Posted by gmays 7 hours ago
Is it something "revolutionary" or just another small brick that will pile up until something really "revolutionary" will happen?
By itself, no consequence. But over time, provided we keep pumping out talented and qualified mathematicians and keep subsidizing costs, we could maybe hit a breakthrough... somewhere... that has real impact.
If you searched for coefficients from -12 to 12, this would be 25^360 = 2 * 10^503 different possibilities. A common reference point is that there are 10^80 atoms in the observable universe. Sure you could probably reduce this a bit with clever tricks, but the starting point makes the method completely unviable, even with the knowledge: A) a counterexample exists, B) it's in 3 variables, C) it's in degree 7 or less, D) it's in integer coefficients, E) those coefficients are 12 or lower.
2) You are starting at 10^500 possibilities. "Much" smaller is not enough, the order of magnitude of the order of magnitude needs to be changed.
3) You still need all of the other assumptions, which were completely unfounded
Impossible.
The coefficients are mostly 1, so biasing toward that would make it much faster.
I learned from poking around that checking the invertibility of a system in C is a much, much harder problem than I thought. Nonetheless if that were no object, let’s say coefficients from -16 to 15 (5 bits) times eight terms times choosing up to cubes (64) times three equations is searchable, especially since you have only the final combination of coefficients in the determinant. It’s not impossible to generate the equations like this Fizzbuzz style.
Edit: no. 2048 possible monomials, to the 24th power, not times 24. Fine, can’t brute force it.
It's not quite the Reimann hypothesis, but many prominent mathematicians have spent years working on this problem. Yitang Zhang wrote his PhD thesis on it.
F1 = x^3y^3z + 3x^2y^4 + 3x^2y^2z + 7xy^3 + 3xyz + 4y^2 + z
F2 = 3x^3y^2z + 9x^2y^3 + 6x^2yz + 12xy^2 + 3xz + y
F3 = -x^3z - 3x^2y + 2x
That’s the counterexample. Low integer coefficients, power 7 in three variables. If someone said it was there, couldn’t we all have written a pretty simple brute force solution for the search space, especially with the constraints that the symbolic determinant had to cancel to a constant?
Maybe they'll find a solution where P=NP.
That could really throw a wrench into the whole internet thing.
It seems they need an expert human driver for now.
In this case, it takes me 12 seconds to see content when first opening the link, and about 18 to re-render content when scrolling.
Yes—for a continuous-time autonomous system
x ˙ =f(x),f(x ∗ )=0,
this is the standard linearization criterion, with J=Df(x ∗ ):
If every eigenvalue of J has strictly negative real part, then x ∗ is locally exponentially asymptotically stable. If at least one eigenvalue has strictly positive real part, then x ∗ is unstable. If no eigenvalue has positive real part but at least one has real part 0, linearization is generally inconclusive. Nonlinear terms or a center-manifold analysis are needed.
The last case really can go either way. For example, all three scalar equations below have Jacobian J=0 at x=0:
x ˙ =−x 3 , x ˙ =x 3 , x ˙ =0.
Yet 0 is respectively asymptotically stable, unstable, and neutrally stable.
A slightly more precise wording is therefore:
If the spectral abscissa
α(J)= λ∈σ(J) max
Reλ
is negative, the equilibrium is locally exponentially stable. If α(J)>0, it is unstable. If α(J)=0, the Jacobian test is inconclusive.
This criterion concerns the Jacobian matrix of a dynamical system at an equilibrium; it is unrelated to the “constant Jacobian determinant” condition in the Jacobian conjecture.