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CONTINUITY OF THE VARIATIONAL EIGENVALUES OF THE p-LAPLACIAN WITH RESPECT TO p
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 83 / Issue 3 / June 2011
- Published online by Cambridge University Press:
- 15 March 2011, pp. 376-381
- Print publication:
- June 2011
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- Article
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Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices
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- Journal:
- Canadian Journal of Mathematics / Volume 56 / Issue 4 / 01 August 2004
- Published online by Cambridge University Press:
- 20 November 2018, pp. 776-793
- Print publication:
- 01 August 2004
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- Article
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We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold
$\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold 
$\text{Sym(}p\text{,}\,\mathbb{R}{{\text{)}}^{++}}\,\times \,\text{Sym(}q\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ block diagonally embedded in $\text{Sym(}n\text{,}\,\mathbb{R}{{\text{)}}^{++}}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when 
$p\,\le \,2$ or 
$q\,\le \,2$.
