Elementary Differential Calculus on Discrete, Continuous and Hybrid Spaces

Abstract

We unify a variety of continuous and discrete types of change of state phenomena using a 
scheme whose instances are differential calculi on structures that embrace both topological 
spaces and graphs as well as hybrid ramifications of such structures.  These calculi include the 
elementary differential calculus on real and complex vector spaces.

One class of spaces that has been increasingly receiving attention in recent years is the class of 
convergence spaces [cf. Heckmann, R., TCS v.305, (159--186)(2003)]. The class of convergence spaces together with the continuous functions among convergence spaces forms a Cartesian-closed category CONV that contains as full subcategories both the category TOP of 
topological spaces and an embedding of the category DIGRAPH of reflexive directed graphs. 
(More can importantly be said about these embeddings.)  These properties of CONV serve to assure that we can construct continuous products of continuous functions, and that there is 
always at least one convergence structure available in function spaces with respect to which 
the operations of function application and composition are continuous.  The containment of TOP and DIGRAPH in CONV allows to combine arbitrary topological spaces with discrete 
structures (as represented by digraphs) to obtain hybrid structures, which generally are not topological spaces.

We give a differential calculus scheme in CONV that addresses three issues in particular. 

1.  For convergence spaces $X$ and $Y$ and function $f: X longrightarrow Y$, the scheme gives necessary and sufficient conditions for a candidate differential $df: X longrightarrow Y$ 
to be a (not necessarily "the", depending on the spaces involved) differential of $f$ at $x_0$.  

2.  The chain rule holds and the differential relation between functions distributes over Cartesian products: e.g. if $Df$, $Dg$ and $Dh$ are, respectively, differentials of $f$ at 
$(g(x_0),h(x_0))$ and $g$ and $h$ at $x_0$, then $Df circ (Dg times Dh)$ is a differential of $f circ (g times h)$ at $x_0$.

3.  When specialized to real and complex vector spaces, the scheme is in agreement with ordinary elementary differential calculus on these spaces.

Moreover, with two additional constraints having to do with self-differentiation of differentials and translation invariance (for example, a linear operator on, say, $C^2$, is its own differential everywhere) there is a (unique) maximum differential calculus in CONV.