I spent most of the 25th and some of the 26th thinking time thinking about natural latents and symmetries. On the 26th I mostly got nothing done, as I was tired and grumpy—it was actually quite unusual, I’m not sure why I was in a funk for at least half the day. Most likely candidate was that I had been not eating well the previous days, on account of underestimating the appetite suppression effect of current medication that I’ve been now taking regularly.
I don’t have good memories of what I did on the 24th. Looking at my comment history, it looks like that’s where I got started on the symmetries thing, which is cool. Looking at my browser history, it looks like I was fishing for examples of basic representation theory, so I was staring at the discrete fourier transform and the more representation theory fourier stuff (like on a finite group). The takeaway for me was mainly that the Hadamard transform that I encountered years ago is in fact what you get when you use (Z/2Z)^n. You (well, at least, I did so, it’s not explicitly listed on the Wiki page but I think I’m right) can interpret this as the subspaces of functions on the vertices of the hypercube, corresponding to the way it is affected by bit flips. I also learned that the Walsh functions (the analogue of sine and cosine in this setting) can be interpreted as taking in a truncated bitstring of a number in the interval [0,1], and then they’ll actually be a complete basis (as long as you allow limits, that is, a schauder basis).
The rest of this, then, is from today (the 27th). Another thing I did today was put my recent and old code on github. If you want to look at monstrosities written by my teenage self, you can now do so. Yes, I used to use three spaces to indent. Yes, I put a hyphen in a directory name. Yes, I put spaces in filenames. Oh, and good luck with the sparse comments.
0: I was thinking about how you could get fractals as unique fixed points. The nicest way is with an iterated function system of contraction mappings that gives the similarities. The iteration map as a function from subsets to subsets will be a contraction mapping in the Hausdorff distance (exercise for the reader—it’s not that hard!), and the Hausdorff distance is complete on the nonempty compact subsets. So you can get unique fixed points on nonempty compact subsets, and iteration converges to it under the Hausdorff distance. e.g. you don’t need to use a triangle to get the Sierpinski triangle.
Also, the way I think about Hausdorff distance now: the distance is ⇐ c when c upper bounds the values of d(x,Y) and d(X,y). Alternatively, X is in the c ball (union of points ⇐ c away from a point in the set) of Y and Y is in the c ball of X.
1: Suppose we have a possibly nonlinear map T: V → W where V is an inner product space, such that <Tx, Ty> = <x, y> for all x,y. That is, it’s ‘unitary but nonlinear’. If T is surjective, then we can actually show that it’s linear: since <T(x+z), Ty> = <x+z, y> = <x,y> + <z,y> = <Tx, Ty> + <Tz, Ty> = <Tx + Tz, Ty> so <T(x+z) - (Tx + Tz), Ty> = 0. If T is surjective then we can make Ty equal the term in other the argument—but then the positive definiteness of the inner product means that T is linear.
If V was merely a normed space, then the Mazur-Ulam theorem says that a surjective isometry must be affine over the reals—or in other words if 0 goes to 0 then it must be linear as a map over the reals. However, if we have complex vector spaces then we might not be linear.
1: The representations used in quantum physics are really projective representations, since quantum states are invariant under scaling. One can typically lift these to representations of the universal cover for lie groups, because you can look at the lie algebra, then deprojectivize, then exponentiate at the cost of losing any global “twist” information.
2: Apparently the Wigner classification of the positive energy finite mass unitary irreps of the Poincare group includes the possibility of massless particles with ‘continuous spin’?? And nobody’s observed fundamental particles like them and only recently have we suggested possible theories for them? What the hell??
I use ManicTime (paid) to track my computer usage and I additionally keep a 7 days log of screenshots at 1 minute interval, which makes it very easy to see what I was doing a couple days ago.
April 24-28
I spent most of the 25th and some of the 26th thinking time thinking about natural latents and symmetries. On the 26th I mostly got nothing done, as I was tired and grumpy—it was actually quite unusual, I’m not sure why I was in a funk for at least half the day. Most likely candidate was that I had been not eating well the previous days, on account of underestimating the appetite suppression effect of current medication that I’ve been now taking regularly.
I don’t have good memories of what I did on the 24th. Looking at my comment history, it looks like that’s where I got started on the symmetries thing, which is cool. Looking at my browser history, it looks like I was fishing for examples of basic representation theory, so I was staring at the discrete fourier transform and the more representation theory fourier stuff (like on a finite group). The takeaway for me was mainly that the Hadamard transform that I encountered years ago is in fact what you get when you use (Z/2Z)^n. You (well, at least, I did so, it’s not explicitly listed on the Wiki page but I think I’m right) can interpret this as the subspaces of functions on the vertices of the hypercube, corresponding to the way it is affected by bit flips. I also learned that the Walsh functions (the analogue of sine and cosine in this setting) can be interpreted as taking in a truncated bitstring of a number in the interval [0,1], and then they’ll actually be a complete basis (as long as you allow limits, that is, a schauder basis).
The rest of this, then, is from today (the 27th). Another thing I did today was put my recent and old code on github. If you want to look at monstrosities written by my teenage self, you can now do so. Yes, I used to use three spaces to indent. Yes, I put a hyphen in a directory name. Yes, I put spaces in filenames. Oh, and good luck with the sparse comments.
0: I was thinking about how you could get fractals as unique fixed points. The nicest way is with an iterated function system of contraction mappings that gives the similarities. The iteration map as a function from subsets to subsets will be a contraction mapping in the Hausdorff distance (exercise for the reader—it’s not that hard!), and the Hausdorff distance is complete on the nonempty compact subsets. So you can get unique fixed points on nonempty compact subsets, and iteration converges to it under the Hausdorff distance. e.g. you don’t need to use a triangle to get the Sierpinski triangle.
Also, the way I think about Hausdorff distance now: the distance is ⇐ c when c upper bounds the values of d(x,Y) and d(X,y). Alternatively, X is in the c ball (union of points ⇐ c away from a point in the set) of Y and Y is in the c ball of X.
1: Suppose we have a possibly nonlinear map T: V → W where V is an inner product space, such that <Tx, Ty> = <x, y> for all x,y. That is, it’s ‘unitary but nonlinear’. If T is surjective, then we can actually show that it’s linear: since <T(x+z), Ty> = <x+z, y> = <x,y> + <z,y> = <Tx, Ty> + <Tz, Ty> = <Tx + Tz, Ty> so <T(x+z) - (Tx + Tz), Ty> = 0. If T is surjective then we can make Ty equal the term in other the argument—but then the positive definiteness of the inner product means that T is linear.
If V was merely a normed space, then the Mazur-Ulam theorem says that a surjective isometry must be affine over the reals—or in other words if 0 goes to 0 then it must be linear as a map over the reals. However, if we have complex vector spaces then we might not be linear.
1: The representations used in quantum physics are really projective representations, since quantum states are invariant under scaling. One can typically lift these to representations of the universal cover for lie groups, because you can look at the lie algebra, then deprojectivize, then exponentiate at the cost of losing any global “twist” information.
2: Apparently the Wigner classification of the positive energy finite mass unitary irreps of the Poincare group includes the possibility of massless particles with ‘continuous spin’?? And nobody’s observed fundamental particles like them and only recently have we suggested possible theories for them? What the hell??
I use ManicTime (paid) to track my computer usage and I additionally keep a 7 days log of screenshots at 1 minute interval, which makes it very easy to see what I was doing a couple days ago.
Screenshot