zmgehlke
> ... In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. ...
I find images of Jupiter absolutely stunning, and that the dynamics behind the auroras is sophisticated (ie, something complex happening above the core) is even more interesting.
I can't wait for Juno to bring us even more results!
Every 1-knot has a sort of 2-knot analog (actually, a whole family of them) made by "revolving" the knot, but there are 2-knots not generated in this manner. (The same holds true for higher dimensional knots.)
As we go higher, there are knot moves not even possible in lower dimensions (such as twists).
Just a thought.
Can someone explain what a DNC is?
Assume I know most CS undergrad topics -- or can at least google them; eg, I know what a Turing machine is but not what a neural Turing machine is.
Books or papers; let's say at the level of math heavy CS/undergrad math degree.
Yes, if you put little R^4 balls around points of the line, you can still untangle it. It's basically still a line. (They might have to be really little balls, but the definitions all use neighborhoods for defining the legality of knot moves, so you can't "shrink" the knot out of existence.)
My actual thought was slightly more complicated, in that particles dont need to necessarily not be a point all the time, since a sphere you're slicing with a plane can appear as a point -- see sibling comment for what inspired the idea.
If we think about a particle going from A to B, over t from 0 to 1, then the common interpretation is that it takes all paths and that those paths can interfere with eachother.
If we think about t=0.5, then the "particle" is not in a definite place, but sort of "smeared" out across its possible paths. The thought was if those possible paths, together, could form an embedded sphere in 3+1D, and in some sense could have similar computational knotting behavior. (And moreso that form shells of equal probability in a solid in 3+1+1D, with the extra dimension coming from the probability associated with each point.)
Im not sure how you'd "read" that, though possibly by "weakly" interacting with it in flight or it might contribute to the "random" outcomes of measured values. Since knotting is low-energy in general, we might not notice that behavior under normal conditions because of thermal interference adding a lot of computation in flight, hence making the values seem random. Similarly, heavily controlled experiments likely don't permit enough freedom to be anything but the trivial knot.
I expect that there is a reason that doesn't work, but figuring out where my hunch fails will help learn more about QM.
This might sound dumb, but how does a point have a wavelength?
(I actually think QM is a conceptual mess of hacked together math, but that's a rant for another day. Here I am being sincere, because maybe (probably, almost certainly) I just don't understand what you mean.)
> I would think that solid spheres
Sorry, I think I was unclear.
I meant sphere in the topological sense of just the "shell" part (the surface), as opposed to a ball, which contains the interior. Think bubble.
You can vizualize a 2-knot fairly easily: tie a knot in a piece of string, hold each end, and spin it. The "sphere" you get by identifying the start and end of a cycle is knotted.
I believe the question there is if non-Abelian braiding statistics can be found to enable universal quantum computation, as Abelian anyons don't enable a univeral quantum computer. (Ideally a 12/5 FQHE, I believe.)
But my reading of the MS paper suggested that they were working with traditional anyons in 2+1D and the question of 3+1D analogs was unresolved. Here Wilzcek seems to suggest that there are no 3+1D analogs.
My question was if the treatment of objects as points, hence having a worldline rather than worldsurface or worldsolid wasn't the cause of that -- you can't knot a line in 4D, but you can knot a surface.
So my question was if the point object model was accurate or a simplification we need to move beyond.
If we think of particles as little spheres (or loops), it should be possible to knot them -- you can knot objects 2 dimensions less than your space. (Hence, there are 1-knots in 3D and 2-knots in 4D, where an N-knot is a N-sphere embedded.)
Is the question of 3+1D "anyons" actually settled, or has no one performed the analysis on the wave equation(s) looking for 2-knots? (My reading of MS papers implied the second, but I'm hardly an expert.)
Ed: As an aside, Im super happy Wilzcek is covering this. I've always found his writing to be rather accessible.