A generalization of the Flow-box Theorem is proven. The assumption of a C 1 vector field f is relaxed to the condition that f be locally Lipschitz continuous. The theorem holds in any Banach space.
This article begins the study of irreducible maps involving finite-dimensional uniserial modules over finite-dimensional associative algebras. We work on the classification of irreducible maps between two uniserials over triangular algebras, and give estimates for the number of middle terms of an almost split sequence with a uniserial end term.
Lenia is a continuous generalization of Conway's Game of Life. Bert Wang-Chak Chan has discovered and published many seemingly organic dynamics in his Lenia simulations since 2019. These simulations follow the Euler curve algorithm starting from function space initial conditions. The Picard-Lindelöf Theorem for the existence of integral curves to Lipschitz vector fields on Banach spaces fails to guarantee solutions, because the vector field associated with the integro-differential equation defining Lenia is discontinuous. However, we demonstrate the dynamic Chan is using to generate simulations is actually an arc field and not the traditional Euler method for the vector field derived from the integro-differential equation. Using arc field theory we prove the Euler curves converge to a unique flow which solves the original integro-differential equation. Extensions are explored and the modeling of entropy is discussed.
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