More assorted ramblings
Dec. 28th, 2010 03:27 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoyp-norms:
So a while ago I thought to ask on MathUnderflow[0] about the question I asked some here some time ago, namely, how do you show that, in two dimensions, the p-norm and q-norm are not isometric for 1/p+1/q=1 unless p is 1, 2, or ∞? This being the only case I couldn't figure out of showing that the p-norm and q-norm in n dimensions are never isometric unless either p=q, or n=2 and you've got the case of the 1-norm and the ∞-norm.
Well, one answer I got was, maybe look at the curvature of the p-unit circle? And in fact looking at the differentiability of this curve does in fact get you an answer.
But I also got another answer, which gave the *right* way to solve the problem in general, and isn't restricted to finite dimension. Hanner's inequalities. They let you compute the modulus of convexity of l^p, and see that (aside from the cases of 1 and ∞) these are all different. Boom. This still leaves you to manually distinguish 1 from ∞, but that's not really so hard.
Kitchen sinks and local fields:
Let me warn you up front, what follows is a bunch of idle speculation about topics I really don't know, fueled mostly by Wikipedia.
OK. So the surreal numbers are kind of an "infinity kitchen sink" for ordered fields. And the surcomplex numbers for characteristic 0 more generally. And I'm told the nimbers basically form one for characteristic 2. Obvious question: What about other characteristics? This is an obvious question so someone must know about this, right? (I suppose an appropriate direct limit would work, though it would be totally unhelpful. And probably a horrendous abuse of set theory. Of course the original definition of the surreals is the latter, too.) Less obvious question: Is there any possibility of a "sur-p-adic numbers"? Which would be algebraically the same as the surcomplex numbers, I mean, but somehow naturally equipped with some sort of valuation so that...? I would expect not, as ISTM that would kind of conflict with the whole "infinity kitchen sink" requirement, but having thought of it, I have to ask it. (Not going to state what led me to think of it, as I assure you, it's even way more muddled.)
I Wanna Be The Guy:
I started playing this again today. From scratch, of course, not like my old saves were recovered from the hard drive crash anyway. Finding it much easier this time around so far. (Still playing on Medium.) Just today I already managed to beat Kraidgief and Dr. Wily! (Note that I never beat it the first time - never managed to beat Mother Brain.) In particular I remember Kraidgief being *much* harder than he seemed now.
I *still* have no idea why I see it said that the first boss of the game is Mike Tyson, because so far as I can tell, *none* of the paths have Mike Tyson as their first boss. When I played before I beat Mecha-Birdo, Kraidgief, Bowser/Wart/Dr. Wily, Dracula, and I fought Mother Brain, but I never so much as found Mike Tyson. Clearly I'm missing something. (I refuse to look up a map. I tried to draw a map of what I know from memory, but since it's been so long I'm not sure it's accurate. Back when I played it previously I made a big point of mapping things.)
The game still crashes a lot. No idea if that's due to Wine or not. I expect not.
Perhaps due to finding it easier, I actually forgot elementary precaution after beating Dr. Wily and went forward, instead of backward to save. Oops. Took me a while to beat him again.
-Harry
[0]AKA math.stackexchange.
So a while ago I thought to ask on MathUnderflow[0] about the question I asked some here some time ago, namely, how do you show that, in two dimensions, the p-norm and q-norm are not isometric for 1/p+1/q=1 unless p is 1, 2, or ∞? This being the only case I couldn't figure out of showing that the p-norm and q-norm in n dimensions are never isometric unless either p=q, or n=2 and you've got the case of the 1-norm and the ∞-norm.
Well, one answer I got was, maybe look at the curvature of the p-unit circle? And in fact looking at the differentiability of this curve does in fact get you an answer.
But I also got another answer, which gave the *right* way to solve the problem in general, and isn't restricted to finite dimension. Hanner's inequalities. They let you compute the modulus of convexity of l^p, and see that (aside from the cases of 1 and ∞) these are all different. Boom. This still leaves you to manually distinguish 1 from ∞, but that's not really so hard.
Kitchen sinks and local fields:
Let me warn you up front, what follows is a bunch of idle speculation about topics I really don't know, fueled mostly by Wikipedia.
OK. So the surreal numbers are kind of an "infinity kitchen sink" for ordered fields. And the surcomplex numbers for characteristic 0 more generally. And I'm told the nimbers basically form one for characteristic 2. Obvious question: What about other characteristics? This is an obvious question so someone must know about this, right? (I suppose an appropriate direct limit would work, though it would be totally unhelpful. And probably a horrendous abuse of set theory. Of course the original definition of the surreals is the latter, too.) Less obvious question: Is there any possibility of a "sur-p-adic numbers"? Which would be algebraically the same as the surcomplex numbers, I mean, but somehow naturally equipped with some sort of valuation so that...? I would expect not, as ISTM that would kind of conflict with the whole "infinity kitchen sink" requirement, but having thought of it, I have to ask it. (Not going to state what led me to think of it, as I assure you, it's even way more muddled.)
I Wanna Be The Guy:
I started playing this again today. From scratch, of course, not like my old saves were recovered from the hard drive crash anyway. Finding it much easier this time around so far. (Still playing on Medium.) Just today I already managed to beat Kraidgief and Dr. Wily! (Note that I never beat it the first time - never managed to beat Mother Brain.) In particular I remember Kraidgief being *much* harder than he seemed now.
I *still* have no idea why I see it said that the first boss of the game is Mike Tyson, because so far as I can tell, *none* of the paths have Mike Tyson as their first boss. When I played before I beat Mecha-Birdo, Kraidgief, Bowser/Wart/Dr. Wily, Dracula, and I fought Mother Brain, but I never so much as found Mike Tyson. Clearly I'm missing something. (I refuse to look up a map. I tried to draw a map of what I know from memory, but since it's been so long I'm not sure it's accurate. Back when I played it previously I made a big point of mapping things.)
The game still crashes a lot. No idea if that's due to Wine or not. I expect not.
Perhaps due to finding it easier, I actually forgot elementary precaution after beating Dr. Wily and went forward, instead of backward to save. Oops. Took me a while to beat him again.
-Harry
[0]AKA math.stackexchange.
no subject
Date: 2010-12-28 07:34 pm (UTC)no subject
Date: 2010-12-29 08:02 am (UTC)no subject
Date: 2011-01-01 06:29 pm (UTC)