Unifying vectors and matrices of different dimensions through nonlinear embeddings

Abstract

Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter $\kappa \in \mathbb{R}$ is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to 4-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, deriving a class of nonlinear diffusion equations. Finally, by means of nonlinear embeddings we introduce CA connections, a class of CMLs that connect any two arbitrary CAs in the limits $\kappa \to 0$ and $\kappa \to \infty$ of the embedding.

30 pages, 5 figures, accepted to Journal of Physics: Complexity

References in corpus (6)

• Gravity and Unification: A review
• From deterministic cellular automata to coupled map lattices
• Weakly coupled map lattice models for multicellular patterning and collective normalization of abnormal single-cell states
• Semipredictable dynamical systems
• Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding
• Diagrammatic approach to cellular automata and the emergence of form with inner structure

Cited by in corpus (1)

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