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Pedro Lauridsen Ribeiro's user avatar
Pedro Lauridsen Ribeiro's user avatar
Pedro Lauridsen Ribeiro's user avatar
Pedro Lauridsen Ribeiro
  • Member for 14 years
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Is the evolution family self-adjoint?
OK, I saw it just after I'd posted my comment, sorry for that. I've already deleted that comment.
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Convexity of distance-to-boundary function
Indeed, the example of a ball (see e.g. Iosif Pinelis's answer below) shows that $d_{\Omega}$ should rather be concave in this case. Anyhow, the answers below seem to take this into account.
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Non-semisimple Lie groups and Higgs bundles
The author probably means $\mathfrak{g}/\mathfrak{h}$ instead of $\mathfrak{g}/\mathfrak{m}$ in context.
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Non-semisimple Lie groups and Higgs bundles
The examples the OP is interested in are a very special kind of non-reductive Lie group - to wit, the semidirect product of a semisimple Lie group with a non-compact Abelian Lie group. Can such issues be circumvented in this case?
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Non-semisimple Lie groups and Higgs bundles
$\mathfrak{h}$ cannot be the Lie algebra of itself but rather that of $H$.
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Non-semisimple Lie groups and Higgs bundles
As far as I know, the (three-dimensional in the OP) Poincaré and Euclidean groups are not reductive, so I have no idea. That's also why I insist that you're actually asking for a notion of $G$-Higgs bundles for $G$ non-reductive - the case of $G$ reductive but not semisimple is already covered by the definition presented in the OP and the examples you're interested in are (I think) non-reductive. You should amend the title and body of your question in order to address that, they are slightly misleading as they stand.
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Non-semisimple Lie groups and Higgs bundles
Don't you mean to ask for a notion of a $G$-Higgs bundle for $G$ non-reductive instead of non-semisimple, since this was your initial hypothesis on $G$? There are (real) reductive Lie groups that are not semisimple (e.g. $G=GL(n,\mathbb{R})$) and your definition applies equally well to them.
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Non-semisimple Lie groups and Higgs bundles
Grammar and punctuation fixes
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The length is bounded
Typo in title corrected
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I want a smooth orthogonalization process
Hmm... According to what seems to be the state of the art in understanding the problem (called the "unbalanced Procrustes problem" in the numerical analysis literature), it certainly appears so - check e.g. L. Eldén, H. Park, A Procrustes Problem on the Stiefel Manifold, Numer. Math. 82 (1999) 599-619 and Z. Zhang, Y. Qiu, K. Du, Conditions for Optimal Solutions of Unbalanced Procrustes Problem on Stiefel Manifold, J. Comput. Math. 25 (2007) 661-671. See also math.stackexchange.com/questions/4492668/…
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I want to find a smooth section of the map from the Stiefel manifold to the Grassmanian manifold
Added remarks on non-northonormal bases and alternatives to follow
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