Intro to Optical Waveguide Analysis [Maxwell's, Schrodinger by K. Kawano, T. Kitoh

By K. Kawano, T. Kitoh

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A graph f is defined by a set of vertices V(r), a set of edges (arrows) E(r) and two maps o:,W : E(f) --+ V(r). Here o:(e) is the beginning (or souTce) of the edge e and w( e) is its end (or range). Two vertices VI, V2 are adjacent if there exists an edge e such that VI = 0:( e) and V2 = w(e) or V2 = o:(e) and VI = w(e). Then we say that the edge e connects the vertices VI and V2. The (edge-)labeled graph is a graph together with a map l : E(r) --+ S, which assigns a label l (e) E S to every edge of the graph.

4. Suppose a collection {

3 can be interpreted as a work of the machine, which being in a state g and receiving as input a letter x, goes into state h and outputs the letter y. 1. Let X be an alphabet. , 7rl , where l. Q is a set (the set of the internal states of the automaton A); 2. : Q x X ---- X is a map, called the output function of the automaton; 3. 7r: Q x X ---- Q is a map, called the transition function of the automaton. An automaton is finite if the set Q is finite. A subset S <;;; Q is a subautomaton of A if for all s E S, x E X the state 7r(s ,x ) belongs to S.

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