tag:blogger.com,1999:blog-10614348.post2783917862248768906..comments2023-01-19T00:50:01.428-08:00Comments on TGD diary: John Baez about Noether's theorem in algebraic approachMatti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-10614348.post-47365142746495292192020-07-06T10:19:46.450-07:002020-07-06T10:19:46.450-07:00https://www.physicsforums.com/threads/implicitly-d...https://www.physicsforums.com/threads/implicitly-differentiating-the-vanishing-real-part-of-the-hyperbolic-tangent-of-one-plus-the-square-of-the-hardy-z-function.991104/#post-6363884<br /><br />its a lost cause i know, but its pretty :) Crow-https://www.blogger.com/profile/11587191916122583498noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-42461954180100417882020-06-30T22:15:59.207-07:002020-06-30T22:15:59.207-07:00See
https://mathworld.wolfram.com/NegativePedalC...See <br /><br />https://mathworld.wolfram.com/NegativePedalCurve.html <br /><br />I mistyped, it must be the negative pedal curve, since the lemniscate is the pedal curve of the hyperbola and the hyperbola is the negative pedal curve of the lemniscateCrow-https://www.blogger.com/profile/11587191916122583498noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-10182319732286900612020-06-30T22:13:35.393-07:002020-06-30T22:13:35.393-07:00This comment has been removed by the author.Crow-https://www.blogger.com/profile/11587191916122583498noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-57844506541016103342020-06-30T19:13:56.088-07:002020-06-30T19:13:56.088-07:00I see what you are saying, the relationship I am ...I see what you are saying, the relationship I am suggesting would be nonlocal but it would still be algebraic . The reason is that, lemniscates are the pedal curves of hyperbolas, and hyperbolas are the pedal curves of lemniscates. My hunch is that the curves I uncovered have this same relationship, I just need to work out the formulas for the pedal coordinate transforms and see how they work out. Maybe my terminology is not correct in calling it algebraic? I would need to evaluate the 1st derivatives of the function to make the coordinate transform. Crow-https://www.blogger.com/profile/11587191916122583498noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-76392316217442160532020-06-30T17:30:24.601-07:002020-06-30T17:30:24.601-07:00Unfortunately the topic is so complex that it woul...Unfortunately the topic is so complex that it would take a lot of time and I do not have it. n any case an alebraic relationship beween real and imaginary parts of complex function does not seem possible. Cauchy-Riemann equations allow only non-local relatioship between real and imaginary parts in f= u+iv. To see that algebraic or more geneal local relationship between u and v is impossible substitute u=f(v) to C-R equations. You get two equations: partial_xu= partial_uf partial_y u and partial_yu= -partial_ufpartial_xu. Together these gives (partial_f)^2=-1, which is contradicton since f is real. Matti Pitkänenhttps://www.blogger.com/profile/13512912323574611883noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-78988173672147585902020-06-30T16:34:52.775-07:002020-06-30T16:34:52.775-07:00Matti, I was reading about this guy who wrote a pa...Matti, I was reading about this guy who wrote a paper on pedal coordinates in 2017 and came up with some new theorem and relates it to the "Dark kepler problem". <br /><br />"one of the main advantages of pedal coordinates is that the operation of making pedal curve(which would in general require solving a differential equation in Cartesian coordinates) can by done by simple algebraic manipulation.<br /><br />https://arxiv.org/pdf/1704.00897.pdf<br /><br />I've updated a paper im writing on the topic at https://fs23.formsite.com/viXra/files/f-1-2-11434891_j9ehLpFN_tanhln1plusZsquared.pdf<br /><br />pedal curves are related to circle inversion. I think it should be possible to show an algebraic relationship between the real and imaginary parts of tanh(ln(1+Z(t)^2))Crow-https://www.blogger.com/profile/11587191916122583498noreply@blogger.com