Unifying vectors and matrices of different dimensions through nonlinear embeddings

TL;DR

This paper introduces nonlinear embeddings that unify vectors, matrices, and tensors of different dimensions, enabling modeling of complex systems and connections between various mathematical structures and physical theories.

Contribution

The paper presents a novel nonlinear embedding framework that unifies objects of different dimensions and applies it to physics, cellular automata, and differential equations.

Findings

Unified mathematical structures for vectors, matrices, and tensors.

Constructed warped models in supergravity dimensional reduction.

Connected cellular automata to coupled map lattices and PDEs.

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.