tag:blogger.com,1999:blog-109897754944870205.post7473815559988599981..comments2024-03-10T23:49:29.804-07:00Comments on Playing and hacking: Supersum? Subproduct?paurea, Gorka Guardiola Múzquizhttp://www.blogger.com/profile/10219436557463757948noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-109897754944870205.post-73201427443396816242016-12-09T03:42:07.986-08:002016-12-09T03:42:07.986-08:00The most principled way is to say: a Lorentz trans...The most principled way is to say: a Lorentz transformation is a linear transformation of four-dimensional space that leaves invariant the Minkowski norm x^2+y^2+z^2-t^2. This is the natural generalization of orthogonal transformations (rotations and reflections) to norms that are not positive-definite. (In fact, spatial rotations and reflections are special cases of Lorentz transformations by this definition.) You can also come up with more physically-motivated definitions by requiring the invariance of the speed of light (and linearity).Adamhttps://www.blogger.com/profile/17742375692890851839noreply@blogger.comtag:blogger.com,1999:blog-109897754944870205.post-40768539977755172662016-12-09T02:58:13.414-08:002016-12-09T02:58:13.414-08:00With respect to tetration, I feel there is somethi...With respect to tetration, I feel there is something missing in the wikipedia ideas (which are forms of interpolation). I think there has to be a canonical way to do tetration for the reals. I have been playing with the Polygamma, but it is too early to see if I would get anything out of it. The smooshing and blowing up the numbers you refer to, reminds me of non-newtonian calculus, which is also related to the approach I took.<br /><br />Do you have any link to whatever Conway did/referred to?paurea, Gorka Guardiola Múzquizhttps://www.blogger.com/profile/10219436557463757948noreply@blogger.comtag:blogger.com,1999:blog-109897754944870205.post-76470863868202048272016-12-09T02:54:57.789-08:002016-12-09T02:54:57.789-08:00Sorry for reposting the response many times, but a...Sorry for reposting the response many times, but apparently preview does not interact well with the math plugin, so I had to remove and repost the comment three times (there is no edit button for comments either...).paurea, Gorka Guardiola Múzquizhttps://www.blogger.com/profile/10219436557463757948noreply@blogger.comtag:blogger.com,1999:blog-109897754944870205.post-6277588318085137942016-12-09T02:52:52.358-08:002016-12-09T02:52:52.358-08:00The operation
$$a\circledast_\alpha b=a+b+ab$$
i...The operation<br />$$a\circledast_\alpha b=a+b+ab$$<br /><br /><br />is what you get if you use $E_\alpha (−x)$ to define the group.<br />I should have defined this for the paper because it is cleaner, but I discovered that $E_\alpha(−x)$<br />also works too late.<br />It is just $a\circledast \alpha b+1=(a+1)(b+1)$. It is a shifted multiplication with -1 being the absorbing element.<br />In geometrical terms I think of this group as defining dilations/contractions and a dilation has a fixed point, which in terms of the group is the absorbing element of the group, which is why -1 is singled out. The absorbing element is self contained, so you can take it out and the group is still a group.<br /><br />–––––<br />The way I have always seen the Lorentz transformation obtained (I think it was Poincaré who did it first) is as the group of transformations which keep the Maxwell equations invariant.<br />Another, more geometrical way is just to write a coordinate transformation in Minkowski space (rotations, boosts, etc...). Minkowski space is hyperbolic, so there is an exponential map (in Riemman geometry terms) relating the tangent bundle and the manifold.<br />I have come across some subgroups of the Lorentz group (boosts, rotations) with remarkable similarities to some of the groups I found in the paper, so it is quite possibly related :-).paurea, Gorka Guardiola Múzquizhttps://www.blogger.com/profile/10219436557463757948noreply@blogger.comtag:blogger.com,1999:blog-109897754944870205.post-25095454537445378502016-12-09T01:39:15.116-08:002016-12-09T01:39:15.116-08:00This is great! Conway once told me that he and a f...This is great! Conway once told me that he and a friend were working on fractional tetration (i.e., a ^^ 0.5 and the like, for ^^ double-arrow) at some point back in the early '90s, and they found two workable ideas: one from imagining "smooshing all the numbers way down, calculating it, and blowing them back up," and one from doing the opposite (to paraphrase him).<br /><br />They wanted to check whether the two operations yielded the same result, so they had a computer do a calculation: and the answer they got was the same to the twenty-fifth digit and then deviated from there.<br /><br />Shortly thereafter, they found out that some other team of two had already done all this back in the late '70s or something, but hadn't published; so they figured their work was unoriginal, and didn't publish either.<br /><br />The wikipedia page for tetration has some good ideas on it these days, but proves pretty clearly that there's no unique qualitatively "best" idea for what the definition of fractional tetration should be: it depends which properties you value more.<br /><br />–––––<br /><br />Recently I was helping a friend through Herstein, and we came across this unsung hero of a group: the rationals delete -1, under the operation a * b = a + b + ab.<br /><br />Who knew? I'm puzzling over the best geometric way to account for this group. Your circle-star-zero looks so similar.<br /><br />–––––<br /><br />Possibly related, possibly unrelated: what's the best way to arrive at the Lorentz transform, or the velocity addition formula (a+b)/(1+ab/c^2) from first principles? Any account of it I've seen goes like this: assume the Lorentz transform. Then we find out all these neat things hold. Can we do it the other way around? Bonus points if you don't use wavefunctions ψ(x-ct).Troglodorhttps://www.blogger.com/profile/03700247687318529837noreply@blogger.com