Timeline for Does Feuter regularity imply derivability in all directions?
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jan 30, 2016 at 22:49 | comment | added | Mirco A. Mannucci | @July: thanks for the refs. I do not know many of them, but I do know some of the recent works by Sabadini, Gentili and Struppa, and somehow I have not seen anywhere a direct answer to the above. But perhaps I have missed something.... | |
| Jan 26, 2016 at 12:45 | history | edited | Mirco A. Mannucci | CC BY-SA 3.0 |
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| Jan 25, 2016 at 19:35 | comment | added | M.G. | (cont.) ...Ward, Graves, Snyder, Szekeres, Rinehart, Ringleb. For the hypercomplex side, you should check out the work of Sabadini, Strupa, Gadea, Masque, Delanghe, Kraußhar, Brackx, Price, Pogorui. Sorry for dumping so many names, but they've all worked or are working on variations and iterations of the above framework really and it's probably an overkill quantity of information for your question. I'll need to sort through my sources to find something specific for what you need, but I'm afraid my POV may not be so suitable for your purpose. | |
| Jan 25, 2016 at 19:27 | comment | added | M.G. | (cont) There is actually tons of literature that remains rather obscured for some reason. The whole story starts with Sheffer in 19th century, and many took over his ideas to generalize complex analysis. In fact, Clifford Analysis was invented partially to remedy the bad behavior in case of noncommutative algebras by dropping certain part of the generalized CR-equations. There is the Kazan school: Shirokov, Shurygin, Vishnevskii, Shpakivskyi, Gaisin, Rosenfeld, Plaksa, Krasnov on the one hand. On the other hand, following Sheffer, there are papers of Fox, Ketchum, Kristzen, Hausdorff, Zorn... | |
| Jan 25, 2016 at 19:19 | comment | added | M.G. | @Mirco Mannucci: If $A$ is a finite-dim. associative algebra over the real or complex numbers and $f:A\to A$ is $A$-differentiable (also called $A$-holomorphic) in a neighbourhood of $z_0\in A$, then $f'(z_0)\in A$ gives an $A$-linear map $A\to A$ via the algebra multiplication. (Translated into a concrete matrix representation, you get the Jacobian or a transpose of it as well as generalized CR-equations.) Some key words: $A$-holomorphy, $A$-analyticity, $A$-differentiability, hypercomplex analysis, paracomplex geometry, hyperholomorphic functions... | |
| Jan 25, 2016 at 19:13 | comment | added | Peter Michor | Is'nt it Fueter de.wikipedia.org/wiki/Rudolf_Fueter | |
| Jan 25, 2016 at 18:38 | comment | added | Mirco A. Mannucci | @July, can you provide some intuition as to why the Jacobian should be in the center of the algebra? Do you have some useful refs? | |
| Jan 25, 2016 at 16:17 | comment | added | M.G. | I can't really comment on the general case of Clifford analysis, but I can say as much. The special case of bicomplex numbers works out just like the complex case, but that's because it too is a special case among the Clifford algebras by being commutative. In general, from the POV of associative algebras, you need the derivative/Jacobian to be in the center of the algebra. If the algebra has a trivial center, well you get the trivial class of functions (dep. on the field). I don't see a way out in the real case (maybe weakly?). OTH, in the complex case you have automatic joint holomorphy.. | |
| Jan 25, 2016 at 13:52 | history | edited | Mirco A. Mannucci | CC BY-SA 3.0 |
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| Jan 25, 2016 at 13:43 | history | asked | Mirco A. Mannucci | CC BY-SA 3.0 |