algebra and exterior product/wedge product is more important than geometric product.
- It shows up in spinor field in physics but this does not concern most people that do not work in quantum physics.
My rudimentary view towards GA:
- It is useful in game physics since rotors can represent n-dimensional rotation in 2^{n-1} numbers instead of n^2 numbers as 2^{n-1} < n^2 when n <= 6. You can use PGA if you want to add translation as well. It is also better in interpolation.
- Outside of this you should just probably just learn exterior algebra instead.
I always dislike the unnecessary use of dual of dual. We should define tensor product as a quotient space of all linear combinations of v₁⊗v₂ as in
https://en.wikipedia.org/wiki/Tensor_product#As_a_quotient_s...
The advantage of not using dual of dual is that it generalizes correctly to infinite dimensional vector spaces and modules.
System F just isn't flexible enough to encode most of mathematics. You need at the minimum dependent type and preferably a more flexible identity type to avoid setoid hell. The closest attempts I know of are HOTT in Coq and Cubical Agda. These systems still have a lot of kinks to be sorted out though, e.g.
https://www.youtube.com/watch?v=TpUOweB2kHU
Symmetric interaction combinators (an instance of interaction nets like 2-state 3-symbol Turing machine is a particular type of Turing Machine) only has 3 agents and 3 rewriting rules. This makes it about as simple as lambda calculus with 3 possible terms (variable, application, and abstraction) and α-equivalence, β-reduction, and η-reduction. The advantage of interaction nets is that each rewriting rule is local and "atomic" unlike subsitution step in β-reduction which can take arbitrarily long time as it is defined recursively.
I think interaction nets [0] is actually a simpler model and can simulate Turing Machine efficiently. I wish courses in Computability/Complexity theory would be taught in interaction nets instead of Turing Machines as the program written would be so much nicer and compose easily. It also has real-life uses in compiling functional languages. [1]
[0]: https://en.wikipedia.org/wiki/Interaction_nets
[1]: https://github.com/HigherOrderCO/HVM
None, but I have written code that requires more data structure and algorithm knowledge than inverting a binary tree (at my job).
Yeah it's not clear what he was asked. Swapping the direction of every edge of a binary tree would result in a DAG that is likely no longer a binary tree though.
> Google: 90% of our engineers use the software you wrote (Homebrew), but you can’t invert a binary tree on a whiteboard so fuck off.
— Max Howell (@mxcl)
If inverting a binary tree means swapping the left and right subtrees of every node, I wouldn't want to work with someone who can't do that either and Google is definitely right to reject him.
I have a similar experience at Oxford. However, the introduction course doesn't even cover IO monad and all of our code are run in repl. No wonder the students regard Haskell as impractical as they did not even write a full program in it.
nuclear fusion reaction with net energy gain?
In some sense, classical probability is just the study of category of Markov Kernels [0]. We can gain much of the insight without a lot of measure theory/functional analysis machineries by restricting to its full subcategory of finite states, which gets us to the category of stochastic matrix.
In this view, quantum probability (of finite states) is just about study of category of classical quantum maps described in Picturing Quantum Processes by Bob Coecke and Aleks Kissinger [1]. I highly recommend the book, which requires hardly any prerequisites other than linear algebra and mathematical maturity.
Once we get into this framework, no philosophical interpretation is required. E.g. we can't reproduce a quantum state by classical data just means any morphism from a (non-trivial) purely quantum state to a purely classical state does not have a left inverse.
[0]: https://en.wikipedia.org/wiki/Markov_kernel
[1]: https://www.cambridge.org/core/books/picturing-quantum-proce...
I'm 1800 rapid on chess.com and can't play blindfold beyond the opening.
I assume Gradient Boosting Machine.