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In this paper, we formalize precisely the sense in which the application of a cellular automaton to partial configurations is a natural extension of its local transition function through the categorical notion of Kan extension. In fact, the two possible ways to do such an extension and the ingredients involved in their definition are related through Kan extensions in many ways. These relations provide additional links between computer science and category theory, and also give a new point of view on the famous Curtis–Hedlund theorem of cellular automata from the extended topological point of view provided by category theory. These links also allow to relatively easily generalize concepts pioneered by cellular automata to arbitrary kinds of possibly evolving spaces. No prior knowledge of category theory is assumed for the most part. [ABSTRACT FROM AUTHOR]

The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form is solved on $$\mathbb{R}{\text{ for }}y \ne 0,\upsilon \ne 0$$ . [ABSTRACT FROM AUTHOR]

The paper describes the general form of functional-differential equations of the first order with m( m ≥ 1) delays which allows nontrivial global transformations consisting of a change of the independent variable and of a nonvanishing factor. A functional equation for u ≠ 0 is solved on $$\mathbb{R}$$ and a method of proof by J. Aczél is applied. [ABSTRACT FROM AUTHOR]