Discrete Differential Manifolds and Dynamics on Networks

Abstract

A `discrete differential manifold' we call a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides us with a convenient framework for the formulation of dynamical models on networks and physical theories with discrete space and time. We present several examples and introduce a notion of differentiability of maps between discrete differential manifolds. Particular attention is given to differentiable curves in such spaces. Every discrete differentiable manifold carries a topology and we show that differentiability of a map implies continuity.

26 pages, LaTeX (RevTex), GOET-TP 88/94

References in corpus (5)

• Discrete differential calculus, graphs, topologies and gauge theory
• Differential calculus and gauge theory on finite sets
• Noncommutative Geometry and Gauge Theories on Discrete Groups
• Discrete Time from Quantum Physics
• Finite Approximations to Quantum Physics: Quantum Points and their Bundles

Cited by in corpus (14)

• Noncommutative Geometry of Finite Groups
• Discrete Riemannian Geometry
• Differential Calculi on Commutative Algebras
• Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold
• Differential Geometry of Group Lattices
• Some aspects of noncommutative geometry and physics
• Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity
• Presheaves, Sheaves and their Topoi in Quantum Gravity and Quantum Logic
• Polyhedral representations of discrete differential manifolds
• Dynamical Evolution in Noncommutative Discrete Phase Space and the Derivation of Classical Kinetic Equations
• Finitary, Causal and Quantal Vacuum Einstein Gravity
• Sheafifying Consistent Histories
• Generalized Gauge Theories with Nonunitary Parallel Transport: General Relativity with Cosmological Constant as an Example
• Empirical topology in the histories approach to quantum theory