The ultrametric phylogenetic Laplacian has closed-form eigenvalues that aggregate clade-weighted branch lengths and eigenvectors supported on individual clades, enabling linear-time spectral reconstruction and eigenmode analysis of traits.
Ultrametric pseudodifferential operators and wavelets for the case of non homogeneous measure
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abstract
A family of orthonormal bases of ultrametric wavelets in the space of quadratically integrable with respect to arbitrary measure functions on general (up to some topological restrictions) ultrametric space is introduced. Pseudodifferential operators (PDO) on the ultrametric space are investigated. We prove that these operators are diagonal in the introduced bases of ultrametric wavelets and compute the corresponding eigenvalues. Duality between ultrametric spaces and directed trees is discussed. In particular, a new way of construction of ultrametric spaces by completion of directed trees is proposed.
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Spectral Geometry and Heat Kernels on Phylogenetic Trees
The ultrametric phylogenetic Laplacian has closed-form eigenvalues that aggregate clade-weighted branch lengths and eigenvectors supported on individual clades, enabling linear-time spectral reconstruction and eigenmode analysis of traits.