Laplaciaan

Laplaciaan is a theoretical framework in spectral geometry and data analysis that describes a generalized class of Laplacian-type operators designed to unify diffusion and vibrational analysis on spaces that combine continuous manifolds with discrete structures, such as manifolds endowed with embedded graphs or sensor networks distributed on surfaces. The Laplaciaan operators extend the traditional Laplace-Beltrami and graph Laplacian formalisms by incorporating both local differential terms and nonlocal interaction terms through a coupling mechanism that can be tuned to different contexts.

In the Laplaciaan formalism, the operator, often denoted Δ_L, reduces to the standard Laplace-Beltrami operator on smooth domains and to the graph Laplacian on purely discrete networks in appropriate limits. It is characterized by linearity, self-adjointness under a suitable weighted inner product, and a nonnegative spectrum in finite domains. The framework allows for anisotropic diffusion, multiscale coupling, and nonlocal interactions, making it flexible for modeling phenomena that involve both geometry and connectivity.

Applications of Laplaciaan include spectral clustering and dimensionality reduction on hybrid spaces, diffusion-based processes for geometry processing, and analysis of quantum-like networks where both continuous and discrete components influence dynamics. The concept supports a unified perspective on modal decompositions, enabling representation of complex systems through Laplaciaan eigenfunctions.

See also: Laplace operator, Laplace-Beltrami operator, graph Laplacian, spectral theory.