Seems AI have trouble with math proofs atm, but if they can predict chaotic behavior better than people can it's still a significant step. The same issue occurs in medicine where doctors aren't given information by AI as to how it recognizes cancer and heart disease sooner than they can.
But I'm looking precisely for a broad answer to cover bases to research. Does a direct product of measurable Euclidean spaces to a Hilbert-like space preserve measurability? Maybe, maybe not. If one has only countably many dimensions, seems like there is a good chance. I imagine in uncountable...
I said "along" a subset of such a space, which is pretty conventional. What else are you going to integrate if not a function over that space?
The domain issue you pointed it is the entire point of the post. If there are countably many points missing from any finite dimensional subspace that is...
That's something I've actually found to be true. Even now, all we have are statistical measurements, so to a certain degree, you can't be entirely certain about anything. That inability to be entirely certain is how scientific models appear to perpetually evolve.
You can make an argument that...
Why are people making this weird assumption that "math" says anything? People are the ones making arguments. No matter what you write it will only ever be an approximation of relative observations. The only reason we take an interest in it is for practical applications in being able to predict...
The way I was taught to solve many quasi-linear PDEs was by harnessing the initial condition in the characteristic method at ##u(x,0) = f(x)##. What if however I need use alternative initial conditions such as ##u(x,y=c) = f(x)## for some constant ##c##? Can the solution be propagated the same way?
I couldn't help but notice the lack of an explicit formula for the equations. Did an AI actually solve them or just approximate the solution implicitly on its own?
I'm thinking of a more open interpretation where at every point in the domain space, there is an open ball in the inverse image of the codomain that contains that limit point.
Suppose we have an infinite dimensional Hilbert-like space but that is incomplete, such as if a subspace isomorphic to ##\mathbb{R}## had countably many discontinuities and we extended it to an isomorphism of ##\mathbb{R}^{\infty}##. Is there a measure of integrating along any closed subset of...
Actually there was one property I forgot to check for linearity ##G[a f(x)] = a f + \frac{1}{c}a f(x^c) = a G[f]## where ##a## is the constant function ##h(x) = a##.
With group-ness established, how can we then identify the corresponding Lie algebra and Lie derivative?
Allegedly Frechet derivatives are used in optimization problems in mechanics, but I have not found a clear example of this. Does anyone know of an example to go through? I would think because of the significance of Lagrangian mechanics that it could be more related to a variational calculus...
Thanks, that makes more sense. We know at least one function maps to the identity function of most function spaces, being the 0 function. Analytic functions unfortunately do not form a normable space in general, so to analyze this, perhaps what needs to be considered is ##C^{1}[a,b]## where...
Thank you for clarifying. For the simple case of ##f(x) = x## I can see what you're saying that it might not be helpful. What is interesting to consider however is how such a structure might be preserved in other functions or across a function space. For instance, what if one had ##f(x) =...